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Moduli of representations of the fundamental group of a smooth projective variety. II

Moduli of representations of the fundamental group of a smooth projective variety. II MODULI OF REPRESENTATIONS OF THE FUNDAMENTAL GROUP OF A SMOOTH PROJECTIVE VARIETY. II by CARLOS T. SIMPSON Introduction This second part is devoted to the subject of the title, moduli spaces of represen- tations of the fundamental group of a smooth complex projective variety X. We study three moduli spaces for related objects. The Betti moduli space MR(X , n) is a coarse moduli space for rank n representations of the fundamental group of the usual topological space X t~ A vector bundle with integrable connection is a pair (E, V) where E is a vector bundle and V:E-+ E | f~ is an operator satisfying the Leibniz rule and V2-- - 0. The de Rham moduli space MDg(X, n) is a coarse moduli space for rank n vector bundles with integrable connection on X. A Higgs bundle [Hill [Si5] is a pair (E, ~) where E is a vector bundle and q~ : E --> E | f/~ is a morphism of 0x-modules such that q0 ^ q0 = 0. There is a condition of semistability analogous to that for vector bundles, but only concerning subsheaves preserved by ~. The Dolbeault moduli space MDot(X , n) is a coarse moduli space for rank n semlstabte Higgs bundles with Chern classes vanishing in rational cohomology. In all three cases, the objects in question form an abelian category in which we can apply the Jordan-H61der theorem. Let gr(E) denote the direct sum of the subquotients in a Jordan-H61der series for E, and say that E 1 is Jordan equivalent to E~ if gr(E1) _-__ gr(E~). The points of the coarse moduli spaces parametrize Jordan equivalence classes of objects. The constructions of these moduli spaces are reviewed in w 5. The construction of Mu is a classical one from the theory of representations of discrete groups. The cons- truction of MD~ follows from the construction of Part I, w 4, for the case where A "t~ = ~x is the full sheaf of rings of differential operators on X. We give two constructions of MDo ~. One is based on an interpretation of Higgs sheaves as coherent sheaves on T* X, and uses the construction of the moduli space of coherent sheaves constructed in Part I, w 1. The other consists of applying the general construction of Part I, w 4, to the case A"~ Sym'(TX). The three types of objects are related to each other. The Riemann-I-Iilbert corres- pondence between systems of ordinary differential equations and their monodromy repre- CARLOS T. SIMPSON sentations provides an equivalence of categories between vector bundles with integrable connection and representations of the fundamental group. To (E, V) corresponds the monodromy of the system of equations V(e) = 0. The correspondence between Higgs bundles and local systems of [Hil], [Do3], [Co], [Si2], and [Si5] gives an equivalence of categories between semistable Higgs bundles with vanishing rational Chern classes, and representations of the fundamental group. Together, these correspondences give isomorphisms of sets of points Ms(X , n) ~ MD~(X, n) ~ Mnol(X, n). In w 7 we use the analytic results of Part I, w 5, to show that the first map is an isomorphism of the associated complex analytic spaces, and that the second is a homeomorphism of usual topological spaces. There is a natural algebraic action of the groupe C" on the moduli space Mi)ol(X, n), given by t(E, ~0) = (E, t~0), and our identifications thus give a natural action--no longer algebraic--on the space of representations. The fixed points of this action are exactly those representations which come from complex variations of Hodge structure [Si5]. Although MDo 1 is not compact, the properness of Hitchin's map (Theorem 6.11) implies that MDoI(X, n) contains the limits of points tE as t-+ 0. This yields the conclusion that any representation of the fundamental group may be deformed to a complex variation of Hodge structure (Corollary 7.19 below). This theorem was in some sense the principal motivation for constructing the moduli spaces. See [Si5] for more details on some consequences. The reason for the terminologies Betti, de Rham and Dolbeault is that these modull spaces may be considered as the analogues for the first nonabelian cohomology, of the Betd cohomology, the algebraic de Rham cohomology, and the Dolbeault cohomology @~+q=~ H~(X,Y~) of X. The first nonabelian cohomology set Hi(X, Gl(n, C)) is the set of isomorphism classes of rank n representations of nt(X). This has a structure of topological space, but it is not Hausdorff. The universal Hausdorff space to which it maps is the Betti moduli space MB(X, n). To explain the analogies for the de Rham and Dolbeault spaces, we have to digress to discuss the (2ech realizations of the cohomology groups with complex coefficients. The algebraic de Rham cohomology is the hypercohomology of X with coefficients in the algebraic de Rham complex ~o ~ ~: ~ ... If X = LI U, is an affine open covering of X, and if we denote multiple intersections by multiple indices, then the cocycles defining H]~(X, C) consist of the pairs of collections ({g,~}, {a~}) where g,o are regular functions U,~ -+ C, and a, are one-forms on U~, such that g~ = g~v + g,,,t, d(g,,f~) = a= -- a~, and d(a~,) = O. MODULI OF REPRESENTATIONS. II Addition of the coboundary of a collection { s~ ), where s~ are regular functions U, ~ C, changes the pair ({ g~ }, { a~ }) to ({ g~ q- s~ -- s o }, { a~ q- d(s~) }). The group of cocycles modulo coboundaries is H~R(X , C). The nonabelian case has formulas which are more complicated, but which reduce to the above if the coefficient group is abelien. A vector bundle with integrable connection is defined by a pair ({ g~ }, { A, }) , where g~:U,~ ~ Gl(n, C) are the gluing functions for the vector bundle, and A s are n � n matrix-valued one forms defining the connection V = d -k- As. These are subject to the conditions g~v g~ ~ g~v, A~ = g~l d(g~) + g~l A~ g~, and d(A~) + A s ^ A~ ~ 0. A change of local frames by a collection of regular functions s~ : U~ -~ Gl(n, C) changes the pair ({g~),{A~}) to ({s~Ig~s~},{sZ ~A~s~ + sZ x d(s~)}). The set of pairs up to equivalence given by such changes of frames, is the first nonabelian de Rham cohomo- logy set H~(X, Gl(n, C)), the set of isomorphism classes of vector bundles with integrable connection on X. A similar if somewhat looser interpretation gives an analogy between the abelian Dolbeault cohomology group Hi(X, 0x) | H~ ~) and the first nonabelian Dolbeault cohomology set Hx(X, Gl(n, C)), the set of isomorphism classes of Higgs bundles (E, ~) which are semistable with vanishing rational Chern classes. Here E is a vector bundle and ~ ~ H~ End(E) |162 ~)" Such a pair may be given a cocycle description similar to the above (just eliminate the terms involving d). The conditions of semistability and vanishing Chern classes are new. Following this interpretation, we can think of the first nonabelian cohomology as a nonabelian motive in a way analogous to [DM], with its Betti, de Rham and Dolbeault realizations. It would be good to have t-adic, and crystalline interpretations in charac- teristic p. We treat everything in the relative case of a smooth projective morphism X ~ S to a base scheme of finite type over G. This creates some difficulties for the Betti moduli spaces: we have to introduce the notion of local system Od e schemes over a topological space. The relative Betti spaces lVlB(X/S, n) are local systems of schemes over S t~ The de Rham and Dolbeault moduli spaces are schemes over S, whose fibers are the de Rham and Dolbeault moduli spaces for the fibers X~. The interpretation in terms of nonabelian cohomology suggests the existence of a Gauss-Manin connection on NIDR(X/S , n), a foliation transverse to the fibers which when integrated gives the transport corresponding to the local system of complex analytic spaces 1VI"~(X/S, n). We construct this connection in w 8, using Grothendieck's idea of the crystalline site. In w 9, we treat the case of other coefficient groups. If G is a reductive algebraic group, we may define the Betti moduli space bl~(X, G) to be the coarse moduli space CARLOS T. SIMPSON for representations of rq(X) in G. We construct the de Rham moduli space MDg(X , G) for principal G-bundles with integrable connection, and the Dolbeault moduli space Mml(X , G) for principal Higgs bundles for the group G, wtfich are semistable with vanishing rational Chern classes, and extend the results of w167 7 and 8 to these cases. One corollary is a result valid for representations of any finitely generated discrete group T: if G -+ H is a morphism of reductive algebraic groups with finite kernel, then the resulting morphism of coarse moduli spaces M(Y, G) -+ NI(Y, H) is finite (Corol- lary 9.16). Parallel to the discussion of moduli spaces, we discuss the Betti, de Rham and Dolbeault representation spaces RB(X, x, n), RoR(X, x, n), and RDoI(X, x, n). These are fine moduli spaces for objects provided with a frame for the fiber over a base point x ~ X (here we assume that X is connected). There are relative versions for X/S where the frames are taken along a section ~ : S ~ X, and there are versions for principal objects for any linear algebraic group. In w 10, we discuss the local structure of the singularities of the representation spaces, using the deformation theory associated to a differential graded Lie algebra developed by Goldman and Millson [GM]. By Luna's 6tale slice theorem, this also gives information about the local structure of the moduli spaces. The differential graded Lie algebra controlling the deformation theory of a principal vector bundle with integrable connection or a principal Higgs bundle is formal if the object is reductive (in other words, corresponds to a closed orbit under the action of G on R(X, x, G)). By the theory of [GM], this implies that the representation space has a singularity defined by a quadratic form on its Zariski tangent space. Furthermore, the differential graded Lie algebras controling the deformation theories of the flat bundle and the corresponding Higgs bundle are the same. This gives a formal isomorphism between the singularities of the de Rham (or Betti) representation space and the singu- larities of the Dolbeault representation space, at semisimple points which correspond to each other. This isosingularity principle holds also for the singularities of the moduli spaces. The homeomorphism between Mx)r: and MDo , is not complex analytic, so these local formal isomorphisms are not directly related to the global homeomorphism. Finally, in w 11 we discuss the case of representations of the fundamental group of a Riemann surface of genus g/> 2. Hitchin calculated the cohomology in the case of rank two projective representations of odd degree, where the moduli space is smooth [Hil]. We do not attempt to go any further in this direction. We simply treat the most elementary property, irreducibility (which is more or less a calculation of H~ We treat the case of representations of degree zero, so the moduli space has singularities corresponding to reducible representations; we prove that the singularities are normal. The technique is to use the fact that the Betti moduli space is a complete intersection, and apply Serre's criterion (following a suggestion of M. Larsen, and prompted by a question of E. Witten). We have to verify that there are no singularities in codimension one. To prove irredu- cibillty it suffices to prove that the space is connected, which we do by a simple argument derived from Hitchin's method for calculating the cohomology. MODULI OF REPRESENTATIONS. II Relationship with part I It is worth reiterating the nature of the connections between this second part and Part I of the paper. The sections are numbered globally, so we begin with w 6. References to lemmas and such, numbered for sections 1-5, are references to the statements in " Moduli of representations of the fundamental group of a smooth projective variety I " We rely on the technical work done in part I for many of the constructions of moduli spaces, identifications, and criteria for convergence used here. For the most part, we apply statements from the first part, so Part II can be read without having read Part I in a very detailed way, but just having a copy at hand for reference. Origins The correspondence between Higgs bundles and local systems, reflected in the homeomorphism between the Dolbeault and the de Rham or Betti moduli spaces, comes from work of Hitch.in [Hill, Colette [Co] and Donaldson [Do3], as well as [Si2]. The formalism of this correspondence is developed in [Si5]. The original correspondence of this type was the result of Narasimhan and Seshadri [NS] between unitary representations and stable vector bundles. This was subsequently generalized by Donaldson IDol] [Do2], Mehta and Ramanathan [MR1] [MR2], and Uhlenbeck and Yau [UY]. The idea of obtaining a correspondence between all representations into a non- compact group, and vector bundles provided with the additional structure of a Higgs field, comes from Hitchin's paper [Hil] with the appendix [Do3] of Donaldson. Hitchin established the correspondence between rank two Higgs bundles and rank two repre- sentations on a Riemann surface (and his arguments are easily extended to any rank). Independently, I had arrived at a correspondence between certain representations with noncompact structure group (the complex variations of Hodge structure), and certain holomorphic objects involving an endomorphism valued one-form (systems of Hodge bundles) [Sil]. Deligne and Beilinson had also arrived at a correspondence between systems of Hodge bundles and variations of Hodge structure over a Riemann surface (unpublished work). My definitions and very first results were independent of those of Deli_nge and Beilinson, then Deligne explained their work and made some important suggestions. I didn't see, until W. Goldman directed me to Hitchin's paper which had just appeared, that one could hope to obtain a correspondence involving all representations. In light of Hitchin's definition, systems of Hodge bundles could be seen as special types of Higgs bundles, and my arguments in higher dimensions could be generalized to the case of Higgs bundles [Si2]. This provided one direction of the correspondence. The other direction (corresponding to Donaldson's appendix to Hitchin's paper) was provided by the results on equivariant harmonic maps and the Bochner formula obtained by Colette in his thesis [Co]. 2 10 CARLOS T. SIMPSON This correspondence provided the motivation for the construction of the moduli space of Higgs bundles. Hitchin gave an analytic construction in his paper, and he obtained all of the interesting properties, such as the properness of the map given by taking the characteristic polynomial of the Higgs field. In my thesis, I had constructed a moduli space for systems of Hodge bundles, using Mumford's construction for vector bundles on curves. The construction presented in part I grew out of this, but uses methods of geometric invariant theory more suited to higher dimensions, as pioneered by Gieseker [Gi] and Maruyama [Mal] [Ma2]. See the introduction of Part I for further details. When I first discussed this with him, Hitchin advised me that Nitsure had given an algebraic construction for Higgs bundles over a curve [Nil]. Early on, while I was looking at systems of Hodge bundles, J. Bernstein made the comment that a system of Hodge bundles could be considered as a sheaf on the cotangent bundle of the variety. This remark, generalized to the case of Higgs bundles, forms the basis for one of the constructions of the moduli space of Higgs bundles presented inw The discussion of the Gauss-Manin connection in w 8 was prompted by a discussion with S. Mochizuki, wherein he pointed out that the analytic connection provided by the Betti trivialisation of M~R(X/S, n) over S, was not a priori algebraic. The methods used to prove algebraicity are the crystalline methods envisioned by Grothendieck in connection with his construction of the Gauss-Martin connection for abelian cohomo- logy [Gr3]. The existence of the Gauss-Man.in connection was announced in [Si3], and a brief sketch of the proof was given. The material in w 10 about deformation theory is an easy extension of the work of Goldman and Millson [GM ]. Their work was, in turn, based on a deformation theory developed by Schlessinger, Stasheff and Deligne. The proof of irreducibility in w 11 was motivated by an old question posed to me by J. Bloch, and made possible by Hitchin's method of using Morse theory or the C ~ action to calculate the topology of the moduli space (which we use just to show connectedness). My original proof contained a long and technical part showing that the singularities were locally irreducible. E. Witten later posed the question of whether the singularities were normal, and M. Larsen helped by directing me to the place in [Ha] explaining how to use Serre's criterion to prove normality of a complete intersection. The only technical part now needed is an inductive verification that the singularities are in codimension at least two, which makes the argument much shorter. Acknowledgements I would like to reiterate the acknowledgements of Part I in what concerns this second half of the paper. I would particularly like to thank J. Bernstein, J. Carlson, K. Corlette, P. Deligne, S. Donaldson, W. Goldman, P. Griffiths, N. Hitchin, M. Larsen, G. Laumon,J. Le Potier, M. Maruyama, S. Mochizuki, N. Nitsure, W. Schmid, K. Uhlen- MODULI OF REPRESENTATIONS. II 11 beck, E. Witten, K. Yokogawa, and A. Yukie for many helpful discussions about their work and the present work. The early versions were full of mistakes, pointed out by many of these people on innumerable occasions. I am most greatful for this help, sorry that they had to take the time to worry, and hope that in this last revision, I haven't introduced too many new ones. 6. Moduli spaces for representations The Betti moduli spaces We begin with a classical construction from the theory of spaces of representations of discrete groups. Suppose r is a finitely generated group. Fix n. Put R(F, n) ---- Horn(F, Gl(n, C)). It is a scheme over Spec(C) representing the functor which to a C-scheme S associates the set Hom(F, Gl(n, H~ Os))). The scheme R(F, n) can be constructed by choosing generators Y1, ..., Yk for F. Let Rel denote the set of relations among the y~. Then R(F, n) C Gl(n, C) � ... � Gl(n, C) (k times) is the closed subset defined by the equations r(mx, ..., m~) = 1 for r e Rel. It is easy to see that this subset represents the required functor--a representation 0 : I" -+ Gl(n) corresponds to the point (ml, ..., ink) with m s = O('f~). Note that R(F, n) is a closed subset of an affine variety, so it is affine. The group Gl(n, C) acts on R(F, n) by simultaneous conjugation of the matrices. The orbits under this action are the isomorphism classes of representations. Two representations 9 and p' are said to be Jordan equivalent if there exist compo- sition series for each such that the associated graded representations are isomorphic. The theorem of Jordan-H61der says that the associated graded doesn't depend on the choice of composition series; this semisimple representation is an invariant of the repre- sentation, known as its semisimplification. Proposition 6.1. -- There exists a universal categorical quotient R(F, n) -+ M(F, n) by the action of Gl(n, C). The scheme 1VI(F, n) is an affine scheme of finite type over C. The closed points of M(F, n) represent the Jordan equivalence classes of representations. Proof. -- This is well known. The quotient is constructed by taking the coordinate ring A = H~ n), On(r. J, setting B- A ~ to be the subring of invariants, and putting M(F, n) = Spec(B). Hilbert proved that B is finitely generated. Mumford shows in [Mu] that Spec(B) is a universal categorical quotient of Spec(A) ----R(F, n). Finally, Seshadri shows that the closed points of the quotient are in one to one corres- pondence with the closed orbits [Se]. The closed orbits are the orbits corresponding 12 CARLOS T. SIMPSON to semisimple representations, and the closed orbit in the closure of a given orbit is the one corresponding to the semisimplification of the given representation. [] Suppose X is a conected smooth projective variety over Spec(C). Choose x e X and let F = nl(X '~, x). We will use the notation an(x, ~, ~) ~=~ R(r, n) and call this the Betti representation space; and the notation MB(X, n) d~ M(r, n) calling this the Betti moduli space. This terminology is suggested by the terminology of [DM]. The space Mn(X, n) does not depend on the choice of x. More precisely, ff we include the choice of x in the notation then there are canonical isomorphisms "~(x,y) : Mn(X, x, n) - MB(X,y, n) such that v(y, z) v(x,y) = v(x, z), given as follows. We may choose a path from x toy, giving ~I(X '~, x) - ~l(X'~,y) and hence Rn(X , x, n) ~ RB(X,y, n). This isomorphism is compatible with the action of Gl(n, C) so it descends to the desired "~(x,y). Choice of a different path gives a different isomorphism of representation spaces which differs by the action of a section g : Rn(X,y, n) -+ Gl(n, C). By the definition of quotient, the two natural maps RB(X,y , n) X Gl(n, C) ~ Mn(X,y, n) are equal. Hence the two maps from the graph ofg to M~(X,y, n) are the same. Thus the two maps from Rn(X, x, n) to Mn(X,y, n) are the same, so the two isomorphisms v(x,y) are the same. Thus .r is canonically defined; and this independence of the choice of path implies the formula v(y, z) .~(x,y) = "r(x, z). We will identify the spaces obtained from different choices of base point, and drop the base point from the notation for Mn(X, n). Local systems of schemes Suppose T is a topological space. A local system of schemes Z over T is a functor from the category of C-schemes to the category of sheaves of sets over T, denoted (backward) Z : (S E Sch, U C T) ~ Z(U)(S), such that: there exists a covering by open sets T = I.J, U~ such that for any open set V contained in one of the U~, the functor S ~ Z(V)(S) is represented by a scheme Z(V) over 13; and such that if W C V are connected open sets contained in one of the U~ MODUI.I OF REPRESENTATIONS. II then the restriction map Z(V) -~ Z(W) is an isomorphism (note that the restriction morphisms of functors are automatically morphisms of schemes). Choose a point t ~ T. The stalk Z, lim Z(V) t~v is a scheme. Lemma 6.2. -- If Z is a local system of schemes on T, the group ~I(T, t) acls on the stalk Z t by C-scheme automorphisms. If T is connected and locally simply connected, the construction Z ~ Z t is an equivalence between the category of local systems of schemes over T and the category of schemes with action of rq(T, t). Proof. -- Suppose Z is a local system of schemes over T. Let T =- U, U~ be an open covering as in the definition. We may suppose that the U, are connected. We have schemes Z(U~). If v ~ U~ t~ U~ then we have isomorphisms of schemes z(u ) z~ If v ~ U~ n U~ c~ U v then the resulting hexagon commutes. If ~ : [0, 1] --> T is a path with ~(0) ----~(1) =t then we may choose O=s o<s*< ...<s,= 1~[0,1] and ~o, ...,~ such that s,+,]) c We obtain Zoo,0 --- Z(U,i ) ~ Zoc,i § 11" Putting these isomorphisms together we get an isomorphism of schemes Z t "~ Z t . One can check that a homotopic path ~' ,-~ ~ gives the same isomorphism, so we get an action of r~l(T , t) on Z t. Suppose T is connected and locally simply connected, so the universal covering exists. Given a scheme Z t with action of rq(T, t), form the constant local system of schemes Z over the universal covering ~, whose fiber at the base point t is Z t (the sheaf is given by the rule Z(U)(S) = Zt(S ) for connected open sets U; the restriction maps are the identity). The group of covering transformations Aut(~/T)= ~I(T, t) acts on ~. over its action on ~, by the given action on Zt. Now define Z(U)(S) to be the set ofinvariants in Z(U)(S), where U is the inverse image of U in the universal covering. This gives a local system of schemes Z over T. This construction is the inverse of the previous construction. [] We can make a similar definition of local system of complex analytic spaces over a topological space T, and the analogue of the previous lemma still holds. If Z is a local system of schemes then we obtain a corresponding local system of complex analytic spaces Z ~. The stalk Z~" is the complex analytic space associated to Z~. If T is a complex analytic space and Z is a local system of schemes over T then we denote by Z ~ the total analytic space over T constructed as follows. Choose an open a,_=_r 14 CARLOS T. SIMPSON coveting T = O, T~ such that the restriction of Z to T, is a constant local system with stalk Z,. For each connected component of T, n Tj there is an isomorphism Z~ -- Z~., satisfying a compatibility relation for triples of indices. Let Z (~n~ be the space obtained by glueing together the complex analytic spaces Z~ ~ � T, using these isomorphisms. The Betti moduli spaces in the relative case Suppose that f: X -+ S is a smooth projective morphism to a scheme S of finite type over C. Suppose that S and the fibers X, are connected. The associated map of complex analytic spaces fan is a fibration of the underlying topological spaces. Choose base points t ~ S and x ~ X,. Let P = ~I(X~ n, x). Let Aut(P) denote the group of auto- morphisms of P; Inn(P) C Aut(r) the image of the natural map Ad : r -+ Aut(r) (which sends y to the inner automorphism Ad(y)(g) = ygy-1); and Out(F) = Aut(P)/Inn(P). The group 7r1(S =, t) acts on P by outer automorphisms, in other words there is a map ,:l(S =, t) -, Out(r). This may be defined as follows: if ~: [0, 1] -+ S = is a loop representing an element of 7rx(S =, t), then the pullback a*(X an) is a fibration over [0, 1]; it is trivial, so we obtain a homeomorphism X~ n-~ X~ n between the fibers over 0 and 1; this gives a map ~zl(X~, x) - rrl(X~,y ) for some other point y; finally, choose a path joining x and y, to get an automorphism of r ---= nl(X~, x)--which is well defined independent of the choice, up to inner automorphism. The resulting outer automorphism is independent of the homotopy class of the path ~. The group Aut(r) acts on the representation space R(F, n). This descends to an action of Out(r) on the moduli space M(F, n), as inner automorphisms act on the representation space through functions R(F, n) ~ Gl(n, C) and hence trivially on the moduli space. In our case, we have denoted M(F, n) by Mr(Xt, n). Composing this action with the action of nl(S ~n, t) on F, we obtain an action of ~zl(S "n, t) on MB(Xt, n) by C-scheme automorphisms. From Lemma 6.2, we obtain a local system of schemes MB(X/S , n) over the topological space underlying S an. The relative version of the Betti moduli space is this local system of schemes MB(X/S , n). It is independent of the choice of base points t and x. The stalk over s e S is MB(X/S , n)~ = M~(X,, n). Suppose ~:S-+X is a section. Then rrl(S ~, t) acts on 7h(X,, ~(t)) by auto- morphisms. We obtain a local system of schemes RB(X]S , ~, n), which is again inde- pendent of the choice of t. The stalk over s e S is RB(X/S, ~, n), = RB(X,, ~(s) n). The constructions MB(X/S , n) and RB(X/S , ~, n) may be extended to the case of non- connected base S by taking the disjoint union of the spaces over each connected com- MODULI OF REPRESENTATIONS. II 15 ponent of S. The construction Ms(X/S , n) may also be extended to the case where the fibers are not connected. If s ~ S and if X s = X 1 u ... t3 X k is the decomposition of the fiber into connected components, then MB(X/S, n), = M~(X1, n) � ... � M~(Xk, n). The action of ~1(S ~n, s) permutes the factors in the product appropriately. We close the discussion of the Betti moduli spaces by giving the universal and co-universal properties they satisfy. Proposition 6.3. -- Suppose f: X ~ S is smooth and projective with connected fibers, and suppose that ~ : S ~ X is a section. Then for any scheme Y and any open set U C S ~ the set RB(X/S, ~, n)(U)(Y) is equal to the set of isomorphism classes of pairs (L, ~) where L bs a locally constant sheaf of H~ Oy)-modules on f-l(U) and ~ : ~-I(L) ~ H~ 0y)". Proof. -- It suffices to prove this for small open sets U, for example connected open sets over which the topological fibration (X ~n � U, ~) is trivial. In this case, choose s e U. A locally constant sheaf L of H~ d~y)-modules on f-I(U) together with a fram ~ is the same thing as a representation of nl(Xs, ~(s)) in Gl(n, H~ 0y)), hence the same thing as a morphism Y --~ R~(X,, ~(s), n). This is the set of Y-valued points of the local system of schemes over the set U, since the local system of schemes is trivial and U is connected. [] Proposition 6.4. -- Suppose f: X ~ S is a smooth projective morphism. Let M~(X/S, n) denote the functor from C-schemes to sheaves of sets over S ~ which associates to each scheme Y and each open set U C S ~" the set of isomorphism classes of locally constant sheaves of free H~ 0y)- modules of rank n on f-X(U). There is a map of functors from M~(X/S, n) to Ms(X/S , n). If Z is any local system of schemes over S ~" with a natural transformation of functors M~(X/S, n) -+ Z, there is a unique factorization through a map M~(X/S, n) -+ Z. Proof. -- This is a translation to the case of local systems of schemes, of the property that the fiber Ms(X,, n) universally co-represents the functor M~(X,, n). [] Moduli of Higgs bundles Suppose that f: X ~ S is a smooth projective morphism to a scheme of finite type over C. A Higgs sheaf on X over S is a coherent sheaf E on X together with a holomorphic map q~ : E -+ E | f~/s such that q~ ^ q0 ---= 0. Similarly, a Higgs bundle is a Higgs sheaf (E, q~) such that E is a locally free sheaf. Higgs bundles on curves were introduced by Hitchin in [Hi2] and [Hill. The condition ~ ^ q0 = 0 for higher dimensional varieties was introduced in [Si2] and [Si5]. Hitchin gave an analytic construction of the moduli space [Hill (this part of his argu- ment works for any rank). Nitsure gave an algebraic construction of the moduli space of Higgs bundles over a curve [Nil]. 16 CARLOS T. SIMPSON We give two constructions of the moduli space of Higgs bundles, based on two different interpretations. The first is simply to note that a Higgs bundle is a A-module for an appropriate sheaf of rings A. This does not give too much other information, and is based on all of Part 1. The second construction uses only the moduli space of coherent sheaves ofw 1, and it gives some additional information about the moduli space: the properness of Hitchin's map. Lemma 6.5. -- Let A m"~ = Sym'(T(X/S)). Then a I-[iggs sheaf on X over S is the same thing as an Ox-coherent Aaig~-module on X. Proof. -- This follows from the discussion of split almost polynomial rings A at the end ofw 2, Part I (Lemma 2.13). In this case it is easy to see that an action of the symmetric algebra on a sheaf E is the same thing as a map q~ : E ~ E | T'(X/S) such that ~^~=0. [] Fix a relatively very ample Ox(1 ). Define the notions of pure dimension, p-semi- stability, p-stability, ~t-semistability and ~t-stability for Higgs sheaves to be the same as the corresponding notions for Am~"-modules. These coincide with the notions defined in [Si5] for the case when S = Spec(C) (pure dimension d = dim(X) is the same thing as torsion-free). Recall that in the relative case, the conditions of semistability and stability contain the hypothesis that the sheaf is flat over the base S. Let M~g=(X/S, P) denote the functor which associates to an S-scheme S' the set of isomorphism classes ofp-semistable Higgs sheaves E on X' over S' with Hi]bert polynomial P. This is univer- sally co-represented by the moduli space M~,,~(X/S, P) d~ M(Ar,,g~, p) constructed in Theorem 4.7, Part I. The points of MH,~(X/S, P) parametrize Jordan equivalence classes of p-semistable Higgs sheaves with Hilbert polynomial P on the fibers X s. If P has degree dim(X), then these are the same as torsion-free p-semistable Higgs sheaves which were discussed in [Si5]. Let P0 denote the HUbert polynomial of 0 x. Let M~,(X/S, n) denote the functor which to an S-scheme S' associates the set of isomorphism classes ofp-semistable Higgs sheaves E on X' over S' of Hilbert polynomial nP0, such that the Chern classes c~(E~) vanish in H2~(Xs, C) for all closed points s ~ S'. In general, if f: X --~ S is a smooth projective morphism and E is a coherent sheaf on X which is flat over S, then the Chern classes c~(E) are sections of the relative algebraic de Rham cohomology R~*f.(~x/s, d) which are flat with respect to the Gauss-Manin connection. Thus the condition that the Chern classes q(E,)~ H2~(X~, C) vanish depends only on the connected component of S containing s. The functor M~I(X/S , n) is universally corepresented by a scheme MDoI(X/S , n) which is a disjoint union of some of the connected components of Mm~(X[S, nP0) (the fact that MDo1(X/S , n) may be a proper subset of Mmgn(X/S, nP0) was pointed out to me by J. Le Potier [Le]). The points of M~I(X/S, n) correspond to Jordan equivalence classes ofp-semistable torsion-free Higgs sheaves of rank n on the fibers Xs, MODULI OF REPRESENTATIONS. II with Chern classes vanishing in the complex valued (or equivalently, rational) cohomology of X,. We call MDol(X/S , n) the DoIbeault moduli space. There is an open set Mgol(X/S , n) parametrizing p-stable Higgs sheaves, and there a universal family exists 6tale locally. Proposition 6.6. -- Suppose X is a smooth projective variety over S = Spec(C). IrE is a v-semistable torsion free Higgs sheaf with Chern classes equal to zero, then E is a bundle, and is in fact an extension of v-stable Higgs bundles whose Chern classes vanish. Any sub-Higgs sheaf of degree zero is a strict subbundle with vanishing Chern classes. Proof. -- [Si5] Theorem 2. [] Corollary 6.7. -- If X is smooth and projective over a base S, if S' is an S-scheme, and if E is an element of M~ol(X/S , n) (S'), then E is locally free over X'. The points of Mgo,(X/S , n) correspond to direct sums of v-stable Higgs bundles with vanishing rational Chern classes on the fibers X a . Proof. -- This follows from the previous proposition and Lemma 1.27, Part I. [] Remark. -- For Higgs sheaves with vanishing Chern classes, p-semistability (resp. p-stabiIity) is equivalent to ~-semistability (resp. ~-stability). This follows from Pro- position 6.6. Suppose X is smooth and projective over S. A Higgs bundle E on X (flat over S) is of semiharmonic type if the restrictions to the fibers E, are semistable Higgs bundles with vanishing rational Chern classes. Say that E, is of harmonic type if it is a direct sum of stable Higgs bundles with vanishing rational Chern classes. The Higgs bundles of semiharmonic type are those which correspond to representations of the fundamental group in [Si5]. Those of harmonic type correspond to semisimple representations. The closed points of MDo~(X/S, n) parametrize the Higgs bundles of harmonic type of rank n. Suppose X-+ S is a smooth projective morphism with connected fibers, and suppose ~ : S -+ X is a section. Let R(A ~g~, ~, nP0) denote the representation space for framed Amg~-modules constructed in Theorem 4.10, Part I. By Proposition 6.6, allp-semistable Higgs sheaves with vanishing rational Chern classes satisfy condition LF (X) and hence condition LF(~). Let RDol(X/S , ~, n) denote the disjoint union of those connected components of R(A mg~, ~, nP0) corresponding to Higgs sheaves with vanishing rational Chern classes. Then RDol(X/S , ~, n) represents the functor which associates to an S-scheme S' the set of isomorphism classes of pairs (E, ~) where E is a Higgs bundle of semiharmonic type on X' over S' and ~ : ~*(E) -~ 0~, is a frame. We call this scheme the Dolbeault representation space. The C* action Recall that an action of an algebraic group G on a scheme Z is a morphism G � Z ~ Z satisfying the usual axioms for a group action, with the axioms written in 18 CARLOS T. SIMPSON terms of diagrams of morphisms: the two maps G x G x Z ~ Z are the same (asso- ciativity); and the map Z -7 Z induced by the identity element e ~ G is the identity. We can define similarly the notion of an action on a functor Y~: this is a natural trans- formation offunctors G x Y~ ~ Y~ satisfying the same axioms. IfG acts on a functor Y~, and q~:Y~ -+ Y is a natural transformation so that the scheme Y universally corepre- sents Y~, then there is a unique action of G on the scheme Y which is compatible with ~. The morphism G x Y ~ Y is obtained from the natural transformation of functors G x Y~ ~ Y by applying the universality hypothesis, that G x Y corepresents the functor (G � Y) � YI = G � Y". The axioms are checked using the uniqueness part of the definition of universally co-representing a functor. The algebraic group C" acts on the functor M~g~(X/S, P) in the following way. If S' is an S-scheme, t : S' -~ C* is an S'-valued point, and (E, ~0) e M~I,~(X/S , P) (S') is a p-semistable Higgs sheaf with HUbert polynomial P on X' over S', then (E, tq~) is again an element of M~I~gB(X/S, P)(S') (the property of p-semistability is preserved because the subsheaves preserved by tq~ are the same as those preserved by q~). We obtain a morphism of functors giving the group action. By the above discussion, there is a unique compatible action of C" on Mm,,~(X/S, P). This gives an action of t3" on MDoI(X/S, n). Similarly, if the fibers X, are connected and ~ is a section, the formula t((E, 9), ~) = ((E, tg) , ~) gives an action of C* on the Dolbeault representation space RDoI(X/S, ~, n). This commutes with the action of Gl(n, C) and the good quotient RgoI(X/S, ~, n) ~ M,ol(X/S, n) is compatible with the action of C'. The subspace MDol(X/S, n) c~ of points fixed by C" is a closed subvariety. Alge- braically the structure of a lliggs bundle on X, fixed by C* is the following (cf. [Si5] Lemma 4.1). If (E, q~) ~ (E, tq~) for some t e C* which is not a root of unity, letf be the isomorphism. By appropriately combining the generalized eigenspaces off, we get a decomposition E = @ E ~ such that q~ : E ~ -+ E ~ -~ | ~)~x. By the analytic results of [Si5], the points of MDot(X/S, n) c" correspond to Higgs bundles which come from complex variations of Hodge structure. The second construction The idea behind this construction is that a Higgs bundle on X can be thought of as a coherent sheaf 8 on the relative cotangent bundle T*(X/S). Let Z denote a projective completion of T*(X/S), and let D = Z -- T*(X/S) be the divisor at infinity. Choose Z so that the projection extends to a map z~ : Z --> X. Lemma 6.8. -- A Higgs sheaf E on X over S is the same thing as a coherent sheaf g on Z such that supp(E) c~ D = O. This identification is compatible with morphisms, giving an equi- valence of categories. The conditions of flatness over S are the same. For s ~ S, the condition that E, is torsion-free is the same as the condition that g, is of pure dimension d = dim(X,). MODULI OF REPRESENTATIONS, II 19 Proof. -- The projection ~ : T*(X/S) ~ X is an affine morphism, in other words T*(X/S) is the sheafified spectrum of the sheaf of tings ~. 0T.IX!.~ on X. This sheaf of tings is naturally isomorphic to the symmetric algebra on the tangent bundle Sym" T(X]S), so giving a quasicoherent sheaf 8 on T*(X/S) it is equivalent to giving a quasicoherent sheaf E = re. 8 on X together with an action ofSym" T(X/S). But Sym" T(X/S) --=- A ~l~, so by the discussion in w 2, this action is the same as the data of a map r : E ~ E | f~ such that q~ ^ ~ = 0. A coherent Higgs sheaf E is the same thing as a sheaf 8 on T'(X/S) such that ~, 8 is coherent. This condition of coherence means that 8 is coherent on T*(X/S) and the closure of the support of 8 in Z does not meet the divisor at infinity D. A morphism of coherent sheaves d ~ -* o~- is the same thing as a morphism re.(8) -~ ,~.(~-) compatible with the action of the symmetric algebra, or equivalently compatible with q~. Since ~ !~.lx!sl is an affine map, flatness of 8 over S is equivalent to flatness of n.(8) over S. Finally note that the dimension of support of any subsheaf of E is the same on X, as it is on Z,, because of the condition that the support doesn't meet D,. Therefore the conditions of pure dimension d = dim(X,) on X, and Z, are the same. On X,, the condition that a sheaf has pure dimension d = dim(X,) is the same as the condition that it is torsion free. [] Choose k so that 0z(1 ) aej re* ~)x(k)|162 0z(I)) is ample on Z (here we suppose that Z is the standard completion of the cotangent bundle to a projective space bundle). In particular, OT.~XtS~(1) = r~" 0x(k ). Thus, for any coherent sheaf o ~ on Z with support not meeting D, the Hilbert polynomials of 8 and ~. ~ differ by scaling: p( 8, m) = p(~. 8, km). Corollary 6.9. -- The notions of p-semistability, p-stability, ~-semistability, and ~-stability I-[iggs sheaf E on X over S are the same as the corresponding notions for the coherent sheaf 8 for a associated to E in the previous lemma. on Z Proof. -- The sub-Higgs sheaves of E correspond to the coherent subsheaves of ~', since a subsheaf of 8 is the same thing as a subsheaf of re. 8 preserved by the action of the symmetric algebra. Scaling the Hilbert polynomials preserves the ordering and scales the slope. [] Fix a polynomial P of degree d = dim(X/S), and put k" P(m) = P(km). Fix a large N as required by the constructions of w 1 for sheaves on Z. Put ~ ----- 0z(-- N) and V = C k'cPcs~, and let Hilb(V | k ~ P) denote the Hilbert scheme of quotients V| on Z, flat over S, with Hilbert polynomial k'P. Let Q x and Q2 denote the subsets defined in w 1 (not those defined in w 3), and let Q3cQ2 denote the open subset parametrizing quotient sheaves 8 whose support does not meet D. By Theorem 1.19, Part I, and [Mu], a good quotient M(Oz, k" P) = Q2]SI(V) exists. The open set Q3 is Sl(V)-invariant and is set-theoretically the inverse image of a subset of M(Oz, k" P) (since the support of 8, is the same as the support of gr(8,)). Therefore a good quotient Q3/SI(V) exists and it is equal to an open subset which we denote 20 CARLOS T. SIMPSON M(dYT.~x/s~, k* P) of M(dYz, k" P). Theorem 1.21, Part I, Lemma 6.8, and Corollary 6.9 imply that M(Oz.~x/s~, k* P) universally co-represents the functor M~m~(X/S, P), and we have all of the properties of Theorem 1.21, Part I. We may put Mmg,~(X/S , P) = M(d~T.,X/S,, k" P). Define the subset MDo,(X/S, n)C Mmgg.(X]S , nPo) as before. These moduli spaces are the same as those constructed previously, because they co-represent the same functors. Hitch|n~s proper map We will now define a map t?om the space of Higgs bundles to the space of possible characteristic polynomials for 0. This map is the generalization of the determinant map that Hitchin studied in [Hil]. In Hitchin's case it turned out that this map was proper ([Hil] Theorem 8.1), and we will prove the same here also. Roughly speaking this means that the only way for a Higgs bundle to " go to infinity" is for the characteristic polynomial to become singular. For any n let ~e'(X/S, n) ~ S be the scheme representing the functor which to an S-scheme S' associates + H~ ', Sym' fit.;s, ) i=l [Grl] [Mu]. We consider the points of ~e'(X/S,n) as polynomials written t"+a it "-1 + ... +a, with a i~H~ ',Sym ~ 1 ~x'/s')" Let ~1, ..., % denote the symmetric polynomials in an r � r matrix variable A such that det(t -- A) = t ~ + al(A) t "-1 + ... + ~,(A). For example, al(A) = -- Tr(A) and %(A) = (-- 1)" det(A). Let P be a polynomial of degree d = dim(X/S) and rank r = deg(X)n, so that sheaves of pure dimension d and Hilbert polynomial d on the fibers X 8 are torsion-free with usual rank equal to n. Suppose S' is an S-scheme and (E, tp) is a p-semistable Higgs sheaf with Hilbert polynomial P on X' over S'. Then there is an open subset U'C X' such that the intersection of X' -- U' with any fiber has codimension at least 2, and such that E is locally free over U'. Over U', ~p is an f~,/s,-valued endomorphism of a rank n-vector bundle. Furthermore, the endomorphisms obtained by contracting with different sections of T(X'/S') commute with each other. Thus we can evaluate the elementary symmetric polynomials to obtain ~,(q0 Iv') ~ H~ U', Sym'(~)u'/s'))" MODULI OF REPRESENTATIONS. II 21 Since Sym~(D~,/s,) is a locally free sheaf, Hartog's theorem applied over artinian subschemes of S', coupled with the theorem on formal functions and Artin approxi- mation, imply that ~,(~ [o,) extend uniquely to sections which we denote ~(q~) e H~ ', Sym'(f2Jx,/s,)). Define ~(E, e)ei/'(X/S,n)(S ') to be the point corresponding to (~l(q~), ..., %(?)). This construction defines a morphism from the functor M~m.,,(X/S, P) to t/'(X/S, n), and hence a morphism of schemes ~ : Mr,,,~(X/S, P) --~ tP(X/S, n). We call ~(E, e) the characteristic polynomial of (E, 0). The morphism cr was introduced by Hitchin for Higgs bundles on curves in [Hi2] and [Hil]. There are universal sections a~" : X x s t/'(X/S, n) -+ Sym ~ T'(X/S) X 8 ~(X/S, n). Here Sym ~ T~ denotes the total space of the i-th symmetric power of the relative cotangent bundle. There is a multiplication map Sym ~ T'(X/S) Xx Sym J T'(X/S) ~ Sym' ~ ~ T'(X/S) as well as a map corresponding to addition. There is a closed subscheme #'(X/S, n) C T'(X/S) x s r n) defined by the equation t"+a~ l't"-l+... +a~,=0. -,- 1 as points in the This represents a functor which can be seen by considering t ~ and ,,u,l, total spaces of the symmetric powers, then multiplying them together and adding to get a point in Sym" T'(X/S) which is required to be in the zero section. Lemma 6.10. -- Suppose S' is an S-scheme and (E, ~) is a p-semistable Higgs sheaf with Hilbert polynomial P on X' over S', corresponding to a cokerent sheaf o a on T'(X'/S'). Let e(E, 0) : S' --~ $/'(X/S, n) be the characteristic polynomial defined above. Then o ~ is supported set theoretically on #'(X/S, n) x,~x/s, ,~, or S'C T'(X'/S'). Proof. -- In general, if A is a vector space and Sym'(A) acts on a vector space B, then the support of B considered as a coherent sheaf on A ~ is equal to the set of eigen- forms of the action of A. Now let U' C X' be the open set used above in the construction of ,(E, ~?). Then the zeros of the characteristic polynomial ,(E, 0) over U' are the eigen- forms of q0 [o'. Thus Zt/r(X/S, n) X~cx/s, ,~, ol~, ~ S' is the spectral variety of the endo- morphism 91tr'. From the above general principle, o~[T.cw/s ,, is supported on this spectral variety. On the other hand, any section of @ whose support is contained in T*(X'/S') --T*(U']S') restricts to a section of the fiber oa8 supported over the corn- 22 CARLOS T. SIMPSON plement X'~ -- U', (which has dimension less than or equal to d -- 2). The do8 are finite over X',, so such a section is supported in dimension less than or equal to d -- 2. By the hypothesis that g, are of pure dimension d, such a section is zero. Hence any section of d ~ supported in T*(X'/S') -- T*(U'/S') restricts to zero in all the fibers, so it is zero. Since the spectral variety is closed, this implies that all the sections of do are supported in $//'(X/S, n) � ~lx/s, ,~, o(~, ~I S'. [] Remark. -- Using Cayley's theorem, one can see that the support is scheme- theoretically contained in the spectral scheme ~(X/S, n) X rix/s, ,~, ocE, ~ S'. Theorem 6.11. -- The map a : Mmg~(X/S, P) -+ ~r n) is proper. Proof. -- Note, first of all, that all schemes involved are separated. Suppose S' is a curve, s e S' is a closed point, and put S" = S' --{ s }. Suppose that g : S" --> Mmg~(X/S, P) is a map such that the composed map ag extends to a map h : S' ~ ~P(X/S, n). Recall that Mm,,,(X/S, P) is an open set in M(0z, k* P), and that Nl(0z, k* P) is projective over S. The map g extends to a map g' : S' -+ M(~z, k* P). Let q0 : Q.2 -+ M(0z, k* P) be the good quotient of the parameter scheme O~ for sheaves on Z used in the cons- truction of w 1. Then 02 X moz, k*1.~ S' -+ S' is a categorical quotient. Thus there is a quasi-finite morphism of curves Y ~ S' such that s is the image of a pointy e Y, such that Y' ~ Y -- {2 } maps to S" C S', and such that the resulting map Y ---> M(Oz, k* P) lifts to a map Y -+ Q.,. Let do be the resulting sheaf on Z x s Y. If w e Y' then dow is a sheaf corresponding to a point in Mm~,(X/S , P). Thus @w has support contained in T*(Xw). This implies that the support of @IY' is contained in T*(X/S) x s (Y'), so it corresponds to a Higgs sheaf (E', q~') on X X s (Y') over Y'. The map Y' -+ Ma,,g,(X/S, P) corresponding to (E', ~') is equal to the map obtained by composing Y' -+ S" with g. In particular, the characteristic polynomial a(E', qZ) : Y' -+ r n) is the composition ofY' ~ S" with ag. Thus a(E', ~') extends to a map f: Y -+ $/'(X/S, n), equal to the composition of Y -+ S' with h. By the previous lemma, do [:r is supported inf*~(X/S, n) C T*(X/S) � s Y" Since 8 is flat over Y, it has no local sections supported on Zu. Thus do is supported on the closure off* ~r n) in Z X s Y. But since the equation defining SU(X/S,n) is monic, the subscheme $r is closed in Z X s ~e"(X/S, n). Therefore f* $K(X/S, n) is closed in Z x s Y, and do is supported in f* Y//'(X/S, n) C T*(X/S) x s Y. Hence @ corresponds to a Higgs sheaf (E, q~) on X � s Y over Y, restricting to (E', ~') over Y'. As 8 is a p-semistable sheaf, (E, q~) is a p-semistable Higgs sheaf by Corollary 6.9. We obtain a map Y -+ M~I,~(X/S, P) extending the composition Y' -+ S" -+ Ma~,(X/S , P). But this map is also equal to the composition Y-.--S'-+M(Oz, k* P) since the moduli space is separated. In this last map, the image ofy is equal to the image g'(s). From the fact that Y maps into MODULI OF REPRESENTATIONS. II Ma~***(X/S, P) we obtain g'(s) ~ Mmg~(X/S , P). Thus g' maps S' into MH~g~(X/S , P). This is the extended map required to prove properness of a. Remark. -- Hitchin gives an analytic proof of the properness of a in the case when X is a curve [Hil]. Corollary 6.12. -- Any p-semistable torsion-free Higgs sheaf on a fiber X, can be deformed to one which is fixed by the action of C*. A Itiggs bundle of semiharmonic type can be deformed to a Itiggs bundle of semiharmonic type which is fixed by C*, through a family of Higgs bundles of semiharmonic type. Proof. -- Suppose (E, ~) is a p-semistable torsion-free Higgs sheaf on Xs. Write the characteristic polynomial as ~(E, 9) ---- t" + a 1 t "-~ -+- ... + a,. For z ~ C*, the characteristic polynomial of zq~ is ~(E, z~) = t" + za 1 t"-I + ... + z" a,. As z -+ 0 these polynomials approach the limit t" in r n). The orbit of (E, 9) is a map C* ~Mm~,,(Xs, P ) such that the composed map C*-+~r n ) extends to a map A 1 ~ r n). By the theorem, the orbit extends to a map A a -7 Mn~g,~(X/S, P). Let Q, be the parameter scheme used to construct Mln,gs(X/S , P). Then Q.2 x Mmgg,~x~. P) A1 --+ A1 is a good quotient. In this situation, the unique closed orbit lying over 0 ~ A 1 is contained in the closure of the union of orbits corresponding to (E, zq0). Thus (E, 9) may be deformed to a Higgs sheaf corresponding to a closed orbit over 0 E A x. The map A 1 -+ Mn,,,~(X/S, P) is equivariant under the action of C*, so the image of the origin is a fixed point. Since there is a unique closed orbit lying over the origin, this closed orbit is preserved by C'. Thus the Higgs sheaf corresponding to the closed orbit over the origin is fixed up to isomorphism by the action of C*. This proves I the first statement. For the second statement, note that by Proposition 6.6, if the rational Chern classes of E vanish, then all of the Higgs sheaves involved are Higgs bundles. When X is a curve, Hitchin has a beautiful description of the generic fiber of the map a [Hi2]. In terms of our description of the moduli space, the idea is as follows. For a generic point s E S, the corresponding polynomial defines a smooth curve in the cotangent bundle, counted with multiplicity one. A Higgs bundle E in the fiber over that point is a coherent torsion free sheaf on the curve, of rank one. In other words, it is a line bundle. Furthermore, all line bundles of the appropriate degree occur. Thus the fiber is the Jacobian of the curve. See also [Ox]. CARLOS T. SIMPSON Vector bundles with integrable connections In this section we will apply the results of w 4, Part I, to construct a moduli space of vector bundles with integrable algebraic connection. For greatest generality, suppose that S is a base scheme of finite type over C, and that X is smooth and projective over S. A vector bundle with connection on X/S is a vector bundle or locally free sheaf E on X, together with a map of sheaves such that Leibniz's rule V(ae) -- dx/s(a ) e -i- aV(e) is satisfied for any sections a of G x and e of E. Here dx/s : 9 x ~ f2Jxts is the relative exterior derivative. Given a vector bundle with connection, we can extend V to an operator V: E | ~ E| by enforcing Leibniz's rule for forms a, using the usual sign conventions. In particular, the square of V is an operator V2:E_~E � 2 4<) ~x/s. Using Lcibniz's rule and thc fact that (dxts) s = 0, it is easy to sec that V z is d)x-linear. Thus it is givcn by a scction V ~ e H~ | ~/s) callcd thc curvature of V. A vector bundle with integrable connection is a vcctor bundlc with conncction (E, V), such that the curvaturc vanishcs, VS~ 0. Wc could makc a similar definition of cohcrcnt sheaf with intcgrablc connection. Howcvcr, it is a wcll known fact (which wc will not provc hcrc) that if E, is a cohcrcnt shcaf with intcgrablc conncction on X,, then E s is locally frcc, and thc Chcrn classcs of E 8 vanish (hcncc thc normalized Hilbcrt polynomial of Es is thc samc as that of Ox). If E is a cohcrcnt shcaf on X with intcgrablc rclativc conncction, such that E is flat ovcr S, thcn E is locally frcc by Lcmma I. 27, Part I. Bccausc of this, wc may as wcll assumc that thc pure dimcnsion d is cqual to the rclativc dimension of X/S, and that the normalizcd Hilbcrt polynomial P0 is cqual to that of ~x-----othcrwisc thc moduli spaccs arc cmpty. Furthcrmorc, any subshcaf of E~ prcscrvcd by V is again a vcctor bundle with intcgrablc connection, with the samc normalizcd Hilbcrt polynomial P0- Theorem 6.13. -- Suppose X is smooth and projective over S. There is a scheme MvR(X/S , n) quasi-projective over S, universally co-representing the functor M~R(X/S, n) which assigns to an S-scheme S' the set of isomorphism classes of vector bundles with integrable connection (E, V) on X'/S' of a given rank n. Suppose X is smooth and projective with connected fibers over S, and suppose x : S --* X is a section. There is a scheme RvR(X/S, ~, n) quasi-projective over S, representing the functor which assigns to an S-scheme S' the set of isomorphism classes of (E, V, 0~) where (E, V) is a vector bundle with integrable connection on X/S, and ~ : E !~s'~ _Z~ ~, is a frame along the section. MODULI OF REPRESENTATIONS. II 25 Furthermore, with respect to an appropriate line bundle all points OfRm~(X/S , 4, n) are semistable for the natural action of Gl(n, C), and the universal categorical quotient is naturally identified with MD~(X/S , n). Proof. -- Let A "R be the sheaf of rings of all relative differential operators on X over S. It is split almost polynomial, and the sheaf H arising in the description of w 2, 1 . Part I, is equal to Y~x/s, its dual H* is the relative tangent bundle T(X/S). The derivation is the standard one, and the bracket { , ),r is given by commutator of vector fields. The description of ADR-modules given in Lemma 2.13, Part I, coincides with the above definition of vector bundle (or sheaf) with integrable connection. If E is a vector bundle with integrable relative connection on X over S, then any subsheaf of E s preserved by the connection has the same normalized Hilbert polynomial P0, so E~ is p-semistable as a Ang-module. If E is fiat over S then E is a p-semistable AWg-module. The theorem follows from the general construction of moduli spaces given in Theorem 4.7, Part I. For the second paragraph, note that any vector bundle with relative integrable connection automatically satisfies condition LF(~). Hence we may apply Theorem 4.10, Part I, to obtain RD~(X/S, 4, n). [] Dependence on the base point In the Betti case, given two different base points x and y, and a choice of path y from x to y, we obtain an isomorphism T v : RB(X, x, n) ~ R~(X,y, n). This projects to a canonical isomorphism of M~(X, n), justifying dropping the basepoint from the notation for the moduli space. On the other hand, there is no natural isomorphism between RD~(X, x, n) and RD~(X,y, n). Our construction began with a construction of MD~(X, n) independent of the base point. On the complex analytic spaces, the isomorphism T v gives a complex analytic isomorphism (cf. w 7 below). This projects to an algebraic isomorphism (the identity) in the quotient, and on each orbit of the group Gl(n, C), it comes from an algebraic automorphism of groups; however Tv doesn't seem to be algebraic. One might conjecture that, in good cases, the isomorphism class of RD~(X, x, n) is a distinguishing invariant of the point x. 7. Identifications between the moduli spaces The analytic isomorphism between the de Rham and Bettl spaces Suppose f: X-+ S is a smooth projective morphism with connected fibers, and suppose ~ : S -+ X is a section. Recall that R~nI(X/S, 4, n) denotes the complex analytic space over S an associated to the local system of complex analytic spaces R~n(X/S, 4, n). Let RL~(X/S, 4, n) denote the complex analytic space associated to the de Rham representation scheme. 4 CARLOS T. SIMPSON Theorem 7.1. -- (The framed Riemann-Hilbert correspondence.) There is a natural isomorphism of complex analytic spaces R'.='(X/S, n) = Rg%(X/S, n), compatible with the action of Gl(n, C). We will prove this by showing that both spaces represent the same functor from the category of complex analytic spaces over S ~ to the category of sets. [,emma 7.2. -- Suppose Y and Z are topological spaces which can be exhausted by relatively compact open subsets, and suppose Y is locally simply connected. Suppose A is a sheaf of rings on Z. Suppose F is a locally free sheaf of p~l(A)-modules on Y x Z. Then G =pl,.(F) is a locally constant sheaf of H0(Z, A)-modules on Y, and the stalk at y ~ Y is given by G v = H~ � Z, � Proof. -- First of all, suppose that U is a connected open subset of Y. Then P2,.(P;~(A) Iu � z) = A. This implies that if F is free over p~-'(A), U C Y is a connected open subset, and y e U, then P2,.(F !v� z) ~P2,.( F [tul� z) is an isomorphism of free sheaves of A-modules. Suppose that there exists an open covering Z = O~ Z~ such that F is free of rank n on Y � Z~. The previous result implies that ify ~ U C Y and U is connected, then is an isomorphism of locally free sheaves on Z. This gives an isomorphism of spaces of global sections, in other words the restriction maps H~ � Z, F!~j~z) -+ U~ Z, Flcy~� are isomorphisms. This implies the lemma in this case. It follows that the lemma is true for sheaves F such that there exist open coverings Y = [J~ Y~ and Z = [.J~ Z~ with F free on each Y~ � Z,. Finally we treat the general case of a locally free F. Suppose that Z ~ is an increasing sequence of relatively compact open subsets exhausting Z. Then, for any relatively compact open set Y'C Y, F [y,x zi satisfies the hypotheses of the previous paragraph. def f ~ p Thus G ~ = p:,.@iyxzl) is locally constant when restricted to any Y'. This implies that G ~ is locally constant. The stalk at y e Y is G; := H~ I � Z', � MODULI OF REPRESENTATIONS. II 27 Finally, we have G --=- liln G i, .z.-- and since Y' is locally simply connected, the inverse limit of a system of locally constant sheaves is again locally constant. This proves that G is locally constant. The stalk G v is the inverse limit of the stalks G~v, hence equal to the desired space of global sections. /_,emma 7.3. -- The complex analytic space R~=)(X/S, 4, n) over S an represents the functor which to each morphism S' ~ S ~" of complex analytic spaces associates the set of isomorphism classes of pairs (o~,~) where o~" is a locally free sheaf off '(Os,)-modules of rank n on X' = X ~ � S', and ~ : ~-1(o~') ~- ~, is aflame over the section 4. Proof. -- Let X t~ denote the topological space underlying X '~. Note that the quantities appearing in the statement of the lemma are local over S an and depend only on the structure off: Xt~ S = as a fibration of topological spaces with a section over S ~. Thus we may suppose that X~P=Xo � S ~ and ~(s) = (x,s) for xeX o. Let P = nl(X0, x). Then 4, n) = x=(r, n) x s =, so the set of S~-morphisms from S' to R~')(X/S, 4, n) is equal to the set of morphisms from S' to Ran(r, n). The scheme R(I', n) is affine. It follows that the complex analytic morphisms from S' to the associated analytic space R'~(I TM, n) are given by the homomorphisms of C-algebras H~ n), O.,r..,) -+ Ho(s ', 0s, ) (this can be seen by embedding R(P, n) in an aHine space). In particular, R~'(X/S, 4, n)(S') = R~(r, n)(S') = R(r, n)(Spec(H~ ', Os,)) ) = Hom(P, Gl(n, H~ ', d)s,)) ). Let Px and pz denote the first and second projections on X0 � S'. Suppose o~ is a locally free sheaf ofp~-l(Os,)-modules on X 0 � S'. Let N = p,,.(o~'). This is a sheaf of H~ ', 0s,)-modules on X 0. Lemma 7.2 implies that ~ is locally constant, with fiber if, = #-I{~)xs' over x cX o. The monodromy of the locally constant sheaf ff is a representation F -* Aut~o(s,,e~.,)(ff~). If [~ : ~'(o~') ~ O~, then by Lemma 7.2 the fiber is fr _-__ H~ ', Os,) ~, so the mono- dromy of fr gives a representation r Ol(n, Ho(s ', This gives a map from the set of isomorphism classes of(#', ~) to Hom(P, Gl(n, H~ ', Ss'))- 28 CARLOS T. SIMPSON For the inverse map, note that X 0 is locally simply connected, so a universal covering X0 exists. Choose a base point ~ over x eX 0. Set ~,~ = p~-1(0~,) on ~ x S'. The identity gives "~ : "~1~ � s' ~ ~'. A representation r -+ Gl(n, Ho(S ', 8s,)) gives an action of I" on ,~ over the action on ~ � S'. We can use this to descend to a locally free sheaf ofp~-L(Os,)-modules .~ on X 0 � S', with the required frame ~. This is the inverse of the previous construction. We obtain an isomorphism between the set of S'-valued points of RI~")(X/S, ~, n), and the set of (oq~, ~) on X' over S' as desired. [] This lemma provides half of the proof of the theorem. For the other half, we begin with a lemma and some corollaries. Lemma 7.4. -- Suppose that V C C, ~ is an open disc and S'C V is a complex analytic subspace, such that all embedded components pass through a single point s ~ S'. Suppose U is an open disc centered at the origin in C2. If (E, V) # a holomorphic vector bundle with integrable connection on U � S' over S', then the map v : H~ x S', E> ~ -+ H~ 0 ) � S', E [~o~ ~) is an isomorphism (here the exponent V means the space of covariant constant sections). Proof. -- This is well known ifS' is a point. It follows that it is true ifS' is an artinian complex analytic space (the same as an artinian scheme), for the result in that case follows from the same result for Pl..(E, V) on U. It is also well known if S' is a smooth complex analytic manifold. We show how to deduce the theorem when S' may be nonreduced, for example. Choose a point s ~ S' containing all irreducible embedded components of S. Let S',, be the m-th infinitesimal neighborhood of s in S'. The map v is injective: suppose e is a section with V(e) = 0 and ,J(e) ~ 0; then from the result for artinian spaces, e Is;, = 0 for all m, and this implies that e = 0. With the same hypotheses on S', suppose also that dim(U) = 1. We show that v is surjective in this case. The holomorphic bundle E is trivial, so it has an extension to a trivial bundle E ~ on U � V. Let d denote the constant connection on EeL We may write v = dl~� + A(u, t) du where A(u, t) is a holomorphic section of End(E) over U � S. There exists an extension of A to a holomorphic section A ~ of End(E ~x) on U � V, and we can then put V e~ = d + Ae*(u, v) du. MODULI OF REPRESENTATIONS. II 29 This is an integrable holomorphic connection on E ex relative to V (it is integrable because dim(U) = 1 implies ~ x v/v = 0). Suppose e 0 e H~ 0 } � S', E I{o} x s'), and choose an extension to e~ x e H~ 0 } � V, E I{0} � v). By the result for smooth base spaces, there exists e eXeH~ X V, Eex) vex with v(e "*) =e~ x. Putting e=W[v� gives a section with V(e) = 0 and v(e) = e 0. This proves that v is an isomorphism in the case of relative dimension 1. Now proceed by induction on the relative dimension k, assuming that the theorem is known for relative dimension k -- 1. Let U 1 denote the disc of dimension k -- 1 obtained by intersecting U with one of the coordinate planes. By the inductive hypothesis, there exists a section e i in H~ � S', E Ivl x s;,) v with e i restricting to e 0 on { 0 ) � S'. Let h : U � S' -+ U i � S' denote the vertical projection. Let V t denote the projection of V into a relative connection for the map h. The map h is smooth of relative dimension 1, so by the previous result, there exists a section e in H~ � S', E) v~ restricting to e i on Ui x S'. In order to show that V(e) = 0 we use the infinitesimal neighborhoods S~. There exist sections e" in H~ � S~,Elvxs~,) v such that e~l,0~� xs~. By the uniqueness result for U~ � S~ over S~, e m is equal to e~ on the subspace U1 x S~. By the uniqueness for U X S~ over Ui � S~, ely� = era" But V is Os,-linear , so V(e) Iv x Sm= V(e,,) = 0. This is true for any infinitesimal neighborhood, so V(e) = 0. This shows that ~ is surjective in relative dimension k. [] Keep the same hypotheses as in this lemma. Suppose E is a trivial bundle of rank n. We can choose n sections el, ..., e, in H~ � S', E) v such that v(e,) form a frame for E I~o} � s'- The lemma implies that H~ X S', E) v ~ H~ ', Os, ) | (Cel Q '' 9 e Ce,). Conversely, if the e~ are a collection of sections such that this formula holds, then the ~(e~) form a frame for E I~0} x s'. We claim that the e~ form a frame for E over U. It suffices to show that over each closed point (u, s) e U � S', the e~ are a basis for the fiber of E. But this follows from the above statement and its converse applied to { u } � S' instead of{O) X S'. Corollary 7.5. -- Suppose el, ..., e, are sections chosen as above. Then the map Ml(r ,) | (eel e ... e Ce.) -+ E v is an isomorphism of sheaves on U � S'. Proof. -- This is injective, because the ei are a frame for the holomorphic bundle E. For surjectivity, suppose e is a section of E v over an open set V C U X S'. Then V can be covered by subsets U' � S" of the form considered above. The restriction of { e~ } to a section { u } x S" is a frame for the restriction of the bundle E, so the previous argument applies. There exist a, e H~ '', 08,, ) with E aiei = e on U'� S". This shows that the map of sheaves is surjective. [] 30 CARLOS T. SIMPSON Corollary 7.6. -- Suppose f: X' -+ S' is a smooth morphism of complex analytic spaces. Suppose (E, V) is a vector bundle with integrable holomorphic connection relative to S'. Let ~- = E v denote the sheaf of sections e ore such that V(e) = 0. Then ~" is a locally free sheaf of f-a(Os,)- modules. Proof. -- Let k denote the relative dimension of X' over S'. We can cover X' by a collection of open subsets of the form U x S" where U C (l k is an open disc, and S"C S' is a subset satisfying the hypotheses of the lemma. By the corollary, ~-[u � s" is free overf-l(Os,). Thus ~- is locally free. [] Lemma 7.7. -- The complex analytic space R~(X/S, ~, n) over S ~ represents the functor which to each morphism S' ~ S ~ of complex analytic spaces associates the set of isomorphism classes of pairs (o~', ~) where o~" is a locally free sheaf of f-x(Os,)-modules of rank n on X' = X ~ xs~, S' , and [3 : ~-1(o ~') ~ O~, is a frame over the section 4. Proof. -- Note that by Lemma 5.7, Part I, the argument of Theorem 6.13, and the analogue of Lemma 2.13, Part I, for the complex analytic case, R~a(X/S , 4, n) represents the functor which to each morphism S' -+ S "~" of complex analytic spaces, associates the set of isomorphism classes of triples (E, V, a) where E is a holomorphic vector bundle over X' = X ~ X San S', V is a holomorphic integrable connection on E relative to S', and ~ : ~'(E) ~ 0~, is a frame. We have to identify this functor with the functor given in the lemma. First, note that the trivial bundle ~x' has a natural connection dx,/s, , the exterior derivative with values projected into DAx,/s,. This connection is f-l(Os,)-linear. If ~" is a locally free sheaf off- l(Os,)-modules on X', then is a locally free sheaf of Ox,-modules , and it has a relative holomorphic integrable connection V = 1 | A frame ~ : ~-1(5) --- @~, yields 0c: ~'(E) ~ ~,. This gives a map from the set R~(X/S, 4, n)(S') to the set R~(X/S, 4, n)(S'). Suppose (E, V) is a holomorphic vector bundle with integrable connection on X' over S'. By Corollary 7.6, the sheaf ~ = E v is a locally free sheaf off- l(Os,)-modules. Suppose , is a frame for E along 4. Let ~ denote the composed map ~-~(E ~) ~ ~-~(E) -+ ~'(E) -~ ~,. The arguments from the proof of Lemma 7.4 show that this is an isomorphism. This completes the construction of the inverse to our previous map, so we obtain an iso- morphism between the set of (E, V, 0c) and the set of (~', 9). [] Proof of Theorem 7.1. -- Lemmas 7.3 and 7.7 show that the spaces R~'~I(X/S, 4, n) and R~a(X/S , 4, n) both represent the same functor. Thus they are naturally isomorphic. The isomorphism between functors is compatible with the group action, so the iso- morphism between spaces is too. [] MODULI OF REPRESENTATIONS. II 31 Proposition 7.8. -- (The Riemann-Hilbert correspondence.) Suppose f: X -7 S is a smooth projective morphism. Then there is a natural isomorphism M~='(X/S, n) - M~(X/S, n). If f has connected fibers and ~ is a section, then this isomorphism is compatible with the isomorphism given by Theorem 7.1. Proof. -- Suppose first of all thatfhas connected fibers and a section ~ exists. Then MD~(X/S, n) is a good quotient of RDR(X/S , ~, n) by the action of Gl(n, C). Propo- sition 5.5, Part I, implies that M~aR(X/S, n) is a universal categorical quotient of R~R(X/S , ~, n) in the category of complex analytic spaces over S "n. On the other hand, M~(X,, n) is the good quotient of R~(X,, ~(s), n) by the action of Gl(n, C), so again by Proposition 5.5, M~(X,, n) is a universal categorical quotient of R ~"(Y~ ~.~,,, ~(s), n). The space M~I(X/S, n) is, locally over S ", of the form M~(X~, n) X S '~. The property of being a universal categorical quotient is preserved under taking the product with another space, as well as localization in the quotient space (hence by localization in S'~), so M]3~(X,, n) � S "~ is a universal categorical quotient of R]3"(X,, ~(s), n) � S '~. The isomorphism of Theorem 7.1 is compatible with the action of Gl(n, C), so it induces an isomorphism of universal categorical quotients M~I(X/S, n) _-_ M~R(X/S , n). Suppose that X is a disjoint union of components, each of which has connected fibers over S and admits a section. The resulting moduli spaces M~(X/S, n) and M~a(X/S, n) are then products of spaces obtained by taking quotients of representation spaces. The above isomorphism for each factor gives the desired isomorphism. In general, we can make a surjective dtale base change S' -~ S such that X'/S' satisfies the hypotheses of the previous paragraph. The isomorphism ~n t t ', n) = descends to give the desired isomorphism. [] The homeomorphlsm between the de Rham and Dolbeault spaces Recall some facts from [Si5]. These results are based on non-linear partial diffe- rential equations, and in particular on the works iNS] [Co] iDol] [Do2] [Do3] [Hil] [Si2] [UY]. There is a notion of harmonic metric for a vector bundle with integrable connection (flat bundle) or a Higgs bundle on a smooth projective variety X. Given a flat bundle and a harmonic metric, one obtains a Higgs bundle, and vice versa. The structures of Higgs or flat bundles obtained from the harmonic metric do not depend on the choice of harmonic metric. The conditions for the existence of a harmonic metric are as follows. A flat bundle has a harmonic metric if and only if it is semisimple [Co] [Do3]. A Higgs bundle has a harmonic metric if and only if it is a direct sum of ~t-stable 32 CARLOS T. SIMPSON Higgs bundles with vanishing rational Chern classes [Hil] [Si2]. A harmonic bundle consists of a flat bundle and a Higgs bundle related by a C oo isomorphism such that there exists a common harmonic metric relating the structures. The set of isomorphism classes of harmonic bundles is exactly the same as the set of flat bundles parametrized by points of the moduli space MD~(X , n). It is also the same as the set of Higgs bundles para- metrized by points of the moduli space MDo~(X , n). We obtain an isomorphism of sets of closed points between these two moduli spaces [Si5]. If X ~ S is a smooth projective morphism, we can take the isomorphisms of sets given in each fiber all together to get an isomorphism between the sets of closed points of MDR(X/S, n) and Mr, o~(X/S, n). Recall that the superscript 5,I t~ denotes the topo- logical space underlying the complex analytic space M". We will show that our isomor- phism of sets gives a homeomorphism of topological spaces MDR(X/S, top n) ~ MDol(X/S to. , n). Wc recall a weak compactness property for harmonic bundles, following the notation of [Si5] (except that the Higgs field which was denoted by 0 there is denoted by ~ here, to conform with Hitchin's original notation). Suppose X -+ S is smooth with connected fibers, and suppose ~ : S ~ X is a section. Suppose { s i } is a sequence of points converging to t in S. Choose a standardized sequence of diffeomorphisms ~F~ : X~. -~ Xt, such that ~'~(~.(s0) = ~(t). Choose a family of metrics on X,, which, when transported via ~F~, are uniformly bounded in any norm with respect to a metric on X t. Use these metrics to measure forms on X~. Proposition 7.9. -- Fix q > 1. Suppose V~ is a harmonic bundle on X,i with harmonic metric K, for each i, such that the coefficients of the characteristic polynomials of the Ifiggs fields % are uniformly bounded in L 1 norm. Then there is a harmonic bundle V, a subsequence { i' }, and isomorphisms r~,, : ~F~,,.(V~,) ~ V of Coo bundles satisfying the following properties. There is a harmonic metric K for V with ~,,(K~) = K, and ifO represents any of the operators O, -0, ~, or combinations thereof, the differences dif(O, i') def ~i', .(Oi') -- O converge to zero strongly in the operator norm for operators from L~ to L ~ Proof. -- This is essentially the same as Lemma 2.8 of [Si5], which is based in turn on Uhlenbeck's weak compactness property [Uh] and the properness of Hitchin's map [Hil]. There are a few new twists. The main difference is that the underlying spaces X,/are varying. In particular, the differences dif(O, i) are differential operators, so they must be measured with respect to operator norms. We recall the proof with this in mind. First of all, the hypothesis that the characteristic polynomials are bounded in L 1 norm implies that they are bounded in C o norm, since the coefficients are holomorphic sections of certain bundles on X~. Fix p large. The bound for the coefficients of the characteristic polynomials implies that I q~i [Ki are uniformly bounded [Si5], Lemma 2.7. The curvatures MODULI OF REPRESENTATIONS. II 33 are therefore uniformly bounded in C ~ Uhlenbeck's weak compactness theorem [Uh] gives a unitary bundle (V, K) with unitary connection 0 + 0 (oftype L[) and a sequence of unitary isomorphisms ~: tF~,.(V~) ~ V such that dif(a + i) = + - a -- converges to zero weakly in L[. In particular, for p big enough this converges strongly to zero in C O (note that dif(O + O, i) is a O-th order operator), and dif(O + O, i) ~ 0 strongly in the operator norm for Hom(L~, L~). We have (o, + = o so (0 + ~) (~,. W) = dif(0 + 0, i) (~,,. q~). Since ~ are unitary isomorphisms, l~q,..q~]~ are uniformly bounded. Thus dif(a+0, i) (~,.q~) ~0 strongly in C o , so (0+0) (:q~,.qh) ~0 strongly in C o . We have to be slightly careful, since ~,. ~ are one-forms--this doesn't constitute an estimate for the full covariant derivatives. The ~.. ~ are of type (1, 0) but each for a different complex structure. More precisely, let T~'~ Tr denote the subbundle of forms of type (1, 0) with respect to the complex structure of X,i as transported to X t by ~,. Then B,,. q~, e H~ End(V) | T~'o) 9 We can choose an open subset U C X~ and a sequence of open immersions ~, : U -+ X t which converge to the identity in any norm, but such that ~;(T~ '~ = T 1'~ U is the subbundle of forms on U of type (1, 0) with respect to the holomorphic structure of X t. We may also choose a sequence of unitary isomorphisms ~ :~(V) ~ V converging to the identity in any norm, so for example ~(0 q- 0) -- 0 -- 0 converges to zero in L]'. Then ~,q)~ ~,,. q~, are End(V)-valued (1, 0)-forms on U with (a + uniformly bounded in (3 ~ Now we can conclude (from the elliptic estimates for 0) that ~, ~ ~q~,. q~ are uniformly bounded in L~ on any relatively compact subset of U. This argument, done for a collection of open sets U covering X~, implies that ~,.(qh) are uniformly bounded in L~. By going to a subsequence, we may suppose that ~,. qh approach a limit q~ weakly in L~ (hence strongly in (30). The limit satisfies (0 + 0) (q~) = 0 and ~ A q~ = 0. Finally, in the conclusion of Uhlenbeck's weak compactness theorem, ~..(F~i+~i) approach Fe+~ weakly in LL In particular (even taking into consideration the change of complex structure), the component of type (0, 2) is the weak limit of the components of type (0, 2), which are zero. Therefore 0* = 0, so (V, 0, ~) is a Higgs bundle. The B~,.(~) approach the K-complex conjugate ~ weakly in L~, and the operator 0 is the one associated to 0 by the metric K. If we set D = 0 + 0 q- ~ q- then the differences dif(D, i) = aq,, .(D,) -- D 5 34 CARLOS T. SIMPSON are 0-th order operators converging to zero weakly in L~. In particular, they converge to zero strongly in C O and hence they converge to zero in the operator norm for operators from L~ to L ~. The weak convergence in L~ implies that the curvature D ~ is the weak D z = 0, proving that V together limit of the curvatures ~i,.(i), which are zero. Thus D z with all of its operators and its metric, is a harmonic bundle. We know from above that dis i) -+ 0 strongly in C O for O = 9 and O = 0 + 0. The same argument as for 9 works for O = ~. We have to extract the cases of O = 0 and O = 0 from the case O = O + 0. Let p~,o denote the projection onto T~ '~ and let pa, o denote the projection for the complex structure of X,. Then pl, o + px, o in any norm. We have ~,,.(~,) = p~,~ ,~, .(~, + 0,). Thus dif(0, i) = p~,0 dif(0 + 0, i) + (p~,o _ p,.o) (0 + 3). Since dif(0 + ~, i) -+ 0 in C O and the PI' 0 are bounded, the first term converges to zero in the operator norm. Now 0 + 0 is a bounded operator from L~ to L ~, and P~' 0 _ px, 0 converges to zero in C C, hence in the operator norm of Hom(L q, Lq). Therefore their composition, the second term, converges to zero in the operator norm of Hom(L~, L~). The same argument works for dif(0, i). This proves the proposition. [3 Let J denote the standard unitary metric on C". Let n) c R o,(x/s, 4, n) denote the subset consisting of triples (s, E, 6) where s e S, E is a Higgs bundle of harmonic type on X,, and [3 : E~,) ~ C ~ is a frame, such that there exists a harmonic metric K for E with ~(K~cs~ ) = J. Note that the harmonic metric K is uniquely deter- mined once it is fixed at one point ~(s) [Si2]--we call this K the chosen harmonic metric. Endow R~o~(X/S, ~, n) with the topology induced by the analytic topology of R Ax/s, n). Suppose s~ is a sequence of points approaching t in S. Choose a standardized sequence of diffeomorphisms ~:X~ ~-X~ such that ~,(~(si)) = ~(t). Corollary 7.t0. -- Suppose (E~, ~) are points in R~ot(X,~, ~(s~), n) which remain inside the inverse image of a compact subset 0fM~ol(X/S, 4, n). Then after going to a subsequence, there is a point (E, 6) in RDaol(X~, ~(t), n) and a sequence of bundle isomorphisms ~ : W~, .(E~) ~ E Suck that: the ~ preserve the chosen harmon# metrics; the operators ~i,.(-O~) and ~,,.(9~) converge to-O and ~ in the operator norm for operators from L~ to Lq; and finally, the frames ~(~) converge to 6. Proof. ~ Since the points remain in the inverse image of a compact subset of M~o~(X/S , ~, n), the eigenforms of the Higgs fields q0 i for Ei are uniformly bounded (this is because of the existence of the map ~ sending (E,, %) to the characteristic polynomial of ~,--see the discussion above Lemma 6.10). From the previous proposition, we can go to a subsequence and obtain a Higgs bundle E with harmonic metric K and MODULI OF REPRESENTATIONS. II 35 a sequence of bundle isomorphisms ~ with the desired convergence properties. Since the unitary group is compact, we may, by going to a further subsequence, assume that the frames ~(~i) converge to a unitary frame ~. Then (E, ~) is a point in n). [] Corollary 7.11. -- The subset R~0~(X/S , ~, n) C R ~"gm.=~ t'q 4, n) is closed. Proof. -- Suppose (s,, E~, ~) is a sequence of points in Rgol(X/S, ~, n), converging to a point (t, E', ~) in R~o~(X,, ~(s~), n). The images in M~o~(X~, ~(s~), n) converge, so they lie in a compact set. Apply the previous corollary to obtain a point (t, E, ~) in R~o~(X/S, 4, n) and a sequence of bundle isomorphisms ~. By Theorem 5.12, Part I, for the case of A ~~ the points (s~, E~, ~) converge to (t, E, ~3). Since R~(X/S, ~, n) is separated, (t, E', ~') = (t, 2, ~). Thus the limit is in ll~(X/S, 4, n). [] Corollary 7. lB. ~ The subset R~ot(X/S, ~, n) is proper over M~oa(X/S , n). Proof. -- Suppose (s~, E,, ~) is a sequence of points in R~ol(XIS, 4, n) lying over a compact subset of M~o1(X]S , ~, n). First, we may choose a subsequence so that the points s~ converge to a point t. Then we can apply Corollary 7.10 and Theorem 5.12, Part I, for the case of A D~ to obtaixt a subsequence which has a limit (t, E, ~3) in RL,(x/s, 4, n). [] We do the same thing for the de Rham spaces. Let n) c R.dX/S, n) denote the subset consisting of triples (s, E, ~) where s e S, 2 is a semisimple vector bundle with integrable connection on X,, and ~ : EL(,) ~ C" is a frame, such that there exists a harmonic metric K for E with ~(K~(~)) = J. The harmonic metric K is uniquely determined once it is fixed at the point ~(s) [Co], and we again call K the chosen harmonic metric. Endow R~R(X/S , ~, n) with the topology induced by the analytic topology of 4, n). Suppose s~ is a sequence of points approaching t in S. Choose a standardized sequence of diffeomorphisms tF, : X,i ~ X t such that tF,(~(s~)) = ~(t). Lemma 7.13. -- Suppose (E~, ~) are points in RJgR(X,;, ~(s,), n) which remain inside the inverse image of a compact subset of M~R(X/S , 4, n). Then after going to a subsequence, there is a point (2, ~) E RJvx(Xt, ~(s,), n), and a sequence of bundle isomorphisms ~q, " ~F,..(2~) ~ 2 such that: the ~ preserve the chosen harmonic metrics; the operators ~q~. .('O~) and ~,..(V~) converge to -0 and V in the operator norm for operators from L~ to Lq; and the frames ~q~(~) converge to ~. Proof. ~ This is die same as for Corollary 7. I0, except that we have to show that the characteristic polynomials of the Higgs fields ~0~ of the harmonic bundles corres- ponding to E~, are bounded. We follow the argument of ([Si4], Lemmas 3 and 5). 36 CARLOS T. SIMPSON Let F = 7rx(Xt, ~(t)), which is also equal to r.l(X,i , ~(s,)) via the diffeomor- phisms tF~. The condition that the points lie over a compact subset of M])~(X/S, 4, n) implies that the monodromy representations of tFi.o(E,, V~) lie over a compact subset of M(F, n) (by Theorem 7.1). The first thing we note is that it is possible to choose frames ~3~ for E, such that the monodromy representations of (E~, [3~) lie in a compact subset of R(F, n). The argument (from [Si4], Lemma 3) is that the subset of zeros of the moment map in R(F, n) is proper over M(P, n) [Ki] [KN] [GS]; our monodromy representations come from harmonic bundles, so they are semisimple--lying in the closed orbits--thus by appropriate choice of frames we can assume they correspond to points in the set of zeros of the moment map. Let p~ denote the monodromy reprcscntations corresponding to (E,, ~). Since they are bounded, it is possible to choose initial p~-equivariant maps from the universal covers X, to Gl(n, C)/U(n), which have uniformly bounded energy (note that the diffeomorphisms ~ are uniformly bounded in any norm). See [Si4], Lemma 5, for a description of how to do this (the process described there works the same way for any rank). Finally, the harmonic equivariant map has lower energy, and the energy is equal to the L z norm of q~,. Thus [1 ~o, Ilr~2,x,i , are uniformly bounded. This implies that the eigenforms of q~i are uniformly bounded in L ~ norm. The eigenforms of % are multi- valued holomorphic sections of f~ which do not depend on our choices of frame ~. Xs i The maximum norm of an eigenvalue of a holomorphic matrix is a subharmonic function, so the eigenforms of % are uniformly bounded in C ~ Thus the characteristic polynomials of the Higgs fields q~ are uniformly bounded in C ~ The rest of the proof is the same as that of Corollary 7.10. [] Corollary 7.14. -- The subset liaR(X/S, 4, n) C R~(X/S, 4, n) is closed. Proof. --- The same as for Corollary 7.11, but using Theorem 5.12, Part I, for the case of A DR. [] Corollary 7.15. -- The subset RaDR(X/S, 4, n) is proper over M~(X/S, n). Proof. -- The same as for Corollary 7.12, but using Theorem 5.12, Part I, for the case of A ua. [] There is an isomorphism of sets Rvol(X/S, 4, n)---RDR(X/S, ~, n). Over each fiber X,, this comes from the equivalence between the category of semistable Higgs bundles with vanishing Chern classes, and the category of flat bundles, constructed in [Si5]. This equivalence of categories is compatible with pullback to a point ~(s) ~ Xo, so it gives an isomorphism between the sets of isomorphism classes of framed objects in the two categories. In other words we get an isomorphism between the set of points in RDol(X,, ~(s), n) and RDR(X,, ~(S), n). Putting these together for all s we obtain the isomorphism of sets stated above. MODUI,I OF REPRESENTATIONS. II 37 Lemma 7.16. -- This isomorphism of sets induces a homeomorphism between the subsets R~(X/S, ~, n) and R~dX/S, ~, n). Proof. -- We prove continuity of the map from the Dolbeault space to the de Rham space. Suppose s~ is a sequence of points approaching t in S. Choose a standardized sequence of diffeomorphisms ~,:Xs/~ X~ such that ~,(~(s~))--~(t). Suppose (si, E~, [~) are points in RDol(Xsi,~(si),n), a converging to a point (t,E, [5') in , , ~l~--~, ~(si), n). Choose any R~ol(X t ~(s,) n). The points lie over a compact set in M s" t3~ subsequence. Apply Proposition 7.9 to obtain a harmonic bundle V over X, and (after going to a further subsequence) a sequence of bundle isomorphisms ~ : ~..(Et) ~- V, such that the transported structures of harmonic bundle on E, converge to the structure of harmonic bundle on V. Then the convergence statements of Lemma 7.13 hold for the operators d" and V giving the structures of vector bundle with integrable connection: the ~..(d~') and ~,.(V~) converge to d" and V in the operator norm for operators from L~ to L ~, and the frames ~i(~) converge to a frame ~. By Theorem 5.12, Part I, for the case A Ira, the points (s,, (E,, d[', V~), ~,) converge in R~t(X/S, ~, n) to the point (t, (V, d", V), ~). Similarly, (s~, (El, 0,, q~), ~) converge to (t, (V, 0, q~), ~) in RaDo,(X/S, ~, n). But this implies that (t, (V, 0, q~), ~) -- (t, E', ~'), so (t, (V, d", V), fS) is the point in Ravrt(X/S, ~, n) corresponding to (t, E', ~'). We have shown that every subsequence has a further subsequence where the corresponding points converge to the correct limit. This proves that the sequence of points in R~a(X/S, ~, n) corresponding to the original sequence of points (s,, E,, ~,) converges to the point corresponding to (t, E', ~). Thus the map from the Dolbeault space to the de Rham space is continuous. The proof of continuity of the map from the de Rham space to the Dolbeault space is exactly the same. n Note that the unitary group U(n) = Aut(C", J) acts on the representation spaces, and preserves the subsets R~o,(X/S, ~, n) and R~It(X/S, ~, n). This action is continuous in the analytic topology. Lemma 7.17. -- The moduli spaces M ~~ vot~,,l~,, ~, n) amt M~(X/S, ~, n) are the topological quotients of the representation spaces R~o,(X/S, ~, n) and R~(X/S, ~, n) by the action of U(n). Proof. -- Let N = R~o,(X/S, ~, n)/U(n) denote the topological quotient. Since U(n) is compact, N is separated (Hausdorff). Two points in R~,(X/S, ~, n) map to the same point in M~o,(X/S, ~, n) if and only if the underlying harmonic bundles are isomorphic, thus if and only if the points are related by a unitary change of frame. Thus the map f: N ~ Mvo,(X/S , 4, n) is one-to-one. The map from the representation space to the moduli space is continuous and proper, and since N is the topological quotient, this implies that the map f is continuous and 38 CARLOS T. SIMPSON proper. Therfore f is a homeomorphism identifying the moduli space with the quotient. The proof for the de Rham spaces is the same. [] We state our next corollary as a theorem. Theorem 7.18. -- The isomorphism of sets induced by the equivalence of categories given in [Si5] is a homeomorphism top MDo~(X/S , n) = to, ~ MD~(X/S , n) of the topological spaces underlying the usual analytic spaces. Proof. -- It is easy to reduce to the case where X --> S has connected fibers and admits a section. Then we may refer to the previous discussion. The modull spaces are identified, in the previous lemma, as topological quotients of R~(X/S, ~, n) and R~(X/S, ~, n). But Lemma 7.16 says that the identification between the representation spaces given by the equivalence of categories of [Si5] is a homeomorphism. This gives a homeomorphism between the quotients. [] Remark. -- Combining this with Proposition 7.8, we obtain a homeomorphism M~I(X/S , n) ~ M~~ n) where the right hand side denotes the topological space underlying M~n~(X/S, n). Corollary 7.19. -- If X is a smooth projective varie~, then any representation of the funda- mental group of X can be deformed to a representation which comes from a complex variation of Hodge structure. Proof. -- By Corollary 6.12, any point in M,o,(X , n) can be deformed to a fixed point of the action of C*. These fixed points correspond to representations which come from complex variations of Hodge structure [Si5]. By the continuity result of the theorem, any connected component of M~(X, n) contains a point parametrizing a complex variation of Hodge structure. But the inverse image of a connected component in Ms(X , n), is connected in RB(X , n), since M~(X, n) is a universal categorical quotient of RB(X, n) by a connected group. A point in the closed orbit over a fixed point of C* comes from a complex variation of Hodge structure. Thus in any connected component of the space of representations, there is a representation which comes from a complex variation of Hodge structure. [] Remark. -- This has topological consequences that were explained in [Si5]. The corresponding result is also true for principal bundles (cf. w 9 below). Counterexample We show that the isomorphism of sets R,g(X, x, n) ~ Rvot(X, x, n) given by the equivalence of categories constructed in ([Si5] Lemma 3.11) is not, in general, continuous. MODULI OF REPRESENTATIONS. II 39 In fact, the isomorphism on the open subset of stable points (which is continuous), has no continuous extension over the whole representation space. Let X be an elliptic curve, with nonvanishing differential dz. Let E = 0 x | 0 x be the trivial bundle of rank 2, with the canonical identification ~ : E, - C ~. Let 0,__ (0 ~ at dz Suppose t is real and approaches 0. Then the point (E, 0t, ~) approaches the point (E, 0o, ~). This limit is independent of the choice of a. However, we will see that the associated representations approach a limit that depends on a. Let gt = 9 Then , (; 0) Ot = gi- 10t g~ = at d Now the metric for (E, 0~) is the usual constant metric, and the associated flat connection is given by the matrix 0 ~ d~)" (;atdz+ Thus the flat connection associated to (E, 0~) is given by the conjugate of this matrix by gt: at ds dz + a~ gt at dz + ~ d~. g[ 1 = (: ~ (i at dz + ~ d~. ~ d~ (note that the entries ofg t are constant so there is no need to differentiate in conjugating the connection). Since we assumed that t was real, this connection matrix approaches (0 ~ as t -+ 0. The limit depends on arg(a). Thus the map between the space of representations and the space of (E, ~) cannot be continuous. It might still be the case that there is a homeomorphism between the topological quotient spaces RDR(X/S, too ~, n)[Gl(n) and top RDoI(X/S , ~, n)/Gl(n), which are non- Hausdorff spaces. Philosophically it would be important because of the interpretation of the topological quotient spaces as non abelian first cohomology spaces. This is an interesting problem for further study 9 40 CARLOS T. SIMPSON 8. The Gauss-Manln connection Suppose f: X ~ S is a smooth projective morphism. We have constructed the relative de Rham moduli spaces MDR(X/S , n). On the other hand, the relative Betd space is in fact a local system of schemes M~(X/S, n). The associated analytic total space M~(X/S, n) has a connection, namely a compatible system of trivializations over artinian subspaces of S ~''. The isomorphism of Theorem 7.1 gives a connection on M~R(X/S , n). We will show that this comes from an algebraic connection on MDI~(X/S , n). We will call this connection the Gauss-Manin connection because it is the analogue for nonabelian cohomology of the usual Gauss-Manin connection on the relative abelian de Rham cohomology. For constructing the algebraic connection, we follow the ideas of Grothendieck's construction for the case of abelian cohomology. Crystalline interpretation of integrable connections The first step is to give an interpretation of vector bundles with integrable connection on X/S as crystals. The advantage of this is that if S' is an S-scheme which contains a closed subscheme S o defined by a nilpotent ideal, and we set X~-----X' X s, So, then a crystal on X'/S' is canonically the same thing as a crystal on X'0/S'. The set of crystals on X'0/S' depends only on the restricted map S' 0 -~ S, so the functor M~R(X/S, n) is itself a crystal on S. The resulting stratifications for MD~(X/S , n) and RDR(X/S, ~, n) provide the Gauss-Manin connections on these schemes over S. This argument shows that the notion of a crystal can be useful in characteristic zero too. We will begin with an intermediate interpretation of vector bundles with connection on a smooth X/S, then proceed to describe what is meant by a crystal (in the present simple case). The contents of this discussion are based on the ideas of Grothendieck [Gr3], by now well known. We present them here for the convenience of the reader, since most of the literature on crystals has concentrated on characteristic p. Our terminology may not be completely standard. Suppose as usual, that X/S is smooth and projective. Denote by (X � X) ^ and (X � X � X) ^ the formal neighborhoods of the diagonals in products of X. We have projections denotedp~ or p~j in an obvious manner. Lemma 8.1. -- Suppose E is a vector bundle on X. Then an integrable connection V is the same thing as an isomorphism ~ : p'~ E-~ p~ E on (X � s X) ^, such that the restriction of to the diagonal is the identity, and on(Xx.~X xsX) ^. MODULI OF REPRESENTATIONS. II 41 Proof.-- Given such an identification 9, we obtain a connection V as follows. Let J denote the ideal of the diagonal in X � s X. If e is a section of E, then set V(e) = p*2(e) -- ~(p~(e)) (modJ2). It is an element ofp~ E | (j/j2), and considered as a module on the diagonal X, j/jz is (by definition) equal to the module of relative differentials ~lis. Note that p~ E/J = E on the diagonal, so V(e) is an element of E | ~x/s. From the discussion below, it will be clear that V is an integrable connection. We would like to see that this construction gives a correspondence between q~ and V. This statement does not depend on the fact that X is projective. We can cover X by open sets V which are finite ~tale covers of open sets U in affine space A~, and it suffices (by considering the direct image from V to U, and the 0v-module structure over U) to verify the lemma for vector bundles on U. We may further assume that E is a trivial bundle, E ~ d~. The isomorphism q~ is then given by a function g(x,y) with values in Gl(n), defined for x,y e U with y infinitesimally close to x (more precisely it is defined on the formal scheme (U � U)^). The conditions on g are that g(x, x) = I, and that g(y, z) g(x,y) = g(x, z). Given such a function g, we can write g(x,y) = 1 + A(x) (x --y) + O((x _y)2), where A(x) is an n � n matrix-valued one-form on U. Then V(e) (x --y) = e(y) --g(x,y) e(x) = e(y) -- e(x) -- A(x) e(x) (x -- y), in other words V = d -- A. This shows that V is a connection. The cocycle condition for g, taken when (y -- z) is a first order infinitesimal, becomes a differential equation: g(x,y) + a(y) (y -- z) g(x,y) = g(x, z) g(x, z) -- g(x,y) = A(y) g(x,y) (y -- z) or d~g(x,y) = Av(y)g(x,y). The subscripts indicate that the differentials are of the form dy. This equation uniquely determines g given the initial conditions g(x, x) = 1, so V determines 9 uniquely. To complete the proof we have to show that ? or g exists if and only if~7 is integrable. Note that if we can solve the equation d~ g(x,y) ---- A~(y)g(x,y) with initial conditions g(x, x) = 1, then the solution will satisfy the cocycle condition. This is because both g(y, z) g(x,y) and g(x, z) satisfy the same differential equation in z, and they are equal when z = y, so they are equal for all values of z. Change variables by setting t =y -- x. Set A(x + t) = ~ A~(x, t) dt,. This is a formal power series in t with coefficients which are regular functions of x ~ U. The differential equation (really a system because there are several tl, ..., tin) becomes Og(x, t) -- = A~(x, t) g(x, t). 6 42 CARLOS T. SIMPSON This is an ordinary differential equation for g(x, t) which is a formal power series in t with coefficients which are regular functions in x E U. The initial conditions are g(x, O) = 1. It has a solution if and only if it satisfies the integrability condition 0A~ 0A~ --+A~Aj-- +A iA~. Ot~ ~t~ The solution may be constructed inductively to higher and higher order in t. This integrability condition is equivalent to (d -- A) ~' = 0, so the function g exists if and only if V is integrable. This completes the proof of the lemma. [] Crystals of schemes Suppose that X is a scheme of finite type over S, not necessarily smooth. Define a category Inf(X/S) as follows. Its objects are pairs (U C V) consisting of an X-scheme U --+ X and an S-scheme V, with an inclusion U ~ V over S, which makes U into a closed subscheme defined by a nilpotent sheaf of ideals (by this we mean a sheaf of ideals I such that I k = 0 for some k). Such a nilpotent inclusion is sometimes referred to as an infinitesimal thickening. A morphism f: (UC V) ~ (U'C V') consists of a morphism f: V -+ V' of S-schemes, such that the restriction f: U ~ U' is a morphism of X-schemes. Let Inf'(X/S) denote the full subcategory of Inf(X/S) consisting of objects (U C V) such that there exists a morphism V -+ X compatible with the map from U. This morphism is not, however, considered part of the data of (U C V). Remark. -- If X/S is smooth, then any object of Inf(X/S) is, locally in the Zariski topology, isomorphic to an object of Inf'(X/S). This is because the infinitesimal tiffing property for smooth morphisms guarantees the local existence of V ~ X. A crystal of schemes F on X/S is a specification, for each (U C V) in Inf(X/S), of a V-scheme F(UCV) ~V; and for each morphism f:(UCV)-+(U'CV'), an isomorphism +(f) : r(v c V) c V')); such that d~(gf) =f'(+(g)) +(f). A crystal of vector bundles, or just crystal for short, is a crystal of schemes F with structures of vector bundles for F(U C V), such that the +(f) are bundle maps. Equivalently, it is a specification of locally free sheaves F(U C V) on V, with isomorphisms of locally free sheaves ~(f) 9 F(U C V) -~f* F(U' C V'). A stratification of schemes F on X/S is the same sort of thing as a crystal of schemes, but with F(UC V) defined only for (UC V) in the restricted category Inf'(X/S). Similarly for a stratification of vector bundles. According to the above remark, if X/S is MODULI OF REPRESENTATIONS. II smooth then stratifications are the same as crystals. In general, a crystal gives a strati- fication but not necessarily vice versa. We will also use the following terminology. If F ~ X is a morphism of schemes, a relative integrable connection for F on X over S is a stratification of schemes with F as the value over X. The corresponding notion for vector bundles is the same as the usual notion of vector bundle with relative integrable connection. This follows from Lemmas 8.1 above and 8.2 below. Suppose F and G are crystals or stratifications of schemes on X/S. A morphism u : F ~ G consists of a specification of morphisms of schemes u : F(U C V) -+ G(U C V), compatible with morphismsfin Inf(X/S) in the sense that +(f) u = u~b(f). A morphism of crystals or stratifications of vector bundles is the same, with the condition that the u should be morphisms of vector bundles, in other words linear. Lemma 8.2. -- A stratification of schemes on X/S is the same thing as a scheme F(X) ~ X, together with an isomorphism x , satisfying a cocycle condition. This condition says that the two resulting isomorphisms P~3(~) P~(~?) and P~s(~) between the restrictions of F(X) � s X � s X and X � s X � s F(X), are equal. A stratification of vector bundles on X/S is the same as above but where F(X) has a structure of vector bundle over X and e? is an isomorphism of vector bundles. Proof. -- Suppose F is a stratification of schemes on X. This gives a scheme F(X) over X. Let (X x s X) c") denote the n-th infinitesimal neighborhood of the diagonal in X � s X, and similarly in triple products. These are objects in the category Inf'(X/S). The maps PI.~,I and P2.~,~ from (X � s X) ~"~ to X give, by definition, isomorphisms F((X X)'"') p;,,.,(F(X)) and F((X � X)'"') ~ p~,,,,(F(X)). Composing these, we get isomorphisms 9 . ~ s, ~,. p,,,,,(r(x)) = p2,,.,(r(x)). Since the pullback isomorphisms defining the stratification F are functorial and satisfy an associativity, we have r xaX~ ~") ~ ~n for n ~< m, so these isomorphisms fit together into an isomorphism cp between the two pullbacks to the formal scheme (X � s X) A. This provides the desired ~. The associativity rule for the pullback maps implies that on (X � s X � s X) ^ the two isomorphisms p; F(X) x x) ^) and p; F(X) ~ pl~ V((X xs X) ^) ~ F((X � X x 8 X) ^) 44 CARLOS T. SIMPSON are equal. Similarly in other combinations. Thus, all of the resulting isomorphisms between p~ F(X) and p; F(X), are equal. This provides the cocycle condition. Suppose given, on the other hand, an isomorphism ~ satisfying a cocycle condition as described in the hypotheses. For every object U C V in the category Inf'(X/S), choose a map i v : V -~ X compatible with the map U ~ X. Define F(UC V) = iV(F(X)). Suppose f: V -+V'. Then iv, fl U is equal to i v ]u, although they may not be equal on V. Since U C V is defined by a nilpotent ideal, the pair (iv, iv, f) maps V into (X xs X) ^. Note that (iv, iv, f)'p'~ F(X) = F(V). while (iv, i v, f)* p~ F(X) =f* F(V'). Our hypothesis gives q~ :p~ F(X) ~ p~ F(X). Thus we may define q~(f) = (iv, iv, f)" (~), to obtain ~p(f) : F(V) ~-f* F(V'). Given f: V ~ V' and g : V' -~ V", we obtain a map (iv, iv, f, iv,,gf) :V -+ (X � X � X) ^. The cocycle condition for ~ implies that the two possible maps i v F(X) -~ (g f)" iv,, F(X) are equal. In other words, f'(~(g)) ~(f) = ~(gf). This shows that we have defined a stratification of schemes. These two constructions are essential inverses, so we get an equivalence of categories. [] Corollary 8.8. -- Suppose X/S is smooth. A vector bundle with integrable connection on X/S is the same thing as a crystal of vector bundles on X/S. Proof. -This follows immediately from the previous two lemmas, and the contention that crystals and stratifications are the same if X/S is smooth. This contention follows from the remark several paragraphs ago, that any object of Inf(X/S) is locally in Infr(X/S). In order to define F(U C V) for (U C V) e lnf(X/S), cover V by Zariski open sets V, which are in Inf'(X/S). Then use the isomorphisms which are provided on overlaps V~, to glue together the objects F(V~), forming F(V). [] Remark. -- Suppose Z/S is another S-scheme, and j:Z-~ X is a morphism of S-schemes. We obtain a functor j:lnf(Z/S) ~ Inf(X/S) in an obvious way. In fact, Inf(Z/S) is a subcategory of Inf(X/S). If F is a crystal of schemes or vector bundles on X, then the restriction is a crystal of schemes or vector bundles j" F on Z/S. The equivalences of categories given by the preceding lemmas and corollary are compatible with pullbacks. The following proposition was Grothendieck's main observation. MODUL1 OF REPRESENTATIONS. II Proposition 8.4. -- Suppose S o C S is a closed subscheme defined by a nilpotent sheaf of ideals. Suppose X is an S-scheme. Let X 0 = X � s So, still considered as an S-scheme. Let j : X 0 ~ X denote the inclusion. Then the pullback functor F ~j" F is an equivalence from the category of crystals of schemes on X/S to the category of crystals of schemes on X0[S. The same is true for crystals of vector bundles. Proof. -- We have a functor a : Inf(X0/S ) ~ Inf(X/S) defined by a(u c v) = (u c v), and a functor b : Inf(X/S) + Inf(X0[S ) defined by b(U C V) -=- (U o C V). The compo- sition ba is equal to the identity. On the other hand, if (U C V) ~ Inf(X/S) then there is a natural map (UoC V) ~ (UC V), so we get a natural morphism ab--+I. The functors a and b (and this natural morphism) preserw; the schemes V. We obtain functors a', from the category of crystals of schemes on X/S to the category of crystals of schemes on X0]S , and b', from the category of crystals of schemes on X0]S to the category of crystals of schemes on X/S. We have a" b'----I, and there is a natural morphism from b" a" to the identity. Note that (b' a" F)(UC V)= F(UoC V). The natural morphism is given by the pullback (using j: (UoC V) ~ (UC V)), ~?(j) : (F(UoC V) -7 F(U C V). But ~(j) is an isomorphism of schemes. A morphism of crystals of schemes which is an isomorphism over each element of Inf(X/S), is an isomorphism of crystals of schemes--the inverse will also be a morphism. Hence b" a" F ~ F. Thus a and b give an equivalence of categories. [] Representability One can define, in exactly the same way as before, the notions of crystal offunctors or stratification offunctors. These mean that for any object (U C V), F(U C V) is a functor of schemes Y -~ V. The set of such functors forms a pre-stack. In fact, given any stack or pre-stack ~ over the category of schemes, one can define a notion of crystal of r The above lemmas, done tor the stacks of schemes or vector bundles, remain valid. (Any comments about glueing are valid only for stacks, not pre-stacks.) Lemma 8.6. -- Suppose F ~ is a stratification of functors on X]S. Suppose F~(X) is repre- sented or universally co-represented by a scheme F(X). Then we obtain a stratification of schemes F, such that for any V E Inf'(X/S), F(V) represents or universally co-represents F~(V). Proof -- Use the characterization of Lemma 8.2. Note that p~ F(X) represents or universally co-represents the functor p] F~(X), and so forth. Hence the isomorphism of functors on (X � s X) ^ translates into an isomorphism between the pullback schemes. The cocycle condition for the isomorphisms of functors implies the cocycle condition for isomorphisms of schemes. [] Remark. -- It is in this lemma that we are forced to go from crystals to stratifications. 46 CARLOS T. SIMPSON The Gauss-Manln connections Suppose S is a scheme over C, and X/S is a smooth projective family. Suppose : S -+ X is a section. Define crystals of functors M~ and R~,~, on S/C as follows. For (S oC S') in Inf(S/C), define M~f~(S' 0C S') to be equal to the set of isomorphism classes of crystals of vector bundles of rank n, on X'0/S'. Define Rc~r,(S 0 C S') to be equal to the set of isomorphism classes of pairs (E, ~) where E is a crystal of vector bundles of rank n, on X0/S' , and Here 1 is the trivial crystal on X'o/S' ~ S0/S'. These crystals of functors restrict to strati- fications of functors. By Lemmas 8.1 and 8.2, we have M~r,s---M~oR(X/S,n), and R~r~ ~ R~R(X/S, 4, n). The first is universally co-represented by MD~(X/S, n), and the sccond is rcprescntcd by RDR(X/S, ~, n). By the previous lemma, we obtain stratifications of schemes M~t~t(X/S, n) and l~t,~t(X/S, ~, n) on the stratifying site Inf'(S/C). By Lemma 8.2, these data are equivalent to the data of isomorphisms q~ :p~ MDR(X/S, n) ~ p~ MDR(X/S, n) and q~ :p~ RDr~(X/S , ~, n) ~ p~ RDR(X/S, ~, n) on (S xeS)^, satisfying the cocycle condition on (S xcS xeS)^. These are the Gauss- Manin connections. We can make the same definitions as above for the category of complex analytic spaces. The algebraic connections induce analytic connections on R~R(X/S , 4, n) and M~(X/S, n). We would like to show that these agree with the connections coming from the Betti realizations. Recall that the Betti objects Rs(X/S, 4, n) and MB(X/S, n) are local systems of schemes over S ~. The associated spaces R~"'(X/S, ~, n) and M~]n'(X/S, n) are, by definition, products locally over S. In other words, if s e S then there exists a neighbor- hood U of s such that (with the subscript U denoting the inverse image of U) S'~~ 4, n)~ = U x rt~~ ~(s), ,,) and M~"~(X/S, n)v = U � M]3"(X ,, n). A product space of the form U � Z has an analytic relative integrable connection, given by the natural equalities of objects over U � U p;(U � xz)=u � � Thus R~~ ~, n)u and M'8"'(X/S, n)v have analytic relative integrable connections. The local product structures over open sets U and V agree over connected components of U n V, so the connections agree over U c~ V. These then glue together to give analytic relative integrable connections on R~'~'(X/S, ~, n) and M~"~(X]S, n). MODULI OF REPRESENTATIONS. II 47 Theorem 8.6. -- The isomorphisms R~'~'(X/S, ~, n) ~ R~a(X/S, ~, n), M~'(XIS, n) ~ M~R(X/S, n) identify the connections coming from the locally constant structure of the Betti objects, with the Gauss-Manin connections constructed above for the de Rham objects. Proof. -- It suffices to treat the case where S = Spec(A) with A an artinian local (]-algebra of finite type, and X/S is smooth, connected and has a section ~. Let s ~ S denote the closed point. The Gauss-Manin connections are equivalent to trivializations RDR(X/S, ~, n) _---- S x RDR(X0, ~(s), n) and MDR(X/S , n) _--__ S � MDR(X., n). In order to show that the associated trivializations of analytic spaces agree with the trivializations R~'(X/S, ~, n) = S x R~(X., ~(s), n) and M~='(X/S, n) = S x M~"(X,, n), it suffices to treat the cases of the representation spaces, since the maps R(X/S, ~, n) ---> M(X/S, n) are universally submersive. For the representation spaces, it suffices to show that if f: S' ~ RDR(X]S, ~, n) is a point with values in an artinian scheme S' = Spec(A'), which has constant projection on the second factor in the above product decomposition, then the resulting monodromy representation =l(Xs, ~(s)) ---> Gl(n, A') takes values in Gl(n, C). Tile point f corresponds to a vector bundle with integrable relative connection (E,V) on X'----X � and frame ~:E]~s,~ d)~,. The fact that the projection on the second factor of the product decomposition given by the stratification is trivial, implies that there is an open set U C X' (containing the image of ~) and trivializations -~ : U ~ S' � U s with ~(~(S')) - S' � ~(s) and (E, v, p)1~, ~ (p~ ~.)'((~:s, v, p,)]~,.). The local system of relatively constant sections of (P9 x)'((Es, V,)Iu,) is just the tensor product of the local system of constant sections of (E,, V,) 1u, with A'. Thus the mono- dromy representation of (P2 x)*((E,, V,, ~s)Iu,) takcs values ~(Us, ~(s)) -> Gl(n, C) C Gl(n, A'). Note that the map on fundamental groups is a surjecdon ~dus, r ~ ~(x,, ~(s)) ~ 1 The trivialization of (E, V, [~)Iu implies that the monodromy representation takes values in Gl(n, s [] 48 CARLOS T. SIMPSON Remark. -- The above proof gives the following criterion: an artinian scheme- valued point f." S' ~RDR(X/S, 4, n) has constant projection on the second factor RDR(X,, ~(s), n) if and only if there exists an open set I~C X' (containing the image of 4) and trivializations .r : U -~ S' � U 8 with "r(~(S')) = S' � ~(s) and For, if such an open set and trivializations exist, then the monodromy representation takes values ill Gl(n, C). Thus the pointf kas constant projection on the second factor for the stratification of the Betti spaces. But since the de Rham and Betti spaces are analytically isomorphic, and this isomorphism is compatible with the stratifications, the pointfhas constant projection on the second factor for the stratification of R~R(X/S , 4, n). Hence it has constant projection for the algebraic stratification. 9" Principal objects Suppose X is a scheme of finite type over C. In what follows, we will use the term tensor category to denote an associative commutative C-linear tensor category with unit object. A tensor functor is a functor together with natural isomorphisms of preservation of the tensor product, compatible with the associative and commutative structures [Sa] [DM]. Suppose G is a complex linear algebraic group. Let Rep(G) denote the tensor category of complex linear representations of G. Let Vect(X) denote the tensor category of vector bundles (considered as locally free sheaves) over X. A morphism u : E ~ F of objects in Vect(X) is strict if coker(u) is a locally free sheaf. In this case, the kernel and image of u are locally free sheaves. A principal right G-bundle over X is a morphism P ~ X together with a right action of G on P such that there exists a surjective dtale morphismf: X' ~ X and a G-equivariant isomorphism P � X' � G. If P is a principal right G-bundle over X, let P � V be the locally free sheaf in the Zariski topology obtained by descending the sheaf {(p, v) eP(Y) � (V| } y eXet ~ (pg, v) ,-~ (p, gv) for g ~ G(Y) from the ~tale topology X et to the Zariski topology. We obtain a functor Pl, : Rep(G) ~ Vect(X) by setting pp(V) = P � V. This has the following properties: that 0P is strict, in other words ifu : V ~ W is a morphism in Rep(G) then Op(u) is a strict morphism in Vect(X) ; MODULI OF REPRESENTATIONS. II 49 that 0i, is exact, that is pp(ker(u)) = ker(pp(u)) and pp(coker(u)) = coker(pe(u)) ; and that pp is faithful. Furthermore, for any closed point x e X the functor V ~ 9e(V)~ is a fiber functor [Sa] [DM]. Nori has proved the following converse: Proposition 9.1. -- Suppose p : Rep(G) ~ Vect(X) is a strict exact and faithful tensor functor. Then there exists a principal right G-bundle P over X and an isomorphism of tensor functors p ~- 9P; and P is unique up to unique isomorphism. Proof. -- [No]. [] Principal Higgs bundles Suppose X -+ S is a smooth projective morphism to a scheme of finite type over C. Let g denote the Lie algebra of G with G acting by the adjoint representation. A principal Itiggs bundle on X over S, for the group G, is a principal right G-bundle P ~ X together with a section 0 of (P X e' g) | f2~x/s such that [0, 0] = 0 in (P � g) | f2x/s 9 This is the relative version of one of the definitions given in [Si5]. Given such an object and a representation V of G, we get a relative Higgs bundle p~(V) = P � V. Say that P is of semiharmonic type if the Chern classes of the restrictions of P to fibers X, are zero in rational cohomology, and if there exists a faithfifl representation V such that pp(V) restricts to semistable Higgs bundles on the fibers. In this case, the same is true for any other representation (cf. [Si5], remarks after Lemma 6.13). The category of semistable Higgs bundles with vanishing Chern classes (Higgs bundles of semiharmonic type) has a natural structure of tensor category--the tensor product of two semistable Higgs bundles is again semistable [Si5]. Lemma 9.8. -- The construction P ~ pp provides an equivalence between the categories of principal Higgs bundles of semiharmonic type for the group G, and strict exact faithful tensor functors p from Rep(G) to the category of Higgs bundles of semiharmonic type on X over S. Proof. -- This follows from the previous proposition--see [Si5], remarks after Lemma 6.13. [] Lemma 9.3. ~ Suppose E is a Higgs bundle of semiharmonic type on X over S. Fix a number k. There is a projective S-scheme N(E, k) ~ S representing the functor which associates to each S-scheme f: S' ~ S the set of quotient Higgs bundles f*(E) -+ F ~ 0 of rank k such that the Chern classes of F vanish on fibers of X' ~ S' (note that any such F is a semistable Higgs bundle on X' over S', hence of semiharmonic type). Suppose that the fibers X~ are connected, and : S --> X is a section. Then the morphism N(E, k) -+ Grasss(~*(E), k) is a closed embedding. Proof. -- Let P0 denote the Hilbert polynomial of d~ x over S. Let Hilb(E, kPo ) denote the Hilbert scheme parametrizing quotient sheaves E-+ F ~ 0 fiat over S, with Hilbert polynomial kpo. Denote the kernel by O ~ K ~E ~ F ~0. Let 50 CARLOS T. SIMPSON N(E, k)C Hilb~ kpo ) denote the closed subscheme representing the condition that the map 0 : K -+ F |162 f2~'s is zero (see the first paragraph of the proof of Theorem 3.8, Part I). The points of N(E, k) with values in f: S' -+ S correspond to quotient Higgs sheavesf'(E) -+ F -+ 0 on X' over S', such that F is fiat over S' with Hilbert polynomial P. If F is such a quotient, then for any s E S' the fiber F s = F Ix; is a quotient Higgs sheaf of E 8 = E Ixs with normalizcd Hilbert polynomial equal to that of E s. Let K8 denote the kernel of E 8 -+ F,. Then K s is a sub-Higgs sheaf of E~ with the same normalized Hilbert polynomial. By Proposition 6.6, K~ is a strict subbundle with vanishing Chern classes, hence F s is locally free and has vanishing Chern classes. By Lemma 1.27, Part I, this implies that F is locally free. Thus the points of N(E, k) correspond to quotient Higgs bundles F which are locally free of rank k and have Chern classes restricting to zero on the fibers. This is the desired parametrizing space. Note that Hilb(E, kpo ) is projective over S and N(E, k) is a closed subset, hence it is also projective. Suppose X has connected fibers over S and ~:S -)-X is a section. Associating, for each quotient E -+ F ~ 0, the quotient vector bundle ~'(E) -+ ~'(F) --~ 0, gives a morphism N(E, k) ~ Grass(~*(E), k). It is proper, since N(E, k) is proper over S. Suppose F 1 and F, the quotients given by points of N(E, k) (S), such that ~*(F1) = ~*(F,) as quotients of ~*(E). Let K 1 denote the kernel of E --~ F t. Then K 1 and F 2 are Higgs bundles on X over S with the same normalized Hilbert polynomials. By Proposition 6.6, they both satisfy condition LF(X). The morphism +:K1 ~ F2 has ~*(+) = 0, so by Lemma 4.9, Part I, for the case ofA ~'gg~, + = 0. Thus F 2 is a quotient of FI; similarly in the other direction, F 1 is a quotient of F 2 so F1 = 1: 8. This shows that the map N(E, k) (S) -+ Grass(~'(E), k) (S) is injective. The same is true for points with values in any S-scheme S'. A morphism which is proper and injective on the level of points is a closed embedding. [] Suppose that the fibers X s are connected, and ~ : S -+ X is a section. Suppose G C H is a subgroup. Suppose P is a principal Higgs bundle for the group H which is semistable with vanishing Chern classes, on X over S. Suppose b:S-~ ~'(P) is an S-valued point. We say that the monodromy of (P, b) is contained in G if the following condition holds: for every linear representation V of H, and every subspace W C V preserved by G, there exists a strict sub-Higgs bundle of semiharmonic type FC P � such that ~'(F) ={b} � WC ~'(V � If S is a point, we define the monodromy group Mono(P, b) to be the intersection of all algebraic subgroups G C H such that the monodromy of (P, b) is contained in G. Note that the monodromy group jumps down under specialization. Lemma 9.4. -- Suppose O C H. Suppose P' is a principal Higgs bundle of semiharmonic type on X over S, for the group G. Then the principal Higgs bundle P = P' � c H obtained by MODULI OF REPRESENTATIONS. II extending the structure group to H is also of semiharmonic type. This construction gives an identi- fication between: (1) the set of isomorphism classes of pairs (P', b') where P' is a principal Higgs bundle of semiharmonic type for the group G and b' is an S-valued point of 4*(P'); and (2) the set of isomorphism classes of pairs (P, b) where P is a principal Higgs bundle of semiharmonic type for the group H and b is an S-valued point of 4*(P'), such that the monodromy of (P, b) is contained in G. Proof. -- The Chern classes of P are induced by those of P', hence they vanish. To check sernistability of P, choose a faithful representation V of H. This restricts to a faithful representation of G, and we have P x ~ V = P' x ~ V. By the assumption of semistability of P', this is semistable, so P is semistable. Our construction gives a functor from the category of objects (1) to the category of objects (2). To go in the opposite direction, let Rep(G, H) denote the category whose objects are pairs (V, W) where W is a representation of H and V is a G-invariant subspace; and whose morphisms are the G-equivariant morphisms between the subspaces V. Forgetting W gives an equivalence of categories Rep(G, H) -~ Rep(G). On the other hand, suppose we have a principal H-bundle P with a point b e 4*(P) (S), such that the monodromy is contained in G. By definition, for any (V, W) E Rep(G, H) there is a unique sub-Higgs bundle F(V, W) C P x ~ W of semiharmonic type with 4*(F(V, W)) = { b } X V. Given (V, W) and (V', W') and a G-equivariant morphism f: V-+V' we obtain a G-invariant subspace L C W @ W' giving the graph of the map f. The hypothesis of monodromy in G implies that there exists a sub-Higgs bundle L(f) C F(V, W) | F(V', W') which restricts to L on the section 4- This gives the graph of a morphism F(V, W) --+ F(V', W') restricting to f over the section 4 (and the morphism is unique by Lemma 4.9, Part I). We obtain a functor from Rep(G, H) to the category of Higgs bundles of semiharmonic type on X over S, commuting with the functor of taking the fiber along 4. Composing with the inverse of the above equivalence of categories gives a functor from Rep(G). This has a natural structure of neutral tensor functor (one can define a tensor operation (V1, W1) | (V~, W,) = V~ | V,, W~ @ W2) on Rep(G, H) as an intermediate in the definition of the tensor structure). By Lemma 9.2, this gives a principal G-bundle P' as desired. [] Lemma 9.5. -- Suppose E is a Higgs bundle of semiharmonic type, of rank n on X over S. Then the frame bundle P of E has a natural structure of principal Higgs bundle of semiharmonic type for the group Gl(n, C) on X over S. The Higgs bundle is recovered as E = P x alC"'c~ C". This construction provides an identification between the sets of isomorphism classes of (E, ~) and (P, b). Proof. -- Define a category Rep(Gl(n, C), std) whose objects are pairs (V, T a' ~(C")) where the second element refers to the tensor product (C")|174 | and V C T "' b(C") is a Gl(n, C)-invariant subspace. The morphisms are equivariant morphisms of the subspaces V. This category is equivalent to Rep(Gl(n, C)) (and it even has a tensor operation compatible with the tensor product on Rep(Gl(n, C))). Suppose 52 CARLOS T. SIMPSON V CTa, b(C ") is a Gl(n, C)-invariant subspace. Then for any n-dimensional vector space U we obtain a subspace V s T"'b(U) which does not depend on the choice of basis. The same construction holds for vector bundles, so we get a subbundle F C T ~' b(E) with ~(F)= V. The construction of V is also compatible with infinitesimal auto- morphisms, so the subbundle F is preserved by 0. There is a complementary subspace V  and a corresponding complementary subbundle F l. The tensor product T~ is also of semiharmonic type [Si5], so any direct factor such as F is of semi_harmonic type. Morphisms of representations V give rise to morphisms of the Higgs bundles F, and it is compatible with tensor product, so we obtain a tensor functor from Rep(Gl(n, C), std) to the category of Higgs bundles of harmonic type on X. Lemma 9.2 gives the desired principal bundle P. [] Suppose G C Gl(n, C). Suppose E is a Higgs bundle of semiharmonic type on X over S, of rank n, and suppose ~: ~'(E) ~ ~. Let P denote the frame bundle of E, and b the point corresponding to ~. We say that the monodromy of (E, ~) is contained in G if the monodromy of (P, b) is contained in G in the sense defined above. If S is a point, the monodromy group Mono(E, ~) is again the intersection of all subgroups G C Gl(n, C) such that the monodromy of (E, ~) is contained in G. Theorem 9.6. -- Suppose ~:S ~ X is a section. There is a scheme Rr~I(X/S, ~, G) over S representing the functor which associates to any S-scheme S' the set of pairs (P, b) where P is aprincipal Higgs bundle for the group G on X' = X x s S' over S', semistable with vanishing Ghern classes, and b : S' ~ ~'(P) is a section over ~. If f: G ~-~ H is a closed embedding, then f induces a closed embedding ]R~I(X/S , ~, G) ~ ]R~I(X/S, ~, H). Proof. ~ By Lemma 9.5, RI~,(X/S, ~, Gl(n, C)) d_a R~,(X/S, ~, n) does the job for the group Gl(n, C). Suppose now that the existence of RDoI(H ) = RDoI(X/S, ~, H) is known, and that G C H is an algebraic subgroup. Suppose that V is a representation of H and W is a subspace preserved by G. Let (P~', b my) denote the universal principal object on X x s RDo~(H), and let E'~l" = p,~sv � V denote the universal Higgs bundle asso- ciated to the representation V. Let Y"= V|162 ORDoI~r~ ~ and let ~i':~*(Em')- denote the frame given by the point b ~. Let k =dim(V)--dim(W), let YW = W | dYR~(~) denote the corresponding subobject of ~, and let o~-lu~ : R~I(H ) ~ GraSSRDolirn(~r k) denote the section corresponding to the quotient ,v'/~. Let N(E ~j*, k) C Grasssr, l~Hj(~ , k) denote the closed subscheme given by Lemma 9.3 and transported by the frame ~*. Define the closed subscheme C(V, W) d~ a~/~(y(E,~,, k)) C RDo,(H). MODULI OF REPRESENTATIONS. II By Lemma 9.3, this subscheme represents the condition on points g:S'-+Roo~(H), that there exists a quotient Higgs bundle F' of harmonic type of g*(E"~*), with g.(~t~) (~*(F')) = g*($f/W'). This is the same as the condition that there exists a strict sub-Higgs bundle F of harmonic type with g.(~l,) (4*(F')) = g*(W'). Set R.o,(X/S, 4, G) = fl C(V, W) (V, W) where the intersection is taken over all representations V of H and subspaces W preserved by G. It is a closed subscheme of RDo~(H ) which represents the functor associating to an S-scheme S' the set of (P, b) where P is a principal Higgs bundle of semiharmonic type for the group H on X' over S', and b is a point, such that the monodromy of (P, b) is contained in G. By Lemma 9.4, RDoI(X/S , 4, G) also represents the functor associating to S' the set of (P', b') where P' is a principal Higgs bundle of semiharmonic type for the group G and b' is a point. Every linear algebraic group G is a subgroup of Gl(n, C) for some n, so we obtain all of the required spaces Rvo~(X/S, 4, G). The last statement is immediate from this construction. [] Remark. -- The last statement of the theorem, applied to G CGI(n, C), gives RDoa~X;S, ~, G) C Rr~,(X]S, 4, n), because Rvo~(X/S, 4, Gl(n, C)) = R,o~(X/S, 4, n). Our next task is to study the universal categorical quotients of these representation spaces. Assume from now on that G is reductive. Note that G acts algebraically on R~(X/S, 4, G). IfG C Gl(n, C) is a faithful representation, then G acts on Rr~(X/S , 4, n) through its inclusion in Gl(n, C), and this induces the natural action on the subscheme RDoI(X/S, 4, G). Choose a Gl(n, C)-linearized line bundle .Lf on RDoI(X/S, 4, n), such that every point is semistable for the action of Gl(n, C) (cf. Theorem 4.10, Part I). By Mumford's criterion involving one parameter subgroups [Mu], every point is also semistable for the action of G. Thus every point of the closed subset RDoI(X/S, ~, G) is semistable for the action of G with respect to the linearized line bundle .2 ~ By [Mu], we may form the universal categorical quotient MDoI(X/S, 4, G) d~ Rml(X/S, 4, G)/G. Proposition 9.7. -- Suppose that X -+ S is smooth and projective. There exists a space MDo,(X/S, G) which universally co-represents the functor associating to S' ~ S the set of iso- morphism classes of principal tIiggs bundles P of harmonic type on X' over S' for the group G. If the fibers X~ are connected and 4 : S --~ X is a section, then there is a natural isomorphism between MDol(X/S , G) and the universal categorical quotient MDol(X/S , 4, G) constructed above. In this case, the points 0f MDo~(X ~, G) parametrize the closed G-orbits in RDo~(Xs, 4(s), G). Proof. -- Choose an dtale morphism S' -~ S with S' connected, such that each connected component X~ of X' admits a section 4~. Then MDo,(X'/S' , G) ---- I-IMDo,(X/[S', ~,, G) iRDol~X/U,~,O). 54 CARLOS T. SIMPSON universally co-represents the appropriate functor. If{ S'~ } is a collection of 6tale S-schemes covering S, then the collection of spaces M,o~(X'JS'~, G) constructed in this way is provided with descent data (since the functors they co-represent are provided with the corresponding descent data). They descend to give M,o~(X/S , G) which co-represents the desired functor. [] Theorem 9.8. -- Suppose G is a reductive group and f:G---~ Gl(n, C) is a faithful representation. Suppose (P, b) ~ RDo,(X,, ~(s), G) maps to (E, ~) ~ RDo1(X,, ~(s), n). Then (P, b) is in a closed G-orbit in RDo~(X~, ~(s), G) /f and only if (E, ~) in a closed Gl(n, C)-orbit in RDo~(X,, ~(s), n), or equivalently the monodromy group of E is reductive, or equivalently E is semisimple. Proof. -- The subobjects of (E, ~) correspond to the subspaces of C ~ preserved by the monodromy group of E. The monodromy group is reductive if and only if the representation C ~ is completely reducible, thus if and only if E is semisimple. The statement of the theorem is true if G = Gl(n, C): in Theorem 4.10, Part I, as applied in w 6, we have identified the closed Gl(n, (])-orbits in RDol(Xs, ~(s), n) as corresponding to the semisimple representations. Certainly if E is semisimple, then its G-orbit is closed, since the G orbit is a closed subset of the Gl(n, C)-orbit. Suppose E is not semisimple, so H = Mono(E, ~) is not reductive. Let U be the unipotent radical of H. There exists a one parameter subgroup C* -+ G and a family of morphisms ft : H --~ G for t ~ C such that f,(g) = tgt -1 for t ~ C*, and such that the imagef0(H) is not conjugate to H. To see this, we apply the theorem ofMorozov [Mo] --see also [BT]--to conclude that since H is not reductive, it is contained in a proper parabolic subgroup Q., and its radical U intersects the unipotent radical of Q. Now we may assume that O is defined by a torus C* --> G. The Levi component of Q. is the centralizer of this torus, and the torus acts with positive weights on the unipotent radical of Q. In particular, if q ~ Q. then the limit lim,_0 tq t-1 exists. These limits give the map f0, which completes the family ft(q) = tq t-1 defined for t ~e 0. The limits of the elements of the unipotent radical of Q. are the identity element, so there is u ~ H whose limit is the identity. In particular f0(H) has dimension smaller than that of H. This is the required family. The construction of the previous paragraph gives a morphism of group schemes over A 1, f: H � fik I ~ G � A 1. Let (P', b') be the principal Higgs bundle for the group H with P = P' � ~ G. The map f gives an associated relative principal Higgs bundle Pf = P' � on X x A 1 over A 1, with an Al-valued point b I. For t ~ Ak 1, Mono(P/, bf) =f,(H), and furthermore (Pf, b[) ~ (P, tb) for t + 0. Thus (P[, b{) are points in the G-orbit of (P, b), which approach the limit (P0 I, b0 f) as t ~ 0. This limit is not in the same orbit, since its monodromy group, f0(H), is not conjugate to H. Thus the G-orbit is not closed. [] MODULI OF REPRESENTATIONS. II 55 The de Rham spaces We can define spaces RDlt(X/S, ~, G) and MDR(X/S , G) in the same way as above, and obtain the same results. Suppose X + S is a smooth projective morphism to a scheme of finite type ovre C. Let g denote the Lie algebra of G with G acting by the adjoint representation. A principal bundle with integrable relative connection on X over S, for the group G, is a principal right G-bundle P ~ X together with an integrable connection V. For purposes of brevity, we can define an integrable connection as being a G-invariant structure of stratitication of schemes for P on X/S in the sense of w 8. Given (P, V) and a representation V of G, we get a vector bundle with integrable relative connection pp(V) = P � The construction P ~pp provides an equivalence between the categories of principal bundles with integrable relative connection, and strict exact faithful tensor functors p from Rep(G) to the category of vector bundles with relative integrable connection. Lemma 9.9. -- Suppose E is a vector bundle with relative integrable connection on X over S. Fix a number k. There is a projective scheme N(E, k) ~ S representing the functor which associates to each S-scheme f: S' -+ S the set of quotients f*(E) ~ F ~ 0 compatible with the connection. Suppose that the fibers X, are connected, and ~:S ~ X is a section. Then the morphism N(E, k) ~ Grasss(~*(E ), k) is a closed embedding. Proof. -- Let A ---- A D~ be the sheaf of rings of all relative differential operators on X over S. We may consider 1" as a A-module. Let P0 denote the Hilbert polynomial of 9 x over S. The Hilbert scheme Hilb(E, kPo ) parametrizes quotient sheaves E -+ F -+ 0 fiat over S with Hilbert polynomial kpo. Let E~lV~ Fun~v ~0 denote the universal quotient on Xun~v = X � kP0), and let K~'vC E un'v denote the kernel. We get a map A~ "iv | K unlv ~ F univ. Let N(E, k) be the closed subscheme representing the condition that this map pulls back to zero. Then N(E, k) parametrizes quotients E ~ F ~ 0 compatible with the action of A, such that F is flat with Hilbert polynomial kpo over the base. Any such quotient restricts to a sheaf with connection on each fiber, hence to a locally free sheaf. By Lemma 1.27, Part I, F is locally free, so it is a vector bundle with integrable relative connection. From the Hilbert polynomial, it has rank k. Thus N(E, k) represents the desired functor. Furthermore, N(E, k) is projective over S and the natural morphism to Grass(~*(E), k) is injective on the level of S'-valued points, by an application of Lemma 4.9, Part I. Hence the map to the Grassmanian is a closed embedding. [] Suppose that the fibers X, are connected, and ~ : S ~ X is a section. Suppose G C H is a subgroup. Suppose P is a principal bundle with relative integrable connection for the group H on X over S. Suppose b : S -~- ~'(P) is an S-valued point. We say that 56 CARLOS T. SIMPSON the monodromy of (P, b) is contained in G if the following condition holds: for every linear representation V of H, and every subspace W C V preserved by G, there exists a strict subbundle preserved by the connection FC P � such that ~*(F) ={b} x WC ~*(P � If S is a point, we define the monodromy group Mono(P, b) to be the intersection of all algebraic subgroups G C H such that the monodromy of (P, b) is contained in G. We obtain the same result as in Lemma 9.4. Note that the concept of " semi- harmonic type " is not needed, since all Am~-modules are automatically p-semistable with vanishing rational Chern classes. Suppose G C H. Suppose P' is a principal bundle with relative integrable connection on X over S, for the group G. Then P ---- P' � H is a principal bundle with relative integrable connection for the group H. This cons- truction gives an identification between: (1) the set of isomorphism classes of pairs (P', b') where P' is a principal bundle with relative integrable connection for the group G and b' is an S-valued point of 4*(P') ; and (2) the set of isomorphism classes of pairs (P, b) where P is a principal bundle with relative integrable connection for the group H and b is an S-valued point of 4*(P'), such that the monodromy of (P, b) is contained in G. Suppose E is a vector bundle of rank n with integrable relative connection on X over S. Then the frame bundle P of E has a natural structure of principal bundle with integrable relative connection for the group Gl(n, C), and E is recovered as P � o11,. c~ t3". This construction provides an identification between the sets of isomorphism classes of (E, ~) and (P, b). Theorem 9.10. -- Suppose 4 : S --~ X is a section. There is a scheme RIm(X/S, 4, G) over S representing the functor which associates to any S-scheme S' the set of pairs (P, b) where P is a principal bundle with relative integrable connection for the group G on X' -~ X � s S' over S', and b : S' -~ 4*(P) is a section over 4. If f: G ~ H is a closed embedding, then f induces a closed embedding R,R(X/S , 4, G) ~ RD~(X/S, 4, H). Proof -- The same as the proof of Theorem 9.6. [] The analogues of Proposition 9.7 and Theorem 9.8 also hold. Relationship with Bettl spaces Suppose X ~ S is smooth and projective, with connected fibers, and suppose 4 : S ~ X is a section. If G is any linear algebraic group, we obtain a local system of schemes R~(X/S, 4, G) on S ~. These are obtained from the fundamental group I" = nl(Xs, 4(s)) by setting R(P, G) = Horn(F, G) ; the fundamental group nl(S, s) MODULI OF REPRESENTATIONS. II 57 acts on P so it acts on R(F, G), and RB(X/S , {, G) is the corresponding local system of schemes. If G is reductive then define MB(X/S , G) to be the local system of schemes whose fibers are the good quotient R(F, G)/G (which exist because the representation space is affine). If X-+ S is any smooth and projective morphism, we obtain My(X/S, G) by the same dcscent as usual. Recall that the superscript " (~"~ " denotes the analytic total space associated to a local system of schemes. Theorem 9.11. -- We have isomorphisms of complex analytic spaces -- (X/S, G) and, if G is reductive, M~R(X/S, G) _-_ M~'~'(X/S, G). These are compatible with the morphisms of funetoriality induced by homomorphisms of algebraic ,Croups, and they are equal to those given by Theorem 7.1 in the case G -=-- Gl(n, C). The monodromy ,croup corresponding to a point in RDR(X,, ~(S), G) is equal to the Zariski closure in G of the image of the representation parametrized by the corresponding point in RB(X,, ~(s), G). Proof. -- Fix an injective homomorphism G C Gl(n, C). Over an analytic base, a relative vector bundle with integrable connection has monodromy contained in G if and only if the corresponding family of representations has image in the subgroup of points with values in G. Thus the subsets a. R ~" rX'~ " Gl(n, C)) RuR(X/S , ~, G) C DR, /~, ~,, and R~'(X/S, ~, G) C R~'(X/S, ~, Gl(n, C)) represent the same functor of analytic spaces S' -~ S a~. Hence they correspond under the isomorphism of Theorem 7.1. We obtain an isomorphism between the moduli spaces by applying Proposition 5.5, Part I. [] Corollary 9.12. -- If G C Gl(n, C) is a closed embedding then the Gauss-Manin connection preserves the subspace RDR(X/S, ~, G) C RDR(X/S, ~, n). If G is reductive, we obtain a Gauss- Manin connection on the universal categorical quotient 1VIDR(X/S, G). Proof. -- The Gauss-Manin connection on the associated analytic space is the same as the connection given by the local trivializations of the Betti spaces. These trivia- lizations are compatible with the subspaces of representations for the group G. Thus the analytic connection preserves the analytic subspace R~(X/S, ~, G). This implies that the algebraic connection preserves the subspace RD~(X/S, 4, G). The connection descends to the universal categorical quotient by Lemma 8 5. [] 8 58 CARLOS T. SIMPSON Principal harmonic bundles Let J denote the standard metric on C". Recall that we defined in w 7 the space R~oI(X/S , ~, n) whose points over s E S consist of pairs (E, ~) where E is a Higgs bundle on X, and ~ is a frame for E~,) such that there exists a harmonic metric K on E with ~(K~(,)) = J. Similarly, Rarer(X/S, ~, n) was the space whose points over s consist of pairs (E, ~) where E is a vector bundle with integrable connection and ~ is a frame for E~,) such that there exists a harmonic metric K for E with ~(K~,)) = J. Suppose G is a reductive algebraic group. Fix a maximal compact subgroup V C G. Choose an inclusion G ~ Gl(n, C) so that the standard metric J is invariant under V; then V = G c~ U(n). Define Rgo (X/S, 4, = 4, G) ngo,(X/S, 4, n). Endow this space with the topology induced by the usual topology of R~oI(X/S , 4, G). Define 4, O) = 4, G) n 4, n), endowed with the topology induced by the usual topology of R~g(X/S, ~, G). Note that V (with its usual topology) acts continuously on R~I(X/S , 4, G) and R~R(X/S , 4, G). Lemma 9.13. -- The map R~(X]S, 4, G) ~ M~o~(X/S, G) is surjective and proper, and identifies M~(X/S,G) with the topological quotient RJ~(X/S,~G)IV. The map RJ,~(X/S, ~, G) ~ M~(X/S, G) is surjective and proper, and identifies M~(X/S, G) with the topological quotient R JR(X/S, 4, G)/V. Proof. -- We give the proof for the Dolbeault spaces. Recall that R,oI(X/S , 4, G) is a closed subset of RDoI(X/S, 4, n), so R~I(X/S, 4, G) is a closed subset of R~oI(X/S , 4, n). But R~oI(X/S, ~, n) is proper over Mgol(X/S , n), so R~1(X/S, 4, G) is proper over Mgo~(X/S , G). Furthermore, if RJI(X/S, ~, G) ~ Mvo~(X/S, G) is surjective, then M~(X/S, G) is proper over M,o~(X/S , n). To show surjectivity, suppose s ~ S and suppose q is a point of Mvol(X,, G). This can be lifted to a point (E, ~) in a closed G orbit of Rvol(X,, ~(s), G), in which case, by Theorem 9.8, the associated rank n Higgs bundle E is semisimple. Write E = O E~ | A~ where E i are the distinct stable summands of E, and A s are vector spaces. Choose good metrics K~ for E i. Then the good metrics for E are those of the form ]~ K,| i where L i are any metrics on A~. The monodromy group fixes the decomposition of E and acts irreducibly on the components E~. Choose a metric K = ~ K~| for E. Let Mono(E, 1~(8) ) CGI(E~Is)) denote the monodromy group induced by the identity frame lg(,): Er -~ Eg(,). Let W = Mono(E, 1~(,)) r3 U(E~(,,, K~,)). MODULI OF REPRESENTATIONS. II 59 We claim that this is a maximal compact subgroup of Mono(E, 1r162 ). Let ~ denote complex conjugation in GI(E~I,I) with respect to the metric K,. We will prove that fixes Mono(E, l~lsj), and also that every component of Mono(E, l~c,I ) contains a fixed point of ~. Then W, being a compact real form which meets every component, will be maximal compact. Since Mono(E, 1r is reductive, it is equal to the group of elements fixing a subspace of tensors T C Er174 | | Furthermore we may assume that T is the space of all tensors so fixed, and hence there is a decomposition of Higgs bundles E|174 E') | = (T | Ox) | F with F not containing any trivial subobjects. In particular, Mono(E, l~l,~) preserves the subspace F~c,~. Now the harmonic metric K on E induces a harmonic metric on the tensor product, and it tbllows that the direct sum (T | ~x) | F is an orthogonal direct sum of bundles with harmonic metrics. For any g s GI(E~,~), let g" denote the adjoint with respect to the metric K~,~, defined by the formula (ge, f) = (e, g'f) (we will suppress reference to the metric K~,~ in the notation (., .) for the metric on E~,~ or any tensor power thereof). The complex conjugation ~ is given by s(g) = (g.)_h Suppose g e Mono(E, l~c,, ). Then for t s T andfs F~cs~ , we have (g" t,f) = (t, gf) = O, since gfs F~c,~. Similarly if s, t e T then (g" t, s) = (t, gs) = (t, s). Therefore if t s T, g" t = t. In other words, g* e Mono(E, l~l,: ). Thus s(g) = (g')-~ is also in Mono(E, 1 ~,1). This proves that W is a compact real form of Mono(E, 1 ~,~). We still have to prove that it meets every component; this we do by a standard argument. Suppose g s Mono(E, 1 ~c,~). Then gg* is a positive definite self adjoint matrix, so it can be raised to any real power, and we get a real one parameter subgroup of Gl(n) consisting of the self adjoint matrices (gg*)t, t s R. Furthermore, it is easy to see that (gg')~ preserves any tensor preserved by gg', so this one parameter subgroup is in Mono(E, 1~,~). Furthermore, we have a((gg')*)= (gg')-*. Let f(t)=g-'(gg')~; it Ls in Mono(E, 1~,~). Note that f(0) = g-~ and f(1) = g*. On the other hand, a(f(t)) = g'(gg')-' -= g-~(gg') (gg')-' =f(1 -- t). Thus f(1/2) is fixed by a. We have joined the element g-~ to an element of W by a path of elements of Mono(E, 1~1,~). This shows that every component of Mono(E, 1~r contains an element of W, completing the proof that W is a maximal compact subgroup. The group W preserves the metric K~,~ on E~c,~ , and fixes the factors E~, so it preserves the metrics K~ on E~, ~,~. On the other hand, Mono(E, 1~,~) acts irreducibly on E~. ~,~, and W--being a maximal compact subgroup--does too. Therefore K~. ~,; is, up to scalars, the unique metric on E,, ~1,~ preserved by W. Since W is compact, there exists g ~ G such that gWg- ~ C V. Then we may replace the point (E, ~) by (E, g~) ~ Rgo~(X,, ~(s), G), so we may assume W C V. Now J is a W-invariant metric on E, ~= @ E~. ~,1 | A~. But since K~. ~,~ is the unique W-invariant metric on E~. ~r up to scalars, and the E~, ~,~ are distinct irreducible representations fi0 U,A.RLOS T. SIMPSON of W, there exist metrics L~ on A i such that J = Y, K~, z | L,'. Thus our point lies in R~ol(X,, 4(s), n). Set K' =  K~| L[, and ~(K~,,~) ---- J. This proves that R~ol(X/S , ~, G) --> MDol(X/S, G) is surjective. The map is clearly V-invariant, so finally we must prove that two points in Rs~(x/s, ~, G) which map to the same point in MInt(X/S, G) differ by an element of V. Then the properness and surjectivir>" will imply that M~(X]S, G) is the topo- logical quotient space. Again, we may restrict our attention to the fiber over a point RDol(X,, ~(s), G) corresponds to a semisimple object, in other words s E S. Any point in a it is contained in a closed orbit. But the inverse image of a point in Mvol(X,, G) contains exactly one closed orbit. Thus if two points map to the same point in M~(X,, G), we may assume that the two points are (E, ~) and (E, g~). Then there are two harmonic metrics on E, say E K~ | L i and 52 K i| L;, which map to the metric J via ~ and g~ respectively. Note that the stabilizer of E in Gl(n, C) is Stab(E) = l-I GI(A,). There is an element s E Stab(E) such that gs~ takes the metric (Y, K, | Li)~ to J. Thus gs ~ U(n), so g ~ U(n).Stab(E), We have a unique decomposition Stab(E) = (U(n) c~ Stab(E)). (exp(p) n Stab(E)), where gl(n) = u(n) | p is 1he Cartan decomposition. Thus we may write g = up for u ~ U(n) and p e exp(p) n Stab(E). Furthermore, since V = G c~ U(n), we get a Cartan decomposition g =v| c~g), and we may write g: vp' uniquely for vcV and p' ~ exp(p c~ g). It follows that v -- u and p' = p. In particular, p' ~ exp(p) n Stab(E). Thus (E, p' ~) ~ (E, ~) so (E, g~) ~ (E, v~). Thus our two points differ by an element of the maximal compact group V. This completes the proof for the Dolbeault spaces. "['he proof for the de Rham spaces is the same. [3 Lemma 9. la,. -- The equivalence of categories constructed in [Si5] gi~es horr~eomorphisms of topological spaces R~ot(X/S, ~, G) -- R~Da(X/S, ~, G) and M~,(X/S, G) ~ MD,(X/S, G). Proof. -- The equivalence of categories of [Si5] gives an isomorphism of sets R o,(X/S, n) - 4, We have seen in Lemma 7 16 that this restricts to a horneomorphism of subspaces g~t(X/S, ~, n) ~ R~(X/S, 4, n). Furthermore, the equivalence of categories is a tensor functor, so it preserves the mono- dromy groups. Thus it gives an isomorphism of subsets RL,(X/S, 4, G) = R~(X/S, ~, G). MODULI OF REPRESENTATIONS. II 61 Note that Rgol(X/S , 4, G) and Rg.(X/S, 4, G) are respectively closed subsets of Rawol(X/S, 4, n) and Ro.R(X/S, 4, n), endowed with the subspace topologies. Therefore the above isomorphism gives a homeomorphism of topological spaces R~o,(X/S, 4, G) ~ R~a(X/S, 4, G). Furthermore, this is compatible with the action of V. Thus it descends to a homeo- morphism between the quotient spaces which are iDol(X/S, G) and MD~(X/S , G). To finish the proof, note that this homeomorphism is compatible with descent data for going from the case where the fibers are connected and there exists a section, to the general case where the moduli spaces are constructed. [] Corollary 9.15. -- Suppose G and H are reductive algebraic groups, and G ~ H is an injective homomorphism. Then the induced maps between moduli spaces M,ol(X/S, G) ~ MDoI(X/S , H) and MDR(X/S , G) -> MD~(X/S, H) are proper. Proof. -- We may assume that X -> S has a section 4, and that the fibers are connected. Then R~ot(X/S , 4, G) is a closed subset of RaDol(X/S, 4, H). Therefore the map RaDon(X/S, 4, G) -> MDo~(X/S, H) is proper. But this factors through MDo~(X/S, G), and the map li~ol(X/S , 4, G)--*M,ol(X/S , G) is surjective. Therefore the map MDol(X/S , G) ~ MDo~(X/S , H) is proper. The same proof works for the de Rham spaces. [] Surprisingly, we obtain a result about representations of any finitely generated group. Corollary 9.16. -- Suppose Y is a finitely generated group. Suppose G ~ H is an injective homomorphism of reductive algebraic groups. The resulting morphism of moduli spaces M(Y, G) -> M(Y, H) is finite. Pro@ -- Suppose X is a connected smooth projective variety with basepoint x e X. The previous corollary implies that i,a(X, G) -* MDR(X, H) is proper. By Theorem 9.11, this implies that the map Ms(X , G) ->Ms(X, H) is proper. However, the Betti spaces are affine, and an affine proper map is finite. Thus Ms(X , G) -+ MB(X , H) is finite. If Fg denotes the free group on n generators, and if X is a smooth connected projective curve of genus g with basepoint x, then there is a surjection from ~I(X, x) -+ Fg -+ 1 (this is easy to see by drawing a picture of the Riemann surface X ~ as the surface of a solid with g holes). Thus if Y is any group generated by g elements, there is a surjecfion z~I(X , x) -> Y. The additional relations in Y give closed conditions on the representation space, so R(T, G) C RB(X, x, G) and R(Y, H) C RB(X , x, H) are closed equivariant embeddings. Reductivity of the groups G and H implies that the corresponding maps on good quotients M(Y, G) -> M~(X, G) and M(Y, H) -> M~(X, H) are closed embeddings. This implies that the map M(Y, G) -> M(Y, H) is finite. [] 62 CARLOS T. SIMPSON Lemma 9.17. -- Suppose A is a C-algebra of finite type, and N is a finitely generated A-module. Suppose that a reductive algebraic group G acts algebraically on A and N. Then the module of invariants N ~ is finitely generated over A c'. Proof. -- [Mu]. [] Corollary 9.18. -- Suppose Y is a finitely generated group. Suppose G --~ H is a homo- morphism of reductive algebraic groups with finite kernel. Then the resulting morphism of moduli spaces M(Y, G) ---> M(Y, H) is finite. Proof. -- Let G'C H denote the image of G. From Corollary 9.16, the map M(Y, G') ~ M(Y, H) is finite. The map G -+ G' is finite, and the representation spaces are embedded as closed subsets in products of copies of the groups, so the map R(Y, G) ~ R(Y, G') is finite. The map G ~ G' is surjective, so M(Y, G') is a good quotient of R(Y, G') by the action of G. The previous lemma implies that the map M(Y, G) ~ M(Y, G') is finite. Composing these statements gives the corollary. [] This in turn gives finiteness for the maps of Corollary 9.15. Corollary 9.19. -- Suppose G and H are reductive algebraic groups, and G-+ H is a homomorphism with finite kernel. Then the induced maps between moduli spaces MDol(X/S, G) --> MDo,(X/S, H) and M.~(X/S, G) ~ M.a(X/S, H) are finite. Proof. -- The map M~(X/S, G)~M~'(X/S, H) is finite by the previous corollary, and the Dolbeault and de Rham spaces are homeomorphic to these Betti total spaces. Thus M~ot(X/S , G) --~ M~o,(X/S , H) and M~R(X/S , G) --~ M~(X/S, H) are finite. This implies that the corresponding algebraic maps are finite. O Limits of the C" action There is an action of C* on the category of principal Higgs bundles: z e C* sends (P, ~) to (P, zr If G is a reductive group, we obtain an action of C" on MDol(X/S, G). This is compatible with the morphisms of functoriality induced by morphisms of groups, and is equal to the action defined in w 6 in the case G = Gl(n, C). Corollary 9.20. -- For any point y e M~I(X/S, G) the limit lim,_~0 zy exists, and is a fixed point of the action of C*, in M~(X/S, G). Proof. -- Corollary 6.12 gives this statement for the group Gl(n, C). Choose a faithful representation G CGI(n, C). The map C ~ -+M~(X/S, n) extends to a map Ax-~ MDot(X/S, n), and by the properness of the maps in Corollary 9.15, the orbit C* -+ MDo~(X/S, G) extends to a map A 1 -7 Mvo,(X/S , G). The image of the origin is the desired fixed point of C*. [] MODUL! OF REPRESENTATIONS. II 63 Corollary 9.9.1. -- Suppose X is a smooth connected projective variety with basepoint x, and G is a reductive complex algebraic group. Any representation ~I(X, x) -+ G can be deformed to a representation which comes from a complex variation of Hodge structure. Proof. -- The points in the closed orbit of R,ol(X, x, G) lying over fixed points of C* correspond to the representations of the fundamental group which come from complex variations of Hodge structure [Si5]. The same proof as for Corollary 6.12 now works. [] 10. Local structure We will now review the deformation theory of Goldman and Millson (descended from Deligne, Schlessinger and Stasheff) [GM]. The cases of RDn and R~ are identical to [GM], and the case of RDo I is analogous. A differential graded Lie algebra [GM] is a collection A = (A ~ A 1, ... ) of C-vector spaces, with differentials d : A' ~ A ~ + 1 and a bracket [ , ] : A' |162 A t -+ A ~ + ~ such that the following axioms hold: d~= 0; the bracket is graded-anticommutative, [a, b] = (-- 1) '~+1 [b, a] for a EA' and b ~A~; the differential and bracket are compa- tible, d[a, b] ---- Ida, b] q- (-- 1)' [a, db] if a ~ A'; and the Jacobi identity holds with the appropriate signs. Fix a finite dimensional Lie algebra g. A g-deformation diagram is a pair (A, ~) where A ~ is a differential graded Lie algebra and g : A ~ ~ g is a morphism of Lie algebras. Let H' denote the i-th cohomology of the complex (A', d). We say that (A, g) is finite dimensional if the spaces H' are finite dimensional. We say that (A, e) is rigid if the map s : H ~ -~ g is injective. Denote by h the image of H ~ in g, and let h J- denote a subspace transverse to h (for example, if g is semisimple we can take the perpendicular space with respect to the Killing form). The main examples are as follows. Suppose X is a connected smooth projective variety over Spec(C) with a point x ~ X. Let E be a Higgs bundle of semiharmonic type of rank n, with a frame ~:E. ~ C ". Let g = gl(n, C). Then we can define a g-deformation diagram (A,o1(E), e) with A' equal to the space of smooth/-forms with coefficients in End E, the differential d given by the operator D", and the Lie bracket given by the graded commutator of forms. The map ~ is evaluation at x composed with the frame ~. Let G CGI(n) be a complex algebraic subgroup, and suppose (E, ~) satisfies condition Mono(E, [5) C G (in other words (E, ~) represents a point in R,ol(X , x, G)). Put g = Lie(G). Let P be the associated principal Higgs bundle and Ad(P) = P � the adjoint Higgs bundle. We can define a g-deformation diagram (Avot(P), r) with A' equal to the space of smooth/-forms with coefficients in Ad(P). The Lie bracket comes from the Lie bracket of g and the graded commutator of forms, and the augmentation r is given by evaluation at x using P~ --- G. CARLOS T. SIMPSON Similarly, suppose E is a flat bundle (thought of as a representation of the fundamental group, or equivalently as a holomorphic vector bundle with integrable connection), with frame ~:E~-~ C". Let g = gl(n, C). The g-deformation diagram (AB(E), ~) = (A~(E), ~) has A' equal to the space of smooth /-forms with coefficients in End(E), with differential d given by the flat connection D on E, Lie bracket given by graded commutator of forms, and augmentation r given by evaluation at x. If the monodromy group is contained in G CGI(n, t3) (in other words (E, $) represents a point in Rr~X, x, G) or RDj~(X , x, G)), and P denotes the associated flat principal bundle, then we obtain a Lie(G)-deformation diagram (AB(P), ~) = (ADx~(P), e) where A * are the spaces of forms with coefficients in Ad(P), the differential is again given by D, the Lie bracket comes from that of Lie(G), and the augmentation is given by evaluation at x. The deformation theory associated to a deformation diagram We recall the basic elements of the theory of Goldman and Millson--see [GM] for details. Let Art denote the category of artinian local schemes of finite type over Spec(C). An object S a Art is of the form S = Spec(0s) for a local C-algebra Os of finite length. Let m s denote the maximal ideal of O s. Fix a Lie algebra g and let G be an algebraic group with lie(G) = g. Let G~ S) C G(S) denote the set of S-valued points sending the closed point to the identity in G. The group G~ S) depends only on g, not on the choice of G. We have an exponential map from g| to G~ S), denoted u ~ e ", which is an isomorphism of sets. The formulas giving the group structure of G~ S) in terms of the exponential isomorphism of sets are universal, applying also to the case of infinite dimensional Lie algebras. Suppose (A, e) is a deformation diagram. For S e Art, we obtain a group G~ ~ S) with exponenlial map A ~ | ms ~ G~ A~ S). The Lie algebra A ~ acts on the A ~, and this gives an action of the group G~ ~ S) on A ~ | ms- We denote the composition of this action with the exponential map by (u, a) ~ e-" ae". There is also an expression e-"d(e") e A 1 | ms- The formulas for these actions are the same as those that one calculates from the terminologies in the case of a finite dimensional Lie algebra. Given a g-deformation diagram (A, e) and an artinian scheme S cArt, let F(S, A, s) denote the set of pairs (~, g) with ~ e A 1 | m~ and g ~ G~ such that d(~) + ~[~, ~] = 0. The group G~ ~ S) acts on F(S, A, e) by the formula e" : (r~, g) ~ (Ad(e -u) ~ q- e-" d(e"), e-"*' g). Let R(S, A, r denote the quotient of the set F(S, A, ~) by the action of G~ ~ S). MODUI.I OF REPRESENTATIONS. II 65 Lemma 10.1. -- Suppose the diagram (A, e) is rigid and finite dimensional. Then the functor S ~ R(S, A, e) is pro-represented by a formal scheme R(A, ~). Proof. -- [GM]. [] Lemma 10.2. -- Fix a linear algebraic group G and put g = Lie(G). If (P', b') is a principal Higgs bundle of harmonic type (resp. a principal flat bundle) for the group G, with a point b ' e P'~ , then (ADoI(P), r ( resp. (ADR(E), e)) is a rigid and finite dimensional g-deformation diagram, and the formal scheme R(A, e) is naturally isomorphic to the formal completion of the representation space RD,I(X, x, G) (resp. RD~(X , x, G) or R~(X, x, G)) at the point corres- ponding to (P', b'). Proof. -- This is a simple variant of one of the theorems of Goldman and Millson [GM]--theirs is the statement for the space RB(X, x, G). Note that the formal completions of RB(X, x, G) and RDR(X , X, G) at corresponding points are isomorphic, by the analytic isomorphism given in Theorem 7.1. This isomorphism is compatible with the equality of deformation diagrams. We may thus restrict our attention to the case of A~ and RDo ,. Suppose H is a linear algebraic group and N C H is a normal unipotent subgroup. Fix a principal Higgs bundle ofsemiharmonic type with frame (P', b') for the group H/N. Let Z denote the set of triples (P, b, ~) where P is a principal Higgs bundle of semi- harmonic type for H, b is a frame, and a : (P, b) � a (H/N) ~ (P', b') is an isomorphism. Choose (Po, b0, ~o) e Z. Since H acts on Lie(N), we obtain a Higgs bundle with Lie algebra structure P0 � n Lie(N). Let 00 be the operator giving the holomorphic structure of P0, and let q~o be the Higgs fie.ld. Let Y~' denote the set of pairs (u, ~) with u e N and aq e AI(X, Po � Lie(N)), + + = o, up to equivalence under the action of A~ P0 � Lie(N)) given by the same formula as above. Then there is a natural isomorphism between Z' and Z. The principal Higgs bundle corresponding to (u, ~) is P~ = (P0, 0o + ~o,1, q~0 + ~t,o), and the frame is b~ = b o u. The isomorphism stays the same, 0% = ~o. These constructions are functorial in terms of the pair (H, N). Suppose G is a linear algebraic group and S is an artinian local scheme of finite type over Spec(C). Then we obtain a new group scheme G(S) defined by setting G(S) (T) = Hom(S � T, G). There is a morphism G(S) -+ G, and the kernel G~ is a normal unipotent subgroup. There is a morphism of group schemes over S, : G(S) xS G� equal to the identity in the second factor, and equal to the element ofHom(G(S) � S, G) corresponding to the identity in Hom(G(S), G(S)) in the first factor. 9 66 CARLOS T. SIMPSON Let pt : X x S -+ X be the projection on the first factor. If Pa is a principal object for the group G(S) on X, then P;(Ps) is a principal object for the group G(S) on X x S over S, and @(Ps) a"fp;(Ps) XGts'�162 X S) is a principal object for the group G on X x S over S. There is a quasMnverse: if P is a principal G-bundle on X x S then put P(S) (T) ----Hom(S x T, P); this is a principal G(S)-bundle over X with P = F(P(S)). If P has some extra structure then P(S) is provided with the same extra structure. Thus, the functor r gives an equivalence of categories between principal objects for the group G on X x S, and principal objects for the group G(S) on X. Furthermore, the S-valued points of O(Ps)[~,} � s correspond to the points of (Ps),- This construction works for principal bundles, principal Higgs bundles, and principal bundles with integrable connection. Denote by s o e S the closed point. The restriction of O(Pa) to X X { s o } is naturally identified with Ps x a~sl G (and this identification is compatible with the identification of the S-valued points above { x } x S given above). Consequently, the construction 9 applied to principal Higgs bundles preserves the property of semiharmonic type. Applying the previous construction with H = G(S) and N ---- G~ we obtain a natural identification between: the set of triples (P, b, 0c) where P is a principal Higgs bundle of harmonic type on X x S over S, b is an S-valued point of P[t,}� and ~: (P, b)Ix� t,0} ~ (P', b'); and the set of elements of R(S, ADol, ,-). We obtain an isomorphism of functors of artinian local C-schemes of finite type, giving an isomorphism of formal schemes between R(AI~, ~) and the formal completion of RDo~(X, x, G) at (P', b'). [] Remark. -- Under the isomorphism of functors given above, the G-orbit of (E, [5) goes to the set of elements represented by (0, g). Remark. -- Let H C G be the stabilizer of (E, ~). Then H acts on RDol(G ). Since H d G, it preserves the bundle Ad(P0). Thus H acts by conjugation on the diagram DDoI(E, G) (the action on the Lie algebra g is also by conjugation). Thus H acts on the functor F and the representing formal scheme R(F). Our isomorphism is compatible with these actions of H. A morphismffrom a diagram D 1 to a diagram D~ is a collection off ~ from A~(D1) to A~(D~), such thatf ~ ~ ---- ~ f'-1, such thatf(ab) ----f(a)f(b), and such that ~f~ = ~(s). Given such a morphism we get a map of functors F(S, D~) -+ F(S, D,), and hence a map f: R(D1) ~ R(D2). We say that a morphism f is a quasi-isomorphism of diagrams if fo: H0(D,) HO(D ) f~: H'(Dj.) --% HI(D,,) MODULI OF REPRESENTATIONS. II are isomorphisms, and if f*: H'(I),) is injective. The fundamental step in the theory of deformations of Goldman-Millson-Deligne- Schlessinger-Stasheff is the following statement [GM]. Proposition 10.3. -- If f: D 1 --*D 2 is a quasi-isomorphism of rigid finite-dimensional diagrams then it induces an isomorphism of formal schemes f: R(I)~) -~ R(D,). Proof. -- [GM]. [] We will apply this by using the formality results from ([Si5] w 3). Suppose (A, 5) is a diagram where the differentials 8 are zero. Let C C HI(A ") be the quadratic cone which is the zero set of the map from HI(A ") to H2(A ") given by B ~-* B ^ B. Recall that h I is the perpendicular space of the image of H~ ") in g. Goldman and Millson show that the formal scheme R(A, ~) is equal to the formal completion of (~ � it  at the origin [GM]. Theorem 10.4. -- Let G be a reductive algebraic group. Suppose (P, p) is a point in a closed orbit in RDol(X , G) (resp. RDR(X , G)). Let C be the quadratic cone in HX(Ad P) defined by the map ~ ~ ~q ^ ~ ~ H2(Ad P). Let C denote the cone defined above for the formal deformation diagram (AH, r and, let h a. denote the perpendicular space to the image under ~ of H~ P) in g. Then the formal completion (RI)o,(X,G), (P,p))^ (resp. (RDR(X,G), (P,p))^) /s isomorphic to the formal completion (C � la  0) ^ Proof. -- Suppose (P, b) is a framed principal Higgs bundle of harmonic type. We get a Higgs bundle Ad(P) with Lie algebra structure. The Higgs bundle Ad(P) is a direct sum of stable Higgs bundles with vanishing Chern classes. By the results of [Si5], there is an operator D' on C ~~ Ad(P)-valued forms. Let (A1),(P), 5) be the diagram with A ~ equal to the space of Ad(P)-valued/-forms u such that D'(u) = 0 (in the notation of [Si5]). The map ~ is given by D"= 0 + 9 or equivalently by D = D' + D". Let (Aa(P), *) be the diagram with A ~ equal to the space of harmonic forms, which is equal to Hi(A~ol(P)). Here the maps 8 are zero, in other words (AH(P), 5) is formal. Let (Axm(P), 5) be the deformation diagram for the flat principal bundle (P, D) corresponding to the principal Higgs bundle P by the correspondence of [Si5]. We have natural morphisms (AD (P), ") (AD,(P), 5) --+ (ADa(P), 5) (AD,(P), 5) --+ (An(P), r 68 CARLOS T. SIMPSON By [Si5] Lemma 3.2, these are quasi-isomorphisms. By Proposition 10.3 we get iso- morphisms of formal schemes R(AD,(P), s) ~ R(ADo,(P), a) R(AD,(P), e) -~ R(AD~(P), ~) R(AD,(P), 5) ~ R(Aa(P), s). Finally, the formal scheme R(AH(P), 5) is isomorphic to the formal completion of the cone (C � h 1, 0) A. Now apply Lemma 10.2. [] Remark. -- Following Goldman and Millson, we may apply the Arfin approxi- marion theorem [Ar] to conclude that the isomorphism of formal completions comes from an isomorphism of analytic or ~tale neighborhoods. Remark. -- The stabilizer H of (E, ~) acts on all of the above spaces and diagrams, and in particular, H acts on the cone C and on 11  The quasi-isomorphisms of diagrams are compatible with the action of H. Therefore the isomorphism of formal neighborhoods is compatible with the action of H. The cone C is affine and H acts linearly, so there is a good quotient C/H. Proposition 10.5. -- Suppose P is a principal harmonic bundle. The formal completion of the moduli space MDo,(X , G) (resp. I~IDR(X , G), 1VI~(X, G)) at the point P is isomorphic to the formal completion of the good quotient C/H of the cone C by the action 0flL Proof. - Apply Luna's Etale slice theorem [Lu] to construct an H-stable subscheme Y C R,o~(G ) passing through (E, [~), and such that the map Hi� Y ~li~l(G ) is locally an isomorphism in the analytic or Etale topology. Here H  is an H-stable subspace of G passing through the identity, such that H  � H -~ G is locally an isomorphism. Now we have an isomorphism (H  � Y) ^ ~ (H  � C) ^ of formal schemes, preserving the subscheme (H  � { 0))^. From this we get projections yA ~ CA and C A ~ Y^. Their composition is a map Y^ ~ Y^ such that the scheme theoretic inverse image of the origin is just the origin. An argument of counting dimensions of the local ring modulo powers of the maximal ideal shows that this must be an isomorphism, so we get an isomorphism yA ~ C A, This is compatible with the action of the group H. Let Y/H and C/H denote the good affine quotients. Now H is reductive, since it is the stabilizer of a point in a closed orbit [Lu], and because of this, we have (Y/H) A = yA/H and similarly for C. Thus (Y/H) ^ ~ (C/H)^. But Y/H is equal to the moduli space MDo~(G), locally at E. Thus the formal completion of the moduli space is isomorphic to the formal completion of the affine quotient of the cone C by the action of H. ~3 We get canonical isomorphisms between the formal completions of the spaces llDot(X, x, G) and Rnrt(X , x, G), or Mvol(X, G) and Mo~(X, G), at points corres- ponding to the same harmonic bundle. These isomorphisms are not related to the identification between the sets of points given by the harmonic theory of [Si5]. MODULI OF REPRESENTATIONS. II 69 Theorem 10.6 (Isosingularity). -- For any point y e RDoI(X , x, G) (resp. y~RDR(X ,x,G), y~M,ol(X,x,G), or y ~ MD~(X ,x,G)) there exists a point z E RD~(X , x, G) (resp. y E R,ol(X, x, G), y e MD~(X , x, G), or y ~ MDoI(X , x, G)) and gtale neighborhoods U of y and V of z such that (U,y) ~ (V, z); and the local systems corresponding to y and z have isomorphic semisimplifications. Proof. -- By the Artin approximation theorem [Ar], it suffices to show that the formal completions aty and z are isomorphic. Suppose first of all thaty lies in a closed orbit, so it corresponds to a reductive representation. Then let z be a point in the other space corresponding to the same reductive representation o. Then there are natural isomrphisms of cohomology rings H~oI(X , Ad(0))_-__ H~R(X , Ad(p)) [Si5]. Thus the cones that appear in Theorem 10.4 for y and z are isomorphic. The automorphism groups H are also the same in both cases. The formal completions of the representation spaces are both isomorphic to the formal completion of the cone C � g and the formal completions of the moduli spaces are isomorphic to C/H. Suppose y does not lie in a closed orbit. Lety' denote a point in the closed orbit adhering to the orbit ofy. There exists a point z' in the other representation space, and isomorphic dtale neigh- borhoods U' ofy' and V' of z'. There is a pointyl ~ U', mapping to a point in the orbit ofy. In particular, the formal completion ofU' atyl is isomorphic to the formal completion of the representation space at y. Let z 1 denote the point corresponding to Yl under the isomorphism U'~ V', and let z denote the image in the other representation space of z 1. The formal completion of V' at zx is isomorphic to the formal completion of U' at yx, so the formal completion of the representation spaces at y and z are isomorphic. We may suppose that y' is in the closure of the orbit Hya, so z' is in the closure of the orbit Hz 1. In particular, the closed orbits adhering to the orbits ofy and z correspond to the same reductive representations. [] Remark. -- Ify E R,R(X , G) (resp. y ~ MD~(X , G)) and if z denotes the corres- ponding point in RB(X , G) (resp. MB~X , G)) then there are dtale neighborhoods U ofy and V of z, and isomorphisms (U,y) ~ (V, z). This tbllows directly from the Artin approximation theorem, since the analytic isomorphism of Theorem 7.1 gives an isomorphism of formal neighborhoods. The Zariski tangent space We give a result valid for any representation, not necessarily reductive. Lemma 10.7. -- Suppose (P,p) a RDR(X, G). Then the dimension of the Zariski tangent space to R.R(X, G) at (P,p) is equal to h~R(X, Ad(P)) + dim(g) -- h~ Ad(P)). The same for R,ol(X, G). Proof. -- Let D be the de Rham or Dolbeault deformation diagram corres- ponding to (P, p). The Zariski tangent space of the representation space is equal to 70 CARLOS T. SIMPSON R(Spec(C[t]/t) ~, D). The set F(Spec(C[t]/t~), D) is equal to the set of pairs (6, g) where g e g and ~ e A 1 with d(~) = 0. The action ofG~ ~ Spec(C[t][t~)) amounts to changing (~, g) by adding (d(s), r for s 9 A ~ The quotient by this action is H a | (g/~(H~ [] 11. Representations of the fundamental group of a Rierna.n surface Theorem 11.1. -- If X is a connected smooth projective curve of genus g >1 2, then the moduli spaces M~(X, n), MDR(X , n), Mml(X , n), and the representation spaces R~(X, n), RD~(X, n), and RDoI(X, n), are normal irreducible varieties. Most of the rest of the section is devoted to the proof. First we prove that the schemes are reduced and normal. Note that a normal connected variety is irreducible, so for the second statement it suffices to prove connectedness, which we do afterward. At the end of the section, we give some auxiliary statements about the local structure of the representation space. These were obtained in my original proof of the theorem; they are no longer needed in the present proof but it seemed like a good idea to record them anyway. Nor The idea for this part of the proof was suggested by M. Larsen (cf. Corollary 11.6 below). Suppose X is a connected smooth projective curve of genus g/> 2. Choose a basepoint x e X. Lemma 11.2. -- Every irreducible component of Rs(X, x, n) has dimension greater than or equal to 2gn ~- n* + 1. The Zariski open subset R~(X, x, n) parametrizing irreducible representations is smooth of dimension 2gn 2 -- n 2 + 1. Proof. -- First note that R~(X, x, n) is the subvariety of Gl(n, C) ~ defined by one relation. The relation is a map R : Gl(n, C) ~ ~ Sl(n, C), and RB(X , x, n) : R-l(e). This implies that every irreducible component of Rs(X, x, n) has dimension /> 2gn ~- n~+ I. Suppose 0 is a point in R~(X, x, n). Let V denote the local system corresponding to p and let Ad(p) denote the local system End(V). Then Tr : H'(X, Ad(p)) ~ H'(X, C) are isomorphisms for i = 0 and i = 2. On the other hand, if ~ 9 Hi(X, Ad(p)) then Tr([~, ~]) = 0. Therefore the cone C which appears in Theorem 10.4 is equal to all MODULI OF REPRESENTATIONS. II 71 of Hi(X, Ad(p)). By Theorems 7.1 and 10.4, RB(X , x, n) is smooth at O. Finally, the rank of Ad(p) is n 2 so a calculation of Euler characteristics gives dim(C) = dim(H 1) = (2g -- 2) n ~ + 2. In the notation of the previous section, dim(h 1) = n 2- I, so the dimension of Rs(X,x,n) at p is 2gn ~-n 2+ 1. [] Proposition 11.3. -- The dimension of any irreducible component of RB(X , x, n) is equal to 2gn ~ -- n + 1, all irreducible components are generically smooth, and RB(X , x, n) is a complete intersection. The dimension of the subspace of reducible representations has codimension at least two, except in the case g = 2 and n = 2 when it has codimension at one. Proof. -- We suppose that the proposition is known for any n' < n. We will prove the proposition for representations of rank n. For 1 ,< k < n, let Pk denote the parabolic subgroup of Gl(n, C) consisting of block-upper triangular matrices with 2 blocks, where the first block has size k and the second block has size n -- k. There is an exact sequence 0 ~ C k''-'' ~ Pk -~ Gl(k, C) x Gl(n -- k, C) ~ 1, where the kernel represents the abelian group of block upper triangular matrices with the identity matrix in the diagonal blocks. For each Pk, let G k = Gl(n, C)/P k. It is the Grassmanian of k-planes in C", with dimension k(n -- k). Choose a constructible section q~ : G k ~ Gl(n, C). Let RB(X , x, Pk) denote the space of representations of hi(X) into Pk. We obtain a constructible family of representations of ~h(X, x) into Pk indexed by G k � RB(X, x, Pk), corresponding to the constructible map a : G k � RR(X , x, Pk) --*RB(X,x, Gl(n)) defined by a(y, p) = q~(y) pq~(y)-~. This has the property that a(y, 9) is a representation of ~(X) into the conjugateyP~y-~ (this conjugate doesn't depend on the choice of lifting q~(y)). Let n~r -. Gl(n, C)) denote the space of reducible representations. Since every reducible representation has a fixed subspace and is therefore conjugate to a represen- tation in some Pk, we have U .(G~ � RB(X, x, Pk)) = R~(X, x, Gl(n, C)). l~<k<. In particular, the dimension of R~d(X, x, Gl(n, C)) is bounded by the maximum of the dimensions of G, � RB(X, x, Pk). [_.emma 11.4. -- Suppose that Proposition I1.3 is known for representations of rank n' < n. Then for any I ~ k < n, the dimension ofG k x RB(X, x, P,) is less than or equal to 2gn 2 -- nZ; and if g >t 3 or n >I 3 then the dimension is less than or equal to 2gn 2 -- n ~ -- l. Proof. -- We count dimensions, looking at the morphism RB(X , x, Pk) -* RB(X, x, Gl(k, C)) � R,(X, x, Gl(n -- k, C)) 72 CARLOS T. SIMPSON which associates to a representation p its diagonal parts (Pl, P2). We would like to know the dimension of the space of representations into Pk which have given diagonal part (~, ~). Let 7~, ..., 7~,~ denote the standard generators of the fundamental group of X, and let r(Ta, ..., Yso) denote the relation. If we fill in the diagonal parts of the matrices P(7~) according to the given representations Pl, 02, then to specify the remaining part of the representation we have to choose a vector (Ax, ..., A2~ ) with each A~ in the kernel C k("-k~ of the above exact sequence. Putting the resulting matrices into the relation gives a map r 0 : C k(.-k~ � ... � C k(--k~ _+ Ck(--k~. The kernel of this map is the fiber over (Pl, P2), in other words the space of representations with diagonal parts P1, P2. This is the last part of a complex calculating the group cohomology of rq(X, x) with coefficients in the vector space C ~'"-k~, so the cokernel of the map is H2(rq(X, x), Ck("-k~). The action of ~I(X, x) on the vector space of coefficients comes from the adjoint action on Lie(Pk) using the representation p. This only depends on Pl and P2. More explicitly it can be seen by expressing Ck(,-k~ = Ck| C,-k, with action on C a given by Pl and the action on C "-k given by P2- By Poincar6 duality, the dimension of the H s is the same as the dimension of H~ x), p~ | P2)- Thus the dimension of the fiber over (Pl, 02) is (2g -- 1) k(n -- k) + h~ ?~ | p~). If Pl and 02 are irreducible and not isomorphic, then H~ x), 9] | Ps) = 0. Therefore we can count the dimension of the fiber over (Pl, P,) as (2g -- 1) k(n -- k). By induction, the dimension of the space of choices of (Px, 98) is (2g -- 1) (k s + (n -- k) 2) + 2. The dimension of this part of G, x RB(X, x, Pk) is (2g-- 1) (k s+ (n--k) 2 + k(n - k)) + 2 + k(n -- k) = (2g-- 1) n2+ 1-- ((2g--2) k(n--k) -- 1). In particular, as gt> 2 and n>/ 2, the dimension is at most 2gn 2-n 2. If g/> 3 or n/> 3 then the dimension is at most 2gn 2 -- n 2 -- 2. The set of pairs (Pl, P2) such that both representations are reducible has (by induction) dimension bounded by (2g -- 1) (k 2 + (n -- k)2). For these points we make a coarse counting of the dimension of the fiber over (01, P2): it is less than 2gk(n -- k). The dimension of this part of G k � RB(X, x, Pk) is therefore bounded by (2g-- 1) (k 2+ (n--k) *) + (2g+ 1) k(n--k) = (2g-- 1) n *+ l--((2g--3) k(n--k) + 1). The dimension is at most 2gn 2 -- n 2 -- I for g/> 2 and n t> 2. The set of pairs (01, P~) which are irreducible and isomorphic (hence of rank k = n- k = hi2) has dimension less than or equal to (2g- 1)kS+ 1. The H ~ has MODULI OF REPRESENTATIONS. II dimension 1, so the fiber has dimension (2g- 1)kS-t - 1. The sum of the dimensions is less than or equal to (2g -- 1) n~[2 + 2. For n 1> 2 and g/> 2 this is less than or equal to 2gn 2 - n ~ - 4. The set of pairs (P1, P2) such that one representation is reducible and one repre- sentation is irreducible has dimension bounded by (2g- 1) (kS+ (n- k) ~) + 1. For such a pair, the H ~ discussed above has dimension 0 or I. Therefore the dimension of the fiber over (Pl, P2) is bounded by (2g- 1)k(n- k) + 1. The dimension of this part of the space G k � liB(X, x, Pk) is bounded by (2g-- 1) (k s+ (n--k) 2 § k(n--k)) +2 + k(n--k) = (2g-- 1) n'+ 1 -- ((2g--2) k(n--k)-- 1). In this case, note that n must be at least 3. Therefore, the dimension is at most 2gn ~ -- n 9 -- 2. We have shown, in all the cases, that the dimension of G~ x liB(X, x, Pk) is less than or equal to 2gn ~ -- n 2, and if n i> 3 or g >/ 3 then the dimension is less than or equal to 2gn z- n 2- 1. [] We continue with the proof of the proposition. It follows from the lemma that the dimension of the subspace of reducible representations is less than or equal to (2g -- 1) n ~. From the lower bound of Lemma 11.2, no irreducible component can consist entirely of reducible representations. Therefore the open set of irreducible representations is Zariski dense, so the dimension of each component is equal to (2g -- 1) n ~ + 1. It follows from the equations for liB(X, x, n) given in the proof of Lemma 11.2 that liB(X, x, n) is a complete intersection. Since li~(X, x, n) is smooth, each component is generically smooth. Finally note that, except in the case g = 2 and n = 2, Lemma 11.4 shows that the dimension of the subspace of reducible represen- tations is less than or equal to (2g -- I) n 2 -- 1. This proves the proposition. [] Lemma 11.5. --- The scheme llB(X , x. n) is smooth outside of a subset of codimension >t 2. Proof. -- This follows from the previous proposition except when g = 2 and n = 2. We are reduced to that case, where the space of representations has dimension 13. From the proof of Lemma 11.4, the codimension 1 part of the locus of reducible representations consists of those representations conjugate to an upper triangular representation with distinct diagonal parts. We show that the space of representations is smooth at such points. This statement is invariant under conjugating the representation; so we may fix an upper triangular representation p, with diagonal entries Pl + Oz. The space of semisimple reducible representations has dimension 10, so we may assume that p is not semisimple. For a 2 � 2 representation, this implies that there is a unique sub- representation of rank 1 and a unique quotient. Make the convention that the sub- representation is Pl and the quotient is P2- We claim that there are no nonscalar endomorphisms of the representation p. 10 74 CARLOS T. SIMPSON For iffis an endomorphism of rank 1, then the image off must be the subrepresentation PI, while the coimage must be the quotient p,, contradicting the condition that Pl 4- P2. Thus if ~ is an eigenvalue of an endomorphismf then f-- k = 0 andfis a scalar. Thus, as claimed, H~ Ad(p)) = { 0 } (we say that p is simple). By Lemma 10.7, the dimension of the Zariski tangent space to RB(X , x, n) at is equal to hX(Ad(p)) + n ~ -- h~ Since p is simple, h~ = 1. By Poincar~ duality, h~(Ad(p)) = 1, so hX(Ad(p)) = (2g -- 2) n' + 2 = 10, and the dimension of the Zariski tangent space is 10 + 4 -- 1 = 13. Since this is equal to the dimension of any irreducible component of the space, the local ring is regular and R~(X, x, n) is smooth at p. This completes the proof of the lemma. [] Corollary 11.6. --- The space of representations Rs(X , x, n) # reduced and normal. Proof. -- This was pointed out to me by M. Larsen (he refered me to [Ha], Proposition 11-8.23). By Proposition 11.3, RB(X, x, n) is a complete intersection. The local rings of a complete intersection are Cohen-Macaulay, hence satisfy Serre's condi- tion $2. The previous lemma shows that the space of representations is regular in codimension 1. By Serre's criterion, RB(X , x, n) is reduced and normal. [] Corollary 11.7. -- The representation spaces RDol(X, x, n), RD~(X, x, n), and R~(X, x, n) are normal varieties of dimension 2gn ~ -- n ~ + 1. The moduli spaces M~,, M,R , and M s are normal varieties of dimension 2gn ~ -- 2n ~ + 2. Proof. -- We have shown that RB(X , x, n) is normal of dimension 2gn 2 -- n ~ + 1. By the isosingularity principle (Theorem 10.6 and the following remark), the same is true for the de Rham and Dolbeault spaces. Good quotients of normal varieties are normal. (This can be seen by proving that if A is a ring which is integrally closed in its field of fractions K and a group acts, then A ~ is integrally closed in its field of frac- tions K~ Thus the moduli spaces are normal varieties. To calculate their dimensions, note that there is a Zariski dense open set of points of the representation space where Sl(n, 12) acts with finite stabilizer. The dimension of the quotient is the dimension of the representation space minus the dimension of Sl(n, 12). [] This corollary provides the first half of the proof of Theorem 11.1. To complete the proof, it suffices to prove that these varieties are connected. Connectedness of the representation spaces is equivalent to connectedness of their universal categorical quotients, the moduli spaces. By Proposition 7.8 and Theorem 7.18, the three moduli spaces are homeomorphic. Thus it suffices to prove that MDo,(X , n) is connected. The idea for the proof of connectedness comes from Hitchin's calculation of the cohomology of the moduli space of rank 2 projective bundles with odd degree. In that case, the moduli space is smooth, and Hitchin uses a Morse function, the moment map for the action of S x, to calculate the cohomology. The lowest stratum is the space of 75 MODULI OF REPRESENTATIONS. ]I unitary representations, known to be connected by the work of Narasimhan and Seshadri using a lemma of Atiyah. Hitchin deduces from Morse theory that the moduli space is connected and hence irredtlcible, and in fact he calculates the Betti numbers. It would be good to carry through this program for higher ranks and for the case when there are singularities. We will not attempt this here, but we will do enough to show that the moduli space is connected. Because of the presence of singularities, we will avoid Morse theory and instead proceed by algebraic geometry, using the C* action discussed in w fi (wbAch is the complexificat~on of H~tcbAn's S t action). The relationship betweex: these approaches is that the critical point set of the moment map is equal to the fixed point set of the C* action. In ([Si5] w 4), the fixed point set was identified with the set of complex variations of Hodge structure. We remark that in order to apply Hitchin's method to compute the Betti numbers, one would have to be able to compute the Betti numbers of the moduli spaces of variations of Hodge structure. Actions of C* Suppose that Y is a quasiprojective variety on which G* acts algebraically. Suppose that L is an ample line bundle w~th a tinear]zation of the action. Then C" acts locally finitely on H~ L ~ ~). Therefore we may choose n and a subspace V C H~ L~-) which is preserved by G" and which embedds Y into the projective space P(V*), so that C* acts on this projective space and acts compatibly on the very ample 0(1). Thin action is compatible with the embedding of Y and with the isomorphism L| @~(1). Write V = O V~ where the sum is over integers ~, and t e C* acts by t" on V,. Then the fixed point sets of C* on P(V*) are the subspaces P(V~). The fixed point sets in Y are Y~ ---= Y c~ P(V*), The action oft e C* on L IY~ is given by t ~. Let Z be the closure of Y in P(V*). Then Z is preserved by C*, and its fixed point sets are Z~ = Z tn P(V[). If z ~ Z then there are unique points lim,_,o tz and lim4_~ tz in Z. These are fixed points, kence are in some fixed p()ints set Z~ and Z~ respecfively. We will describe the weights ~~ and ~(z) explicitly (see the discussion near the end of w 1, Part I). Lift z to a point weV*, and write w=Zw, with w, aV*. Then ~~ (resp. ~(z)) is the smallest (resp. largest) integer ~ such that w, 4= 0. From this description, if z e Z then ~~ ,< ~(z) and equality holds if and only if z is a fixed point. Assume that Y has the property that lira t -~0 tx exists in Y for all x e Y. Let f~ ---- ~0(y) be the smallest integer such that Ya is nonempty. Then ~ is also the smallest integer such that Z~ + 0. If x E Y~ and there exists y e Y such that y # x and lim t _~ ~ ty = x, then a > ~. In particular, we obtain the following criterion. Lemma 11.8. -- Suppose that Y has the property that limt_~0 tx exists in Y for all x E Y. Suppose U C Y is a connected subset of the fixed point set of C', and suppose that for any fixed point x not in U, there exists y * x in Y such that lim t _~ ~o ty = x. Then Y is connected. 76 CARLOS T. SIMPSON Proof. -- Suppose Y'is a connected component of Y not containing U. Let ~ = [3~ Choose x e Y~. By hypothesis there existsy # x in Y' such that lim t ~.~o ty = x. On the other hand, z ---- limt_,0 ty is also in Y', say in Y',. Buty is not a fixed point, so o~< ~, contradicting minimality of [3. [] Connectedness Lemma 11.9. -- Suppose E is a stable Iliggs bundle of degree zero, fixed up to isomorphism by the action of C'. Suppose that q~ + O. Then there is a I[iggs bundle F not isomorphic to E, such that lim, _~ ~o tF = E in the moduli space. Proof. -- Since E is a fixed point, it has the structure of system of Hodge bundles, in other words E O E ~ with 9:E~--+E ~-1 ' = | Assume that the indexing is normalized so that 0~<p~< r, and E ~ and E'# 0. Note that r>/ 1 since 9~:0. Furthermore, note that deg(E ~ < 0 and deg(E') > 0, since E is stable of degree zero. Hence Hom(E', E ~ has degree < 0. In particular the Riemann-Roch theorem implies that there exists a nonzero extension class "~ in Extl(E ", E ~ = H'(Hom(E', E~ For each t e C, let M~ be the extension 0 -+E ~ ~ M t -+E" -+0 given by the class t" ~. Let F t be the Higgs bundle M s | Oo< ~<, E ~. The Higgs field 9 is given by the usual maps for 1 < p < r and by the compositions E I -~ E~ | t2~ -+ M~ | and M~ ~ E' ~ E'- 1 | t2~. Note that F 0 ---- E. We have isomorphisms ~t : Ft --- t-' F 1 given as follows. On E ~, 0 < p < r, 9~ is given by multiplication by t ~. On M, qh is the isomorphism fitting into the middle of the diagram E ~ > M t > E' 1" E ~ > M 1 > E'. Note that 90 = t-1 0<p. Thus we have a family of Higgs bundles F, with F 0 = E and F t = t- ' F t for t # 0. Since E is stable, so are Ft, by the openness of the condition of stability. Hence lira t -~o tFt = E. To complete the verification, we will show that the vector bundles underlying E and Fx are not isomorphic. First we show that M 1 + M 0. Let A C E' be the [5-subsheaf, in other words the subsheaf of highest slopc, and highest rank among subsheaves of that slope. We may choose aq to be a nontrivial extension of A by E ~ Note that the degree of any subsheaf of E ~ is < 0, whereas the slope of A is > 0. Thus if f: A r M is an inclusion, we must MODULI OF REPRESENTATIONS. II have p of: A -~ A where p is the projection from M to E'. Thus (p of)'(~) 4: 0. But the mapfis a splitting of (p of)'(~). This contradiction shows that there is no inclusion of A into Mx, and hence there can be no isomorphism M 0 -~ M~. The bundle M 1 is a deformation of M 0. By semicontinuity, h~ M0) ) >/h~ M~)). Furthermore, the inequality is strict: for if not then t ~ H~ Mr) ) would form a vector bundle over the t-line, and 1 ~ H~ M0) ) could be lifted to ft ~H~ withft ~1 as t ~0; then ft:M0~ Mt = Mx for t near 0, which would contradict the conclusion of the previous paragraph. Now in our situation there exists a vector bundle B (the direct sum of the other Hodge components) such that E = M 0@B and F = M IQB. We get h~ E)) = h~ Mo) ) + h~ B)), while h~ F~)) ----- h~ M~)) + h~ B)). Thus h~ E)) > h~ Ft)), so the vector bundles E and F x are not isomorphic. [] Corollary 11.10. -- The moduli space M~I(X , n) is connected. Proof. -- The ample line bundle L on MDo,(X, n) has a linearization of the action of C ~ This is because we constructed MDol(X, n) as the moduli space of some sheaves on the cotangent bundle to X, and the action ofC ~ came from the action of multiplication on T" X, so C" acts functorially on the Hilbert schemes, the Grassmanians, and the line bundles over the Grassmanians. We apply the criterion of Lemma 11.8. Let U be the subspace corresponding to Higgs bundles with 0----0 (these are the ones corresponding to unitary representations). It is isomorphic to the moduli space of vector bundles of rank n on X. The moduli space of vector bundles is projective, so U is a closed subset of MDoI(X, n). The subset U is connected--this fact was used by Narasimhan and Seshadri [NS] and comes from a lemma of Atiyah. Suppose that E is a direct sum of stable components, representing a point in M~t(X , n) -- U fixed by C'. All of the stable components of E are then fixed by C ~ We can write E----E I| z with E1 stable and not unitary. Apply Lemma 11.9 to obtain F 1 with lim~ ~o FI = Ex, and F 1 4: E~. Set F = F 1| E 2. Then lim~..co F = E, but gr(F)4= gr(E). The criterion of Lemma 11.8 now implies that M~I(X ,n) is connected. [] Proof of Theorem 11.1. -- From the homeomorphism given by Proposition 7.8 and Theorem 7.18, the varieties M~(X, n) and MoR(X, n) are also connected. By Corollary 11.7, all the moduli varieties are normal, and a normal connected variety is irreducible. [] 78 CARLOS T. SIMPSON REFERENCES [Ar] M. ARTIN, Algebraic approximation of structures over complete local rings, Publ. Math. LH.E.S., 86 (1969), 23-58. [Be] J. BERSSTEm, Course on ~-modules, Harvard, 1983-1984. [ST] A. BOREL, J. Trrs, Eldments unipotents et sous-groupes paraboliques de groupes rdductifs I, Invent. Math., 12 (1971), 95-104. [Co] K. CORLEa"rE, Flat G-bundles with canonical metrics, J. Diff. Geom., B8 (1988), 361-382. [Del] P. DELIGNE, Equations diff~reutielles ~. points singuliers r~guliers, Lea. Notes in Math., 168, Springer, New York (1970). [De2] P. DgLm~r~, Letter, 1989. [DM] P. DgLmNE and J. MXL~E, Tannakian categories, in Leer. Notes in Math., 900, Springer (1982), 101-228. S. K. DONALDSON, Anti self dual Yang-MiUs connections over complex algebraic surfaces and stable vector [Doll bundles, Proc. London Math. Soc. (3), 50 (1985), 1-26. [Do2] S. K. DONALDSON, Infinite determinants, stable bundles, and curvature, Duke Math. J., 54 (1987), 231-247. [Do3] S. K. DONALDSON, Twisted harmonic maps and self-duality equations, Proc. London Math. Soc., 55 (1987), 127-131. D. GmSEm~R, On the moduli of vector bundles on an algebraic surface, Ann. of Math., 106 (1977), 45-60. [GI] [GM] W. GOLDMAN and J. MILLSON, The deformation theory of representations of fnndamental groups of compact Ifgdder manifolds, University of Maryland preprint (0000). [Grl] A. GROTHESDmCK, Eldments de gdorrdtrie algdbrique, Several volumes in Publ. Math. LH.E.S. [Gr2] A. GROTHENDmCK, Techniques de construction et thdor~mes &existence en gdomdtrie algdbrique, IV : Les schdmas de Hilbert, Sgminaire Bourbaki, Exposd 221, volume 1960-1961. [Gr3] A. GROTHEm~mCK, Crystals and the De Rham cohomology of schemes, Dix exposgs sur la cohomologie des scMmas, North-Holland, Amsterdam (1968). [GS] V. Guillemin, S. Sternberg, Birational equivalence in symplectic geometry, Invent. Math., 97 (1989), 485-522. R. HARTSHOR~E, Algebraic Geometry, Springer, New York (1977). [Ha] N.J. HITCam, The self-duality equations on a Riemann surface, Proc. London Math. Soy. (3), 55 (1987), [Hil] 59-126. N. J. HITCmN, Stable bundles and integrable systems, Duke Math. J., 54 (1987), 91-114. [Hi2] [KN] G. KEMPF, L. NRss, On the lengths of vectors in representation spaces, Leet. Notes in Math., "/82, Springer, Heidelberg (1982), 233-243. [Ki] F. KmWA.,% Cohomology of Quotients in Symplectic and Algebraic Geometr7 , Princeton Univ. Press, Princeton (1984). J. LE POTIER, Fibrds de Higgs et syst~mes locaux, Sgminaire Bourbaki 737 (1991). [Le] [Lu] D. LUNA, Slices &ales, Bull. Soe. Math. France, Mdmoire 83 (1973), 81-105. M. MARm~AMA, Moduli of stable sheaves, I : J. Math. Kyoto Univ., 17-1 (1977), 91-126; Ih Ibid., 18-3 [Mal] (1978), 557-614. [Ma2] M. MARUYAMA, On boundedness of families of torsion free sheaves, J. Math. Kyoto Univ., 21-4 (1981), 673-701. [Mt] MATSUSHIMA, See reference in Geometric Invariant Theory. V. B. MEHTA and A. RAMANATHAN, Semistable sheaves on projective varieties and their restriction to curves, [MRI] Math. Ann., 258 (1982), 213-224. V. B. MEHTA and A. RAMANATHAN, Restrlction of stable shcaves and representations of the fundamcntal [MR2] group, Invent. Math., 77 (1984), 163-172. V. V. MOROZOV, Proof of the regularity theorem (Russian), Usp. M. Nauk., XI (1956), 191-194. [Mo] D. MUMFORD, Geometric Invariant Theory, Springer Verlag, New York (1965). [Mu] M. S. NARASIMHAN and C. S. SESHADRI, Stable and unitary bundles on a compact Riemann surface, Ann. INS] of Math., 82 (1965), 540-564. N. Nivsua~, Moduli space of semistable pairs on a curve, Proc. London Math. Soc., 62 (1991), 275-300. [Nil] N. NiTsums, Moduli of semi-stable logarithmic connections, preprint (1991). [Ni2] MODULI OF REPRESENTATIONS. II 79 [No] M.V. NORI, On the representations of the fundamental group, Compositio Math., 33 (1976), 29-41. [Ox] W. M. OxBtn~v, Spectral curves of vector bundle endomorphisms, preprint, Kyoto University (1988). [Ru] W. RtrOiN, Real and Complex Analysis, Mac Craw-Hill, New York (1974). [Sa] N. SAAVEDaA RaVAZ~O, Cat6gories tannakiermes, Leer. Notes in Math., 9-65, Swinger (1972). [Sell C.S. SESHADRI, Space of unitary vector bundles on a compact Riemann surface, Ann. of Math., 85 (1967), 303-336. [Se2] C.S. SESHADRI, Mum.ford's conjecture for GL(2) and applications, Bombay Colloquium, Oxford University Press (1968), 347-371. [Sil] C. SIMPSON, Yang-Mills theory and uniformization, Lett. Math. Phys., 1~ (1987), 371-377. [Si2] C. SXMPSON, Constructing variations of Hodge structure using Yang-Mills theory and applications to unlformization, Journal of the A.M.S., 1 (1988), 867-918. [Si3] C. SIMPSON, Nonabelian Hodge theory, International Congress of Mathematicians, Kyoto 1990, Proceedings, Springer, Tokyo (1991), 747-756. [Si4] C. SIMPSON, A lower bound for the monodromy of ordinary differential equations, Analytic and Algebraic Geometo,, Tokyo 1990, Proceedings, Springer, Tokyo (1991), 198-230. [Si5] C. SIMPSON, Higgs bundles and local systems, Publ. Math. LH.E.S., 75 (1992), 5-95. [Uh] K.K. UItLENBECK, Connections with I_2 bounds on curvature, Commun. Math. Phys., 88 (1982), 31-42. [UY] K.K. UHL~NBECK and S. T. YAu, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure and Appl. Math., 39-S (1986), 257-293. Laboratoire de Topologie et Gfomdtrie URA 1408, CNRS UFR-MIG, Universit6 Paul-Sabatier 31062 Toulouse Cedex, France Manuscrit refu le 31 aotlt 1992. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

Moduli of representations of the fundamental group of a smooth projective variety. II

Publications mathématiques de l'IHÉS , Volume 80 (1) – Aug 30, 2007

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Springer Journals
Copyright
Copyright © 1994 by Publications Mathématiques de L’I.É.E.S.
Subject
Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
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0073-8301
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1618-1913
DOI
10.1007/BF02698895
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Abstract

MODULI OF REPRESENTATIONS OF THE FUNDAMENTAL GROUP OF A SMOOTH PROJECTIVE VARIETY. II by CARLOS T. SIMPSON Introduction This second part is devoted to the subject of the title, moduli spaces of represen- tations of the fundamental group of a smooth complex projective variety X. We study three moduli spaces for related objects. The Betti moduli space MR(X , n) is a coarse moduli space for rank n representations of the fundamental group of the usual topological space X t~ A vector bundle with integrable connection is a pair (E, V) where E is a vector bundle and V:E-+ E | f~ is an operator satisfying the Leibniz rule and V2-- - 0. The de Rham moduli space MDg(X, n) is a coarse moduli space for rank n vector bundles with integrable connection on X. A Higgs bundle [Hill [Si5] is a pair (E, ~) where E is a vector bundle and q~ : E --> E | f/~ is a morphism of 0x-modules such that q0 ^ q0 = 0. There is a condition of semistability analogous to that for vector bundles, but only concerning subsheaves preserved by ~. The Dolbeault moduli space MDot(X , n) is a coarse moduli space for rank n semlstabte Higgs bundles with Chern classes vanishing in rational cohomology. In all three cases, the objects in question form an abelian category in which we can apply the Jordan-H61der theorem. Let gr(E) denote the direct sum of the subquotients in a Jordan-H61der series for E, and say that E 1 is Jordan equivalent to E~ if gr(E1) _-__ gr(E~). The points of the coarse moduli spaces parametrize Jordan equivalence classes of objects. The constructions of these moduli spaces are reviewed in w 5. The construction of Mu is a classical one from the theory of representations of discrete groups. The cons- truction of MD~ follows from the construction of Part I, w 4, for the case where A "t~ = ~x is the full sheaf of rings of differential operators on X. We give two constructions of MDo ~. One is based on an interpretation of Higgs sheaves as coherent sheaves on T* X, and uses the construction of the moduli space of coherent sheaves constructed in Part I, w 1. The other consists of applying the general construction of Part I, w 4, to the case A"~ Sym'(TX). The three types of objects are related to each other. The Riemann-I-Iilbert corres- pondence between systems of ordinary differential equations and their monodromy repre- CARLOS T. SIMPSON sentations provides an equivalence of categories between vector bundles with integrable connection and representations of the fundamental group. To (E, V) corresponds the monodromy of the system of equations V(e) = 0. The correspondence between Higgs bundles and local systems of [Hil], [Do3], [Co], [Si2], and [Si5] gives an equivalence of categories between semistable Higgs bundles with vanishing rational Chern classes, and representations of the fundamental group. Together, these correspondences give isomorphisms of sets of points Ms(X , n) ~ MD~(X, n) ~ Mnol(X, n). In w 7 we use the analytic results of Part I, w 5, to show that the first map is an isomorphism of the associated complex analytic spaces, and that the second is a homeomorphism of usual topological spaces. There is a natural algebraic action of the groupe C" on the moduli space Mi)ol(X, n), given by t(E, ~0) = (E, t~0), and our identifications thus give a natural action--no longer algebraic--on the space of representations. The fixed points of this action are exactly those representations which come from complex variations of Hodge structure [Si5]. Although MDo 1 is not compact, the properness of Hitchin's map (Theorem 6.11) implies that MDoI(X, n) contains the limits of points tE as t-+ 0. This yields the conclusion that any representation of the fundamental group may be deformed to a complex variation of Hodge structure (Corollary 7.19 below). This theorem was in some sense the principal motivation for constructing the moduli spaces. See [Si5] for more details on some consequences. The reason for the terminologies Betti, de Rham and Dolbeault is that these modull spaces may be considered as the analogues for the first nonabelian cohomology, of the Betd cohomology, the algebraic de Rham cohomology, and the Dolbeault cohomology @~+q=~ H~(X,Y~) of X. The first nonabelian cohomology set Hi(X, Gl(n, C)) is the set of isomorphism classes of rank n representations of nt(X). This has a structure of topological space, but it is not Hausdorff. The universal Hausdorff space to which it maps is the Betti moduli space MB(X, n). To explain the analogies for the de Rham and Dolbeault spaces, we have to digress to discuss the (2ech realizations of the cohomology groups with complex coefficients. The algebraic de Rham cohomology is the hypercohomology of X with coefficients in the algebraic de Rham complex ~o ~ ~: ~ ... If X = LI U, is an affine open covering of X, and if we denote multiple intersections by multiple indices, then the cocycles defining H]~(X, C) consist of the pairs of collections ({g,~}, {a~}) where g,o are regular functions U,~ -+ C, and a, are one-forms on U~, such that g~ = g~v + g,,,t, d(g,,f~) = a= -- a~, and d(a~,) = O. MODULI OF REPRESENTATIONS. II Addition of the coboundary of a collection { s~ ), where s~ are regular functions U, ~ C, changes the pair ({ g~ }, { a~ }) to ({ g~ q- s~ -- s o }, { a~ q- d(s~) }). The group of cocycles modulo coboundaries is H~R(X , C). The nonabelian case has formulas which are more complicated, but which reduce to the above if the coefficient group is abelien. A vector bundle with integrable connection is defined by a pair ({ g~ }, { A, }) , where g~:U,~ ~ Gl(n, C) are the gluing functions for the vector bundle, and A s are n � n matrix-valued one forms defining the connection V = d -k- As. These are subject to the conditions g~v g~ ~ g~v, A~ = g~l d(g~) + g~l A~ g~, and d(A~) + A s ^ A~ ~ 0. A change of local frames by a collection of regular functions s~ : U~ -~ Gl(n, C) changes the pair ({g~),{A~}) to ({s~Ig~s~},{sZ ~A~s~ + sZ x d(s~)}). The set of pairs up to equivalence given by such changes of frames, is the first nonabelian de Rham cohomo- logy set H~(X, Gl(n, C)), the set of isomorphism classes of vector bundles with integrable connection on X. A similar if somewhat looser interpretation gives an analogy between the abelian Dolbeault cohomology group Hi(X, 0x) | H~ ~) and the first nonabelian Dolbeault cohomology set Hx(X, Gl(n, C)), the set of isomorphism classes of Higgs bundles (E, ~) which are semistable with vanishing rational Chern classes. Here E is a vector bundle and ~ ~ H~ End(E) |162 ~)" Such a pair may be given a cocycle description similar to the above (just eliminate the terms involving d). The conditions of semistability and vanishing Chern classes are new. Following this interpretation, we can think of the first nonabelian cohomology as a nonabelian motive in a way analogous to [DM], with its Betti, de Rham and Dolbeault realizations. It would be good to have t-adic, and crystalline interpretations in charac- teristic p. We treat everything in the relative case of a smooth projective morphism X ~ S to a base scheme of finite type over G. This creates some difficulties for the Betti moduli spaces: we have to introduce the notion of local system Od e schemes over a topological space. The relative Betti spaces lVlB(X/S, n) are local systems of schemes over S t~ The de Rham and Dolbeault moduli spaces are schemes over S, whose fibers are the de Rham and Dolbeault moduli spaces for the fibers X~. The interpretation in terms of nonabelian cohomology suggests the existence of a Gauss-Manin connection on NIDR(X/S , n), a foliation transverse to the fibers which when integrated gives the transport corresponding to the local system of complex analytic spaces 1VI"~(X/S, n). We construct this connection in w 8, using Grothendieck's idea of the crystalline site. In w 9, we treat the case of other coefficient groups. If G is a reductive algebraic group, we may define the Betti moduli space bl~(X, G) to be the coarse moduli space CARLOS T. SIMPSON for representations of rq(X) in G. We construct the de Rham moduli space MDg(X , G) for principal G-bundles with integrable connection, and the Dolbeault moduli space Mml(X , G) for principal Higgs bundles for the group G, wtfich are semistable with vanishing rational Chern classes, and extend the results of w167 7 and 8 to these cases. One corollary is a result valid for representations of any finitely generated discrete group T: if G -+ H is a morphism of reductive algebraic groups with finite kernel, then the resulting morphism of coarse moduli spaces M(Y, G) -+ NI(Y, H) is finite (Corol- lary 9.16). Parallel to the discussion of moduli spaces, we discuss the Betti, de Rham and Dolbeault representation spaces RB(X, x, n), RoR(X, x, n), and RDoI(X, x, n). These are fine moduli spaces for objects provided with a frame for the fiber over a base point x ~ X (here we assume that X is connected). There are relative versions for X/S where the frames are taken along a section ~ : S ~ X, and there are versions for principal objects for any linear algebraic group. In w 10, we discuss the local structure of the singularities of the representation spaces, using the deformation theory associated to a differential graded Lie algebra developed by Goldman and Millson [GM]. By Luna's 6tale slice theorem, this also gives information about the local structure of the moduli spaces. The differential graded Lie algebra controlling the deformation theory of a principal vector bundle with integrable connection or a principal Higgs bundle is formal if the object is reductive (in other words, corresponds to a closed orbit under the action of G on R(X, x, G)). By the theory of [GM], this implies that the representation space has a singularity defined by a quadratic form on its Zariski tangent space. Furthermore, the differential graded Lie algebras controling the deformation theories of the flat bundle and the corresponding Higgs bundle are the same. This gives a formal isomorphism between the singularities of the de Rham (or Betti) representation space and the singu- larities of the Dolbeault representation space, at semisimple points which correspond to each other. This isosingularity principle holds also for the singularities of the moduli spaces. The homeomorphism between Mx)r: and MDo , is not complex analytic, so these local formal isomorphisms are not directly related to the global homeomorphism. Finally, in w 11 we discuss the case of representations of the fundamental group of a Riemann surface of genus g/> 2. Hitchin calculated the cohomology in the case of rank two projective representations of odd degree, where the moduli space is smooth [Hil]. We do not attempt to go any further in this direction. We simply treat the most elementary property, irreducibility (which is more or less a calculation of H~ We treat the case of representations of degree zero, so the moduli space has singularities corresponding to reducible representations; we prove that the singularities are normal. The technique is to use the fact that the Betti moduli space is a complete intersection, and apply Serre's criterion (following a suggestion of M. Larsen, and prompted by a question of E. Witten). We have to verify that there are no singularities in codimension one. To prove irredu- cibillty it suffices to prove that the space is connected, which we do by a simple argument derived from Hitchin's method for calculating the cohomology. MODULI OF REPRESENTATIONS. II Relationship with part I It is worth reiterating the nature of the connections between this second part and Part I of the paper. The sections are numbered globally, so we begin with w 6. References to lemmas and such, numbered for sections 1-5, are references to the statements in " Moduli of representations of the fundamental group of a smooth projective variety I " We rely on the technical work done in part I for many of the constructions of moduli spaces, identifications, and criteria for convergence used here. For the most part, we apply statements from the first part, so Part II can be read without having read Part I in a very detailed way, but just having a copy at hand for reference. Origins The correspondence between Higgs bundles and local systems, reflected in the homeomorphism between the Dolbeault and the de Rham or Betti moduli spaces, comes from work of Hitch.in [Hill, Colette [Co] and Donaldson [Do3], as well as [Si2]. The formalism of this correspondence is developed in [Si5]. The original correspondence of this type was the result of Narasimhan and Seshadri [NS] between unitary representations and stable vector bundles. This was subsequently generalized by Donaldson IDol] [Do2], Mehta and Ramanathan [MR1] [MR2], and Uhlenbeck and Yau [UY]. The idea of obtaining a correspondence between all representations into a non- compact group, and vector bundles provided with the additional structure of a Higgs field, comes from Hitchin's paper [Hil] with the appendix [Do3] of Donaldson. Hitchin established the correspondence between rank two Higgs bundles and rank two repre- sentations on a Riemann surface (and his arguments are easily extended to any rank). Independently, I had arrived at a correspondence between certain representations with noncompact structure group (the complex variations of Hodge structure), and certain holomorphic objects involving an endomorphism valued one-form (systems of Hodge bundles) [Sil]. Deligne and Beilinson had also arrived at a correspondence between systems of Hodge bundles and variations of Hodge structure over a Riemann surface (unpublished work). My definitions and very first results were independent of those of Deli_nge and Beilinson, then Deligne explained their work and made some important suggestions. I didn't see, until W. Goldman directed me to Hitchin's paper which had just appeared, that one could hope to obtain a correspondence involving all representations. In light of Hitchin's definition, systems of Hodge bundles could be seen as special types of Higgs bundles, and my arguments in higher dimensions could be generalized to the case of Higgs bundles [Si2]. This provided one direction of the correspondence. The other direction (corresponding to Donaldson's appendix to Hitchin's paper) was provided by the results on equivariant harmonic maps and the Bochner formula obtained by Colette in his thesis [Co]. 2 10 CARLOS T. SIMPSON This correspondence provided the motivation for the construction of the moduli space of Higgs bundles. Hitchin gave an analytic construction in his paper, and he obtained all of the interesting properties, such as the properness of the map given by taking the characteristic polynomial of the Higgs field. In my thesis, I had constructed a moduli space for systems of Hodge bundles, using Mumford's construction for vector bundles on curves. The construction presented in part I grew out of this, but uses methods of geometric invariant theory more suited to higher dimensions, as pioneered by Gieseker [Gi] and Maruyama [Mal] [Ma2]. See the introduction of Part I for further details. When I first discussed this with him, Hitchin advised me that Nitsure had given an algebraic construction for Higgs bundles over a curve [Nil]. Early on, while I was looking at systems of Hodge bundles, J. Bernstein made the comment that a system of Hodge bundles could be considered as a sheaf on the cotangent bundle of the variety. This remark, generalized to the case of Higgs bundles, forms the basis for one of the constructions of the moduli space of Higgs bundles presented inw The discussion of the Gauss-Manin connection in w 8 was prompted by a discussion with S. Mochizuki, wherein he pointed out that the analytic connection provided by the Betti trivialisation of M~R(X/S, n) over S, was not a priori algebraic. The methods used to prove algebraicity are the crystalline methods envisioned by Grothendieck in connection with his construction of the Gauss-Martin connection for abelian cohomo- logy [Gr3]. The existence of the Gauss-Man.in connection was announced in [Si3], and a brief sketch of the proof was given. The material in w 10 about deformation theory is an easy extension of the work of Goldman and Millson [GM ]. Their work was, in turn, based on a deformation theory developed by Schlessinger, Stasheff and Deligne. The proof of irreducibility in w 11 was motivated by an old question posed to me by J. Bloch, and made possible by Hitchin's method of using Morse theory or the C ~ action to calculate the topology of the moduli space (which we use just to show connectedness). My original proof contained a long and technical part showing that the singularities were locally irreducible. E. Witten later posed the question of whether the singularities were normal, and M. Larsen helped by directing me to the place in [Ha] explaining how to use Serre's criterion to prove normality of a complete intersection. The only technical part now needed is an inductive verification that the singularities are in codimension at least two, which makes the argument much shorter. Acknowledgements I would like to reiterate the acknowledgements of Part I in what concerns this second half of the paper. I would particularly like to thank J. Bernstein, J. Carlson, K. Corlette, P. Deligne, S. Donaldson, W. Goldman, P. Griffiths, N. Hitchin, M. Larsen, G. Laumon,J. Le Potier, M. Maruyama, S. Mochizuki, N. Nitsure, W. Schmid, K. Uhlen- MODULI OF REPRESENTATIONS. II 11 beck, E. Witten, K. Yokogawa, and A. Yukie for many helpful discussions about their work and the present work. The early versions were full of mistakes, pointed out by many of these people on innumerable occasions. I am most greatful for this help, sorry that they had to take the time to worry, and hope that in this last revision, I haven't introduced too many new ones. 6. Moduli spaces for representations The Betti moduli spaces We begin with a classical construction from the theory of spaces of representations of discrete groups. Suppose r is a finitely generated group. Fix n. Put R(F, n) ---- Horn(F, Gl(n, C)). It is a scheme over Spec(C) representing the functor which to a C-scheme S associates the set Hom(F, Gl(n, H~ Os))). The scheme R(F, n) can be constructed by choosing generators Y1, ..., Yk for F. Let Rel denote the set of relations among the y~. Then R(F, n) C Gl(n, C) � ... � Gl(n, C) (k times) is the closed subset defined by the equations r(mx, ..., m~) = 1 for r e Rel. It is easy to see that this subset represents the required functor--a representation 0 : I" -+ Gl(n) corresponds to the point (ml, ..., ink) with m s = O('f~). Note that R(F, n) is a closed subset of an affine variety, so it is affine. The group Gl(n, C) acts on R(F, n) by simultaneous conjugation of the matrices. The orbits under this action are the isomorphism classes of representations. Two representations 9 and p' are said to be Jordan equivalent if there exist compo- sition series for each such that the associated graded representations are isomorphic. The theorem of Jordan-H61der says that the associated graded doesn't depend on the choice of composition series; this semisimple representation is an invariant of the repre- sentation, known as its semisimplification. Proposition 6.1. -- There exists a universal categorical quotient R(F, n) -+ M(F, n) by the action of Gl(n, C). The scheme 1VI(F, n) is an affine scheme of finite type over C. The closed points of M(F, n) represent the Jordan equivalence classes of representations. Proof. -- This is well known. The quotient is constructed by taking the coordinate ring A = H~ n), On(r. J, setting B- A ~ to be the subring of invariants, and putting M(F, n) = Spec(B). Hilbert proved that B is finitely generated. Mumford shows in [Mu] that Spec(B) is a universal categorical quotient of Spec(A) ----R(F, n). Finally, Seshadri shows that the closed points of the quotient are in one to one corres- pondence with the closed orbits [Se]. The closed orbits are the orbits corresponding 12 CARLOS T. SIMPSON to semisimple representations, and the closed orbit in the closure of a given orbit is the one corresponding to the semisimplification of the given representation. [] Suppose X is a conected smooth projective variety over Spec(C). Choose x e X and let F = nl(X '~, x). We will use the notation an(x, ~, ~) ~=~ R(r, n) and call this the Betti representation space; and the notation MB(X, n) d~ M(r, n) calling this the Betti moduli space. This terminology is suggested by the terminology of [DM]. The space Mn(X, n) does not depend on the choice of x. More precisely, ff we include the choice of x in the notation then there are canonical isomorphisms "~(x,y) : Mn(X, x, n) - MB(X,y, n) such that v(y, z) v(x,y) = v(x, z), given as follows. We may choose a path from x toy, giving ~I(X '~, x) - ~l(X'~,y) and hence Rn(X , x, n) ~ RB(X,y, n). This isomorphism is compatible with the action of Gl(n, C) so it descends to the desired "~(x,y). Choice of a different path gives a different isomorphism of representation spaces which differs by the action of a section g : Rn(X,y, n) -+ Gl(n, C). By the definition of quotient, the two natural maps RB(X,y , n) X Gl(n, C) ~ Mn(X,y, n) are equal. Hence the two maps from the graph ofg to M~(X,y, n) are the same. Thus the two maps from Rn(X, x, n) to Mn(X,y, n) are the same, so the two isomorphisms v(x,y) are the same. Thus .r is canonically defined; and this independence of the choice of path implies the formula v(y, z) .~(x,y) = "r(x, z). We will identify the spaces obtained from different choices of base point, and drop the base point from the notation for Mn(X, n). Local systems of schemes Suppose T is a topological space. A local system of schemes Z over T is a functor from the category of C-schemes to the category of sheaves of sets over T, denoted (backward) Z : (S E Sch, U C T) ~ Z(U)(S), such that: there exists a covering by open sets T = I.J, U~ such that for any open set V contained in one of the U~, the functor S ~ Z(V)(S) is represented by a scheme Z(V) over 13; and such that if W C V are connected open sets contained in one of the U~ MODUI.I OF REPRESENTATIONS. II then the restriction map Z(V) -~ Z(W) is an isomorphism (note that the restriction morphisms of functors are automatically morphisms of schemes). Choose a point t ~ T. The stalk Z, lim Z(V) t~v is a scheme. Lemma 6.2. -- If Z is a local system of schemes on T, the group ~I(T, t) acls on the stalk Z t by C-scheme automorphisms. If T is connected and locally simply connected, the construction Z ~ Z t is an equivalence between the category of local systems of schemes over T and the category of schemes with action of rq(T, t). Proof. -- Suppose Z is a local system of schemes over T. Let T =- U, U~ be an open covering as in the definition. We may suppose that the U, are connected. We have schemes Z(U~). If v ~ U~ t~ U~ then we have isomorphisms of schemes z(u ) z~ If v ~ U~ n U~ c~ U v then the resulting hexagon commutes. If ~ : [0, 1] --> T is a path with ~(0) ----~(1) =t then we may choose O=s o<s*< ...<s,= 1~[0,1] and ~o, ...,~ such that s,+,]) c We obtain Zoo,0 --- Z(U,i ) ~ Zoc,i § 11" Putting these isomorphisms together we get an isomorphism of schemes Z t "~ Z t . One can check that a homotopic path ~' ,-~ ~ gives the same isomorphism, so we get an action of r~l(T , t) on Z t. Suppose T is connected and locally simply connected, so the universal covering exists. Given a scheme Z t with action of rq(T, t), form the constant local system of schemes Z over the universal covering ~, whose fiber at the base point t is Z t (the sheaf is given by the rule Z(U)(S) = Zt(S ) for connected open sets U; the restriction maps are the identity). The group of covering transformations Aut(~/T)= ~I(T, t) acts on ~. over its action on ~, by the given action on Zt. Now define Z(U)(S) to be the set ofinvariants in Z(U)(S), where U is the inverse image of U in the universal covering. This gives a local system of schemes Z over T. This construction is the inverse of the previous construction. [] We can make a similar definition of local system of complex analytic spaces over a topological space T, and the analogue of the previous lemma still holds. If Z is a local system of schemes then we obtain a corresponding local system of complex analytic spaces Z ~. The stalk Z~" is the complex analytic space associated to Z~. If T is a complex analytic space and Z is a local system of schemes over T then we denote by Z ~ the total analytic space over T constructed as follows. Choose an open a,_=_r 14 CARLOS T. SIMPSON coveting T = O, T~ such that the restriction of Z to T, is a constant local system with stalk Z,. For each connected component of T, n Tj there is an isomorphism Z~ -- Z~., satisfying a compatibility relation for triples of indices. Let Z (~n~ be the space obtained by glueing together the complex analytic spaces Z~ ~ � T, using these isomorphisms. The Betti moduli spaces in the relative case Suppose that f: X -+ S is a smooth projective morphism to a scheme S of finite type over C. Suppose that S and the fibers X, are connected. The associated map of complex analytic spaces fan is a fibration of the underlying topological spaces. Choose base points t ~ S and x ~ X,. Let P = ~I(X~ n, x). Let Aut(P) denote the group of auto- morphisms of P; Inn(P) C Aut(r) the image of the natural map Ad : r -+ Aut(r) (which sends y to the inner automorphism Ad(y)(g) = ygy-1); and Out(F) = Aut(P)/Inn(P). The group 7r1(S =, t) acts on P by outer automorphisms, in other words there is a map ,:l(S =, t) -, Out(r). This may be defined as follows: if ~: [0, 1] -+ S = is a loop representing an element of 7rx(S =, t), then the pullback a*(X an) is a fibration over [0, 1]; it is trivial, so we obtain a homeomorphism X~ n-~ X~ n between the fibers over 0 and 1; this gives a map ~zl(X~, x) - rrl(X~,y ) for some other point y; finally, choose a path joining x and y, to get an automorphism of r ---= nl(X~, x)--which is well defined independent of the choice, up to inner automorphism. The resulting outer automorphism is independent of the homotopy class of the path ~. The group Aut(r) acts on the representation space R(F, n). This descends to an action of Out(r) on the moduli space M(F, n), as inner automorphisms act on the representation space through functions R(F, n) ~ Gl(n, C) and hence trivially on the moduli space. In our case, we have denoted M(F, n) by Mr(Xt, n). Composing this action with the action of nl(S ~n, t) on F, we obtain an action of ~zl(S "n, t) on MB(Xt, n) by C-scheme automorphisms. From Lemma 6.2, we obtain a local system of schemes MB(X/S , n) over the topological space underlying S an. The relative version of the Betti moduli space is this local system of schemes MB(X/S , n). It is independent of the choice of base points t and x. The stalk over s e S is MB(X/S , n)~ = M~(X,, n). Suppose ~:S-+X is a section. Then rrl(S ~, t) acts on 7h(X,, ~(t)) by auto- morphisms. We obtain a local system of schemes RB(X]S , ~, n), which is again inde- pendent of the choice of t. The stalk over s e S is RB(X/S, ~, n), = RB(X,, ~(s) n). The constructions MB(X/S , n) and RB(X/S , ~, n) may be extended to the case of non- connected base S by taking the disjoint union of the spaces over each connected com- MODULI OF REPRESENTATIONS. II 15 ponent of S. The construction Ms(X/S , n) may also be extended to the case where the fibers are not connected. If s ~ S and if X s = X 1 u ... t3 X k is the decomposition of the fiber into connected components, then MB(X/S, n), = M~(X1, n) � ... � M~(Xk, n). The action of ~1(S ~n, s) permutes the factors in the product appropriately. We close the discussion of the Betti moduli spaces by giving the universal and co-universal properties they satisfy. Proposition 6.3. -- Suppose f: X ~ S is smooth and projective with connected fibers, and suppose that ~ : S ~ X is a section. Then for any scheme Y and any open set U C S ~ the set RB(X/S, ~, n)(U)(Y) is equal to the set of isomorphism classes of pairs (L, ~) where L bs a locally constant sheaf of H~ Oy)-modules on f-l(U) and ~ : ~-I(L) ~ H~ 0y)". Proof. -- It suffices to prove this for small open sets U, for example connected open sets over which the topological fibration (X ~n � U, ~) is trivial. In this case, choose s e U. A locally constant sheaf L of H~ d~y)-modules on f-I(U) together with a fram ~ is the same thing as a representation of nl(Xs, ~(s)) in Gl(n, H~ 0y)), hence the same thing as a morphism Y --~ R~(X,, ~(s), n). This is the set of Y-valued points of the local system of schemes over the set U, since the local system of schemes is trivial and U is connected. [] Proposition 6.4. -- Suppose f: X ~ S is a smooth projective morphism. Let M~(X/S, n) denote the functor from C-schemes to sheaves of sets over S ~ which associates to each scheme Y and each open set U C S ~" the set of isomorphism classes of locally constant sheaves of free H~ 0y)- modules of rank n on f-X(U). There is a map of functors from M~(X/S, n) to Ms(X/S , n). If Z is any local system of schemes over S ~" with a natural transformation of functors M~(X/S, n) -+ Z, there is a unique factorization through a map M~(X/S, n) -+ Z. Proof. -- This is a translation to the case of local systems of schemes, of the property that the fiber Ms(X,, n) universally co-represents the functor M~(X,, n). [] Moduli of Higgs bundles Suppose that f: X ~ S is a smooth projective morphism to a scheme of finite type over C. A Higgs sheaf on X over S is a coherent sheaf E on X together with a holomorphic map q~ : E -+ E | f~/s such that q~ ^ q0 ---= 0. Similarly, a Higgs bundle is a Higgs sheaf (E, q~) such that E is a locally free sheaf. Higgs bundles on curves were introduced by Hitchin in [Hi2] and [Hill. The condition ~ ^ q0 = 0 for higher dimensional varieties was introduced in [Si2] and [Si5]. Hitchin gave an analytic construction of the moduli space [Hill (this part of his argu- ment works for any rank). Nitsure gave an algebraic construction of the moduli space of Higgs bundles over a curve [Nil]. 16 CARLOS T. SIMPSON We give two constructions of the moduli space of Higgs bundles, based on two different interpretations. The first is simply to note that a Higgs bundle is a A-module for an appropriate sheaf of rings A. This does not give too much other information, and is based on all of Part 1. The second construction uses only the moduli space of coherent sheaves ofw 1, and it gives some additional information about the moduli space: the properness of Hitchin's map. Lemma 6.5. -- Let A m"~ = Sym'(T(X/S)). Then a I-[iggs sheaf on X over S is the same thing as an Ox-coherent Aaig~-module on X. Proof. -- This follows from the discussion of split almost polynomial rings A at the end ofw 2, Part I (Lemma 2.13). In this case it is easy to see that an action of the symmetric algebra on a sheaf E is the same thing as a map q~ : E ~ E | T'(X/S) such that ~^~=0. [] Fix a relatively very ample Ox(1 ). Define the notions of pure dimension, p-semi- stability, p-stability, ~t-semistability and ~t-stability for Higgs sheaves to be the same as the corresponding notions for Am~"-modules. These coincide with the notions defined in [Si5] for the case when S = Spec(C) (pure dimension d = dim(X) is the same thing as torsion-free). Recall that in the relative case, the conditions of semistability and stability contain the hypothesis that the sheaf is flat over the base S. Let M~g=(X/S, P) denote the functor which associates to an S-scheme S' the set of isomorphism classes ofp-semistable Higgs sheaves E on X' over S' with Hi]bert polynomial P. This is univer- sally co-represented by the moduli space M~,,~(X/S, P) d~ M(Ar,,g~, p) constructed in Theorem 4.7, Part I. The points of MH,~(X/S, P) parametrize Jordan equivalence classes of p-semistable Higgs sheaves with Hilbert polynomial P on the fibers X s. If P has degree dim(X), then these are the same as torsion-free p-semistable Higgs sheaves which were discussed in [Si5]. Let P0 denote the HUbert polynomial of 0 x. Let M~,(X/S, n) denote the functor which to an S-scheme S' associates the set of isomorphism classes ofp-semistable Higgs sheaves E on X' over S' of Hilbert polynomial nP0, such that the Chern classes c~(E~) vanish in H2~(Xs, C) for all closed points s ~ S'. In general, if f: X --~ S is a smooth projective morphism and E is a coherent sheaf on X which is flat over S, then the Chern classes c~(E) are sections of the relative algebraic de Rham cohomology R~*f.(~x/s, d) which are flat with respect to the Gauss-Manin connection. Thus the condition that the Chern classes q(E,)~ H2~(X~, C) vanish depends only on the connected component of S containing s. The functor M~I(X/S , n) is universally corepresented by a scheme MDoI(X/S , n) which is a disjoint union of some of the connected components of Mm~(X[S, nP0) (the fact that MDo1(X/S , n) may be a proper subset of Mmgn(X/S, nP0) was pointed out to me by J. Le Potier [Le]). The points of M~I(X/S, n) correspond to Jordan equivalence classes ofp-semistable torsion-free Higgs sheaves of rank n on the fibers Xs, MODULI OF REPRESENTATIONS. II with Chern classes vanishing in the complex valued (or equivalently, rational) cohomology of X,. We call MDol(X/S , n) the DoIbeault moduli space. There is an open set Mgol(X/S , n) parametrizing p-stable Higgs sheaves, and there a universal family exists 6tale locally. Proposition 6.6. -- Suppose X is a smooth projective variety over S = Spec(C). IrE is a v-semistable torsion free Higgs sheaf with Chern classes equal to zero, then E is a bundle, and is in fact an extension of v-stable Higgs bundles whose Chern classes vanish. Any sub-Higgs sheaf of degree zero is a strict subbundle with vanishing Chern classes. Proof. -- [Si5] Theorem 2. [] Corollary 6.7. -- If X is smooth and projective over a base S, if S' is an S-scheme, and if E is an element of M~ol(X/S , n) (S'), then E is locally free over X'. The points of Mgo,(X/S , n) correspond to direct sums of v-stable Higgs bundles with vanishing rational Chern classes on the fibers X a . Proof. -- This follows from the previous proposition and Lemma 1.27, Part I. [] Remark. -- For Higgs sheaves with vanishing Chern classes, p-semistability (resp. p-stabiIity) is equivalent to ~-semistability (resp. ~-stability). This follows from Pro- position 6.6. Suppose X is smooth and projective over S. A Higgs bundle E on X (flat over S) is of semiharmonic type if the restrictions to the fibers E, are semistable Higgs bundles with vanishing rational Chern classes. Say that E, is of harmonic type if it is a direct sum of stable Higgs bundles with vanishing rational Chern classes. The Higgs bundles of semiharmonic type are those which correspond to representations of the fundamental group in [Si5]. Those of harmonic type correspond to semisimple representations. The closed points of MDo~(X/S, n) parametrize the Higgs bundles of harmonic type of rank n. Suppose X-+ S is a smooth projective morphism with connected fibers, and suppose ~ : S -+ X is a section. Let R(A ~g~, ~, nP0) denote the representation space for framed Amg~-modules constructed in Theorem 4.10, Part I. By Proposition 6.6, allp-semistable Higgs sheaves with vanishing rational Chern classes satisfy condition LF (X) and hence condition LF(~). Let RDol(X/S , ~, n) denote the disjoint union of those connected components of R(A mg~, ~, nP0) corresponding to Higgs sheaves with vanishing rational Chern classes. Then RDol(X/S , ~, n) represents the functor which associates to an S-scheme S' the set of isomorphism classes of pairs (E, ~) where E is a Higgs bundle of semiharmonic type on X' over S' and ~ : ~*(E) -~ 0~, is a frame. We call this scheme the Dolbeault representation space. The C* action Recall that an action of an algebraic group G on a scheme Z is a morphism G � Z ~ Z satisfying the usual axioms for a group action, with the axioms written in 18 CARLOS T. SIMPSON terms of diagrams of morphisms: the two maps G x G x Z ~ Z are the same (asso- ciativity); and the map Z -7 Z induced by the identity element e ~ G is the identity. We can define similarly the notion of an action on a functor Y~: this is a natural trans- formation offunctors G x Y~ ~ Y~ satisfying the same axioms. IfG acts on a functor Y~, and q~:Y~ -+ Y is a natural transformation so that the scheme Y universally corepre- sents Y~, then there is a unique action of G on the scheme Y which is compatible with ~. The morphism G x Y ~ Y is obtained from the natural transformation of functors G x Y~ ~ Y by applying the universality hypothesis, that G x Y corepresents the functor (G � Y) � YI = G � Y". The axioms are checked using the uniqueness part of the definition of universally co-representing a functor. The algebraic group C" acts on the functor M~g~(X/S, P) in the following way. If S' is an S-scheme, t : S' -~ C* is an S'-valued point, and (E, ~0) e M~I,~(X/S , P) (S') is a p-semistable Higgs sheaf with HUbert polynomial P on X' over S', then (E, tq~) is again an element of M~I~gB(X/S, P)(S') (the property of p-semistability is preserved because the subsheaves preserved by tq~ are the same as those preserved by q~). We obtain a morphism of functors giving the group action. By the above discussion, there is a unique compatible action of C" on Mm,,~(X/S, P). This gives an action of t3" on MDoI(X/S, n). Similarly, if the fibers X, are connected and ~ is a section, the formula t((E, 9), ~) = ((E, tg) , ~) gives an action of C* on the Dolbeault representation space RDoI(X/S, ~, n). This commutes with the action of Gl(n, C) and the good quotient RgoI(X/S, ~, n) ~ M,ol(X/S, n) is compatible with the action of C'. The subspace MDol(X/S, n) c~ of points fixed by C" is a closed subvariety. Alge- braically the structure of a lliggs bundle on X, fixed by C* is the following (cf. [Si5] Lemma 4.1). If (E, q~) ~ (E, tq~) for some t e C* which is not a root of unity, letf be the isomorphism. By appropriately combining the generalized eigenspaces off, we get a decomposition E = @ E ~ such that q~ : E ~ -+ E ~ -~ | ~)~x. By the analytic results of [Si5], the points of MDot(X/S, n) c" correspond to Higgs bundles which come from complex variations of Hodge structure. The second construction The idea behind this construction is that a Higgs bundle on X can be thought of as a coherent sheaf 8 on the relative cotangent bundle T*(X/S). Let Z denote a projective completion of T*(X/S), and let D = Z -- T*(X/S) be the divisor at infinity. Choose Z so that the projection extends to a map z~ : Z --> X. Lemma 6.8. -- A Higgs sheaf E on X over S is the same thing as a coherent sheaf g on Z such that supp(E) c~ D = O. This identification is compatible with morphisms, giving an equi- valence of categories. The conditions of flatness over S are the same. For s ~ S, the condition that E, is torsion-free is the same as the condition that g, is of pure dimension d = dim(X,). MODULI OF REPRESENTATIONS, II 19 Proof. -- The projection ~ : T*(X/S) ~ X is an affine morphism, in other words T*(X/S) is the sheafified spectrum of the sheaf of tings ~. 0T.IX!.~ on X. This sheaf of tings is naturally isomorphic to the symmetric algebra on the tangent bundle Sym" T(X]S), so giving a quasicoherent sheaf 8 on T*(X/S) it is equivalent to giving a quasicoherent sheaf E = re. 8 on X together with an action ofSym" T(X/S). But Sym" T(X/S) --=- A ~l~, so by the discussion in w 2, this action is the same as the data of a map r : E ~ E | f~ such that q~ ^ ~ = 0. A coherent Higgs sheaf E is the same thing as a sheaf 8 on T'(X/S) such that ~, 8 is coherent. This condition of coherence means that 8 is coherent on T*(X/S) and the closure of the support of 8 in Z does not meet the divisor at infinity D. A morphism of coherent sheaves d ~ -* o~- is the same thing as a morphism re.(8) -~ ,~.(~-) compatible with the action of the symmetric algebra, or equivalently compatible with q~. Since ~ !~.lx!sl is an affine map, flatness of 8 over S is equivalent to flatness of n.(8) over S. Finally note that the dimension of support of any subsheaf of E is the same on X, as it is on Z,, because of the condition that the support doesn't meet D,. Therefore the conditions of pure dimension d = dim(X,) on X, and Z, are the same. On X,, the condition that a sheaf has pure dimension d = dim(X,) is the same as the condition that it is torsion free. [] Choose k so that 0z(1 ) aej re* ~)x(k)|162 0z(I)) is ample on Z (here we suppose that Z is the standard completion of the cotangent bundle to a projective space bundle). In particular, OT.~XtS~(1) = r~" 0x(k ). Thus, for any coherent sheaf o ~ on Z with support not meeting D, the Hilbert polynomials of 8 and ~. ~ differ by scaling: p( 8, m) = p(~. 8, km). Corollary 6.9. -- The notions of p-semistability, p-stability, ~-semistability, and ~-stability I-[iggs sheaf E on X over S are the same as the corresponding notions for the coherent sheaf 8 for a associated to E in the previous lemma. on Z Proof. -- The sub-Higgs sheaves of E correspond to the coherent subsheaves of ~', since a subsheaf of 8 is the same thing as a subsheaf of re. 8 preserved by the action of the symmetric algebra. Scaling the Hilbert polynomials preserves the ordering and scales the slope. [] Fix a polynomial P of degree d = dim(X/S), and put k" P(m) = P(km). Fix a large N as required by the constructions of w 1 for sheaves on Z. Put ~ ----- 0z(-- N) and V = C k'cPcs~, and let Hilb(V | k ~ P) denote the Hilbert scheme of quotients V| on Z, flat over S, with Hilbert polynomial k'P. Let Q x and Q2 denote the subsets defined in w 1 (not those defined in w 3), and let Q3cQ2 denote the open subset parametrizing quotient sheaves 8 whose support does not meet D. By Theorem 1.19, Part I, and [Mu], a good quotient M(Oz, k" P) = Q2]SI(V) exists. The open set Q3 is Sl(V)-invariant and is set-theoretically the inverse image of a subset of M(Oz, k" P) (since the support of 8, is the same as the support of gr(8,)). Therefore a good quotient Q3/SI(V) exists and it is equal to an open subset which we denote 20 CARLOS T. SIMPSON M(dYT.~x/s~, k* P) of M(dYz, k" P). Theorem 1.21, Part I, Lemma 6.8, and Corollary 6.9 imply that M(Oz.~x/s~, k* P) universally co-represents the functor M~m~(X/S, P), and we have all of the properties of Theorem 1.21, Part I. We may put Mmg,~(X/S , P) = M(d~T.,X/S,, k" P). Define the subset MDo,(X/S, n)C Mmgg.(X]S , nPo) as before. These moduli spaces are the same as those constructed previously, because they co-represent the same functors. Hitch|n~s proper map We will now define a map t?om the space of Higgs bundles to the space of possible characteristic polynomials for 0. This map is the generalization of the determinant map that Hitchin studied in [Hil]. In Hitchin's case it turned out that this map was proper ([Hil] Theorem 8.1), and we will prove the same here also. Roughly speaking this means that the only way for a Higgs bundle to " go to infinity" is for the characteristic polynomial to become singular. For any n let ~e'(X/S, n) ~ S be the scheme representing the functor which to an S-scheme S' associates + H~ ', Sym' fit.;s, ) i=l [Grl] [Mu]. We consider the points of ~e'(X/S,n) as polynomials written t"+a it "-1 + ... +a, with a i~H~ ',Sym ~ 1 ~x'/s')" Let ~1, ..., % denote the symmetric polynomials in an r � r matrix variable A such that det(t -- A) = t ~ + al(A) t "-1 + ... + ~,(A). For example, al(A) = -- Tr(A) and %(A) = (-- 1)" det(A). Let P be a polynomial of degree d = dim(X/S) and rank r = deg(X)n, so that sheaves of pure dimension d and Hilbert polynomial d on the fibers X 8 are torsion-free with usual rank equal to n. Suppose S' is an S-scheme and (E, tp) is a p-semistable Higgs sheaf with Hilbert polynomial P on X' over S'. Then there is an open subset U'C X' such that the intersection of X' -- U' with any fiber has codimension at least 2, and such that E is locally free over U'. Over U', ~p is an f~,/s,-valued endomorphism of a rank n-vector bundle. Furthermore, the endomorphisms obtained by contracting with different sections of T(X'/S') commute with each other. Thus we can evaluate the elementary symmetric polynomials to obtain ~,(q0 Iv') ~ H~ U', Sym'(~)u'/s'))" MODULI OF REPRESENTATIONS. II 21 Since Sym~(D~,/s,) is a locally free sheaf, Hartog's theorem applied over artinian subschemes of S', coupled with the theorem on formal functions and Artin approxi- mation, imply that ~,(~ [o,) extend uniquely to sections which we denote ~(q~) e H~ ', Sym'(f2Jx,/s,)). Define ~(E, e)ei/'(X/S,n)(S ') to be the point corresponding to (~l(q~), ..., %(?)). This construction defines a morphism from the functor M~m.,,(X/S, P) to t/'(X/S, n), and hence a morphism of schemes ~ : Mr,,,~(X/S, P) --~ tP(X/S, n). We call ~(E, e) the characteristic polynomial of (E, 0). The morphism cr was introduced by Hitchin for Higgs bundles on curves in [Hi2] and [Hil]. There are universal sections a~" : X x s t/'(X/S, n) -+ Sym ~ T'(X/S) X 8 ~(X/S, n). Here Sym ~ T~ denotes the total space of the i-th symmetric power of the relative cotangent bundle. There is a multiplication map Sym ~ T'(X/S) Xx Sym J T'(X/S) ~ Sym' ~ ~ T'(X/S) as well as a map corresponding to addition. There is a closed subscheme #'(X/S, n) C T'(X/S) x s r n) defined by the equation t"+a~ l't"-l+... +a~,=0. -,- 1 as points in the This represents a functor which can be seen by considering t ~ and ,,u,l, total spaces of the symmetric powers, then multiplying them together and adding to get a point in Sym" T'(X/S) which is required to be in the zero section. Lemma 6.10. -- Suppose S' is an S-scheme and (E, ~) is a p-semistable Higgs sheaf with Hilbert polynomial P on X' over S', corresponding to a cokerent sheaf o a on T'(X'/S'). Let e(E, 0) : S' --~ $/'(X/S, n) be the characteristic polynomial defined above. Then o ~ is supported set theoretically on #'(X/S, n) x,~x/s, ,~, or S'C T'(X'/S'). Proof. -- In general, if A is a vector space and Sym'(A) acts on a vector space B, then the support of B considered as a coherent sheaf on A ~ is equal to the set of eigen- forms of the action of A. Now let U' C X' be the open set used above in the construction of ,(E, ~?). Then the zeros of the characteristic polynomial ,(E, 0) over U' are the eigen- forms of q0 [o'. Thus Zt/r(X/S, n) X~cx/s, ,~, ol~, ~ S' is the spectral variety of the endo- morphism 91tr'. From the above general principle, o~[T.cw/s ,, is supported on this spectral variety. On the other hand, any section of @ whose support is contained in T*(X'/S') --T*(U']S') restricts to a section of the fiber oa8 supported over the corn- 22 CARLOS T. SIMPSON plement X'~ -- U', (which has dimension less than or equal to d -- 2). The do8 are finite over X',, so such a section is supported in dimension less than or equal to d -- 2. By the hypothesis that g, are of pure dimension d, such a section is zero. Hence any section of d ~ supported in T*(X'/S') -- T*(U'/S') restricts to zero in all the fibers, so it is zero. Since the spectral variety is closed, this implies that all the sections of do are supported in $//'(X/S, n) � ~lx/s, ,~, o(~, ~I S'. [] Remark. -- Using Cayley's theorem, one can see that the support is scheme- theoretically contained in the spectral scheme ~(X/S, n) X rix/s, ,~, ocE, ~ S'. Theorem 6.11. -- The map a : Mmg~(X/S, P) -+ ~r n) is proper. Proof. -- Note, first of all, that all schemes involved are separated. Suppose S' is a curve, s e S' is a closed point, and put S" = S' --{ s }. Suppose that g : S" --> Mmg~(X/S, P) is a map such that the composed map ag extends to a map h : S' ~ ~P(X/S, n). Recall that Mm,,,(X/S, P) is an open set in M(0z, k* P), and that Nl(0z, k* P) is projective over S. The map g extends to a map g' : S' -+ M(~z, k* P). Let q0 : Q.2 -+ M(0z, k* P) be the good quotient of the parameter scheme O~ for sheaves on Z used in the cons- truction of w 1. Then 02 X moz, k*1.~ S' -+ S' is a categorical quotient. Thus there is a quasi-finite morphism of curves Y ~ S' such that s is the image of a pointy e Y, such that Y' ~ Y -- {2 } maps to S" C S', and such that the resulting map Y ---> M(Oz, k* P) lifts to a map Y -+ Q.,. Let do be the resulting sheaf on Z x s Y. If w e Y' then dow is a sheaf corresponding to a point in Mm~,(X/S , P). Thus @w has support contained in T*(Xw). This implies that the support of @IY' is contained in T*(X/S) x s (Y'), so it corresponds to a Higgs sheaf (E', q~') on X X s (Y') over Y'. The map Y' -+ Ma,,g,(X/S, P) corresponding to (E', ~') is equal to the map obtained by composing Y' -+ S" with g. In particular, the characteristic polynomial a(E', qZ) : Y' -+ r n) is the composition ofY' ~ S" with ag. Thus a(E', ~') extends to a map f: Y -+ $/'(X/S, n), equal to the composition of Y -+ S' with h. By the previous lemma, do [:r is supported inf*~(X/S, n) C T*(X/S) � s Y" Since 8 is flat over Y, it has no local sections supported on Zu. Thus do is supported on the closure off* ~r n) in Z X s Y. But since the equation defining SU(X/S,n) is monic, the subscheme $r is closed in Z X s ~e"(X/S, n). Therefore f* $K(X/S, n) is closed in Z x s Y, and do is supported in f* Y//'(X/S, n) C T*(X/S) x s Y. Hence @ corresponds to a Higgs sheaf (E, q~) on X � s Y over Y, restricting to (E', ~') over Y'. As 8 is a p-semistable sheaf, (E, q~) is a p-semistable Higgs sheaf by Corollary 6.9. We obtain a map Y -+ M~I,~(X/S, P) extending the composition Y' -+ S" -+ Ma~,(X/S , P). But this map is also equal to the composition Y-.--S'-+M(Oz, k* P) since the moduli space is separated. In this last map, the image ofy is equal to the image g'(s). From the fact that Y maps into MODULI OF REPRESENTATIONS. II Ma~***(X/S, P) we obtain g'(s) ~ Mmg~(X/S , P). Thus g' maps S' into MH~g~(X/S , P). This is the extended map required to prove properness of a. Remark. -- Hitchin gives an analytic proof of the properness of a in the case when X is a curve [Hil]. Corollary 6.12. -- Any p-semistable torsion-free Higgs sheaf on a fiber X, can be deformed to one which is fixed by the action of C*. A Itiggs bundle of semiharmonic type can be deformed to a Itiggs bundle of semiharmonic type which is fixed by C*, through a family of Higgs bundles of semiharmonic type. Proof. -- Suppose (E, ~) is a p-semistable torsion-free Higgs sheaf on Xs. Write the characteristic polynomial as ~(E, 9) ---- t" + a 1 t "-~ -+- ... + a,. For z ~ C*, the characteristic polynomial of zq~ is ~(E, z~) = t" + za 1 t"-I + ... + z" a,. As z -+ 0 these polynomials approach the limit t" in r n). The orbit of (E, 9) is a map C* ~Mm~,,(Xs, P ) such that the composed map C*-+~r n ) extends to a map A 1 ~ r n). By the theorem, the orbit extends to a map A a -7 Mn~g,~(X/S, P). Let Q, be the parameter scheme used to construct Mln,gs(X/S , P). Then Q.2 x Mmgg,~x~. P) A1 --+ A1 is a good quotient. In this situation, the unique closed orbit lying over 0 ~ A 1 is contained in the closure of the union of orbits corresponding to (E, zq0). Thus (E, 9) may be deformed to a Higgs sheaf corresponding to a closed orbit over 0 E A x. The map A 1 -+ Mn,,,~(X/S, P) is equivariant under the action of C*, so the image of the origin is a fixed point. Since there is a unique closed orbit lying over the origin, this closed orbit is preserved by C'. Thus the Higgs sheaf corresponding to the closed orbit over the origin is fixed up to isomorphism by the action of C*. This proves I the first statement. For the second statement, note that by Proposition 6.6, if the rational Chern classes of E vanish, then all of the Higgs sheaves involved are Higgs bundles. When X is a curve, Hitchin has a beautiful description of the generic fiber of the map a [Hi2]. In terms of our description of the moduli space, the idea is as follows. For a generic point s E S, the corresponding polynomial defines a smooth curve in the cotangent bundle, counted with multiplicity one. A Higgs bundle E in the fiber over that point is a coherent torsion free sheaf on the curve, of rank one. In other words, it is a line bundle. Furthermore, all line bundles of the appropriate degree occur. Thus the fiber is the Jacobian of the curve. See also [Ox]. CARLOS T. SIMPSON Vector bundles with integrable connections In this section we will apply the results of w 4, Part I, to construct a moduli space of vector bundles with integrable algebraic connection. For greatest generality, suppose that S is a base scheme of finite type over C, and that X is smooth and projective over S. A vector bundle with connection on X/S is a vector bundle or locally free sheaf E on X, together with a map of sheaves such that Leibniz's rule V(ae) -- dx/s(a ) e -i- aV(e) is satisfied for any sections a of G x and e of E. Here dx/s : 9 x ~ f2Jxts is the relative exterior derivative. Given a vector bundle with connection, we can extend V to an operator V: E | ~ E| by enforcing Leibniz's rule for forms a, using the usual sign conventions. In particular, the square of V is an operator V2:E_~E � 2 4<) ~x/s. Using Lcibniz's rule and thc fact that (dxts) s = 0, it is easy to sec that V z is d)x-linear. Thus it is givcn by a scction V ~ e H~ | ~/s) callcd thc curvature of V. A vector bundle with integrable connection is a vcctor bundlc with conncction (E, V), such that the curvaturc vanishcs, VS~ 0. Wc could makc a similar definition of cohcrcnt sheaf with intcgrablc connection. Howcvcr, it is a wcll known fact (which wc will not provc hcrc) that if E, is a cohcrcnt shcaf with intcgrablc conncction on X,, then E s is locally frcc, and thc Chcrn classcs of E 8 vanish (hcncc thc normalized Hilbcrt polynomial of Es is thc samc as that of Ox). If E is a cohcrcnt shcaf on X with intcgrablc rclativc conncction, such that E is flat ovcr S, thcn E is locally frcc by Lcmma I. 27, Part I. Bccausc of this, wc may as wcll assumc that thc pure dimcnsion d is cqual to the rclativc dimension of X/S, and that the normalizcd Hilbcrt polynomial P0 is cqual to that of ~x-----othcrwisc thc moduli spaccs arc cmpty. Furthcrmorc, any subshcaf of E~ prcscrvcd by V is again a vcctor bundle with intcgrablc connection, with the samc normalizcd Hilbcrt polynomial P0- Theorem 6.13. -- Suppose X is smooth and projective over S. There is a scheme MvR(X/S , n) quasi-projective over S, universally co-representing the functor M~R(X/S, n) which assigns to an S-scheme S' the set of isomorphism classes of vector bundles with integrable connection (E, V) on X'/S' of a given rank n. Suppose X is smooth and projective with connected fibers over S, and suppose x : S --* X is a section. There is a scheme RvR(X/S, ~, n) quasi-projective over S, representing the functor which assigns to an S-scheme S' the set of isomorphism classes of (E, V, 0~) where (E, V) is a vector bundle with integrable connection on X/S, and ~ : E !~s'~ _Z~ ~, is a frame along the section. MODULI OF REPRESENTATIONS. II 25 Furthermore, with respect to an appropriate line bundle all points OfRm~(X/S , 4, n) are semistable for the natural action of Gl(n, C), and the universal categorical quotient is naturally identified with MD~(X/S , n). Proof. -- Let A "R be the sheaf of rings of all relative differential operators on X over S. It is split almost polynomial, and the sheaf H arising in the description of w 2, 1 . Part I, is equal to Y~x/s, its dual H* is the relative tangent bundle T(X/S). The derivation is the standard one, and the bracket { , ),r is given by commutator of vector fields. The description of ADR-modules given in Lemma 2.13, Part I, coincides with the above definition of vector bundle (or sheaf) with integrable connection. If E is a vector bundle with integrable relative connection on X over S, then any subsheaf of E s preserved by the connection has the same normalized Hilbert polynomial P0, so E~ is p-semistable as a Ang-module. If E is fiat over S then E is a p-semistable AWg-module. The theorem follows from the general construction of moduli spaces given in Theorem 4.7, Part I. For the second paragraph, note that any vector bundle with relative integrable connection automatically satisfies condition LF(~). Hence we may apply Theorem 4.10, Part I, to obtain RD~(X/S, 4, n). [] Dependence on the base point In the Betti case, given two different base points x and y, and a choice of path y from x to y, we obtain an isomorphism T v : RB(X, x, n) ~ R~(X,y, n). This projects to a canonical isomorphism of M~(X, n), justifying dropping the basepoint from the notation for the moduli space. On the other hand, there is no natural isomorphism between RD~(X, x, n) and RD~(X,y, n). Our construction began with a construction of MD~(X, n) independent of the base point. On the complex analytic spaces, the isomorphism T v gives a complex analytic isomorphism (cf. w 7 below). This projects to an algebraic isomorphism (the identity) in the quotient, and on each orbit of the group Gl(n, C), it comes from an algebraic automorphism of groups; however Tv doesn't seem to be algebraic. One might conjecture that, in good cases, the isomorphism class of RD~(X, x, n) is a distinguishing invariant of the point x. 7. Identifications between the moduli spaces The analytic isomorphism between the de Rham and Bettl spaces Suppose f: X-+ S is a smooth projective morphism with connected fibers, and suppose ~ : S -+ X is a section. Recall that R~nI(X/S, 4, n) denotes the complex analytic space over S an associated to the local system of complex analytic spaces R~n(X/S, 4, n). Let RL~(X/S, 4, n) denote the complex analytic space associated to the de Rham representation scheme. 4 CARLOS T. SIMPSON Theorem 7.1. -- (The framed Riemann-Hilbert correspondence.) There is a natural isomorphism of complex analytic spaces R'.='(X/S, n) = Rg%(X/S, n), compatible with the action of Gl(n, C). We will prove this by showing that both spaces represent the same functor from the category of complex analytic spaces over S ~ to the category of sets. [,emma 7.2. -- Suppose Y and Z are topological spaces which can be exhausted by relatively compact open subsets, and suppose Y is locally simply connected. Suppose A is a sheaf of rings on Z. Suppose F is a locally free sheaf of p~l(A)-modules on Y x Z. Then G =pl,.(F) is a locally constant sheaf of H0(Z, A)-modules on Y, and the stalk at y ~ Y is given by G v = H~ � Z, � Proof. -- First of all, suppose that U is a connected open subset of Y. Then P2,.(P;~(A) Iu � z) = A. This implies that if F is free over p~-'(A), U C Y is a connected open subset, and y e U, then P2,.(F !v� z) ~P2,.( F [tul� z) is an isomorphism of free sheaves of A-modules. Suppose that there exists an open covering Z = O~ Z~ such that F is free of rank n on Y � Z~. The previous result implies that ify ~ U C Y and U is connected, then is an isomorphism of locally free sheaves on Z. This gives an isomorphism of spaces of global sections, in other words the restriction maps H~ � Z, F!~j~z) -+ U~ Z, Flcy~� are isomorphisms. This implies the lemma in this case. It follows that the lemma is true for sheaves F such that there exist open coverings Y = [J~ Y~ and Z = [.J~ Z~ with F free on each Y~ � Z,. Finally we treat the general case of a locally free F. Suppose that Z ~ is an increasing sequence of relatively compact open subsets exhausting Z. Then, for any relatively compact open set Y'C Y, F [y,x zi satisfies the hypotheses of the previous paragraph. def f ~ p Thus G ~ = p:,.@iyxzl) is locally constant when restricted to any Y'. This implies that G ~ is locally constant. The stalk at y e Y is G; := H~ I � Z', � MODULI OF REPRESENTATIONS. II 27 Finally, we have G --=- liln G i, .z.-- and since Y' is locally simply connected, the inverse limit of a system of locally constant sheaves is again locally constant. This proves that G is locally constant. The stalk G v is the inverse limit of the stalks G~v, hence equal to the desired space of global sections. /_,emma 7.3. -- The complex analytic space R~=)(X/S, 4, n) over S an represents the functor which to each morphism S' ~ S ~" of complex analytic spaces associates the set of isomorphism classes of pairs (o~,~) where o~" is a locally free sheaf off '(Os,)-modules of rank n on X' = X ~ � S', and ~ : ~-1(o~') ~- ~, is aflame over the section 4. Proof. -- Let X t~ denote the topological space underlying X '~. Note that the quantities appearing in the statement of the lemma are local over S an and depend only on the structure off: Xt~ S = as a fibration of topological spaces with a section over S ~. Thus we may suppose that X~P=Xo � S ~ and ~(s) = (x,s) for xeX o. Let P = nl(X0, x). Then 4, n) = x=(r, n) x s =, so the set of S~-morphisms from S' to R~')(X/S, 4, n) is equal to the set of morphisms from S' to Ran(r, n). The scheme R(I', n) is affine. It follows that the complex analytic morphisms from S' to the associated analytic space R'~(I TM, n) are given by the homomorphisms of C-algebras H~ n), O.,r..,) -+ Ho(s ', 0s, ) (this can be seen by embedding R(P, n) in an aHine space). In particular, R~'(X/S, 4, n)(S') = R~(r, n)(S') = R(r, n)(Spec(H~ ', Os,)) ) = Hom(P, Gl(n, H~ ', d)s,)) ). Let Px and pz denote the first and second projections on X0 � S'. Suppose o~ is a locally free sheaf ofp~-l(Os,)-modules on X 0 � S'. Let N = p,,.(o~'). This is a sheaf of H~ ', 0s,)-modules on X 0. Lemma 7.2 implies that ~ is locally constant, with fiber if, = #-I{~)xs' over x cX o. The monodromy of the locally constant sheaf ff is a representation F -* Aut~o(s,,e~.,)(ff~). If [~ : ~'(o~') ~ O~, then by Lemma 7.2 the fiber is fr _-__ H~ ', Os,) ~, so the mono- dromy of fr gives a representation r Ol(n, Ho(s ', This gives a map from the set of isomorphism classes of(#', ~) to Hom(P, Gl(n, H~ ', Ss'))- 28 CARLOS T. SIMPSON For the inverse map, note that X 0 is locally simply connected, so a universal covering X0 exists. Choose a base point ~ over x eX 0. Set ~,~ = p~-1(0~,) on ~ x S'. The identity gives "~ : "~1~ � s' ~ ~'. A representation r -+ Gl(n, Ho(S ', 8s,)) gives an action of I" on ,~ over the action on ~ � S'. We can use this to descend to a locally free sheaf ofp~-L(Os,)-modules .~ on X 0 � S', with the required frame ~. This is the inverse of the previous construction. We obtain an isomorphism between the set of S'-valued points of RI~")(X/S, ~, n), and the set of (oq~, ~) on X' over S' as desired. [] This lemma provides half of the proof of the theorem. For the other half, we begin with a lemma and some corollaries. Lemma 7.4. -- Suppose that V C C, ~ is an open disc and S'C V is a complex analytic subspace, such that all embedded components pass through a single point s ~ S'. Suppose U is an open disc centered at the origin in C2. If (E, V) # a holomorphic vector bundle with integrable connection on U � S' over S', then the map v : H~ x S', E> ~ -+ H~ 0 ) � S', E [~o~ ~) is an isomorphism (here the exponent V means the space of covariant constant sections). Proof. -- This is well known ifS' is a point. It follows that it is true ifS' is an artinian complex analytic space (the same as an artinian scheme), for the result in that case follows from the same result for Pl..(E, V) on U. It is also well known if S' is a smooth complex analytic manifold. We show how to deduce the theorem when S' may be nonreduced, for example. Choose a point s ~ S' containing all irreducible embedded components of S. Let S',, be the m-th infinitesimal neighborhood of s in S'. The map v is injective: suppose e is a section with V(e) = 0 and ,J(e) ~ 0; then from the result for artinian spaces, e Is;, = 0 for all m, and this implies that e = 0. With the same hypotheses on S', suppose also that dim(U) = 1. We show that v is surjective in this case. The holomorphic bundle E is trivial, so it has an extension to a trivial bundle E ~ on U � V. Let d denote the constant connection on EeL We may write v = dl~� + A(u, t) du where A(u, t) is a holomorphic section of End(E) over U � S. There exists an extension of A to a holomorphic section A ~ of End(E ~x) on U � V, and we can then put V e~ = d + Ae*(u, v) du. MODULI OF REPRESENTATIONS. II 29 This is an integrable holomorphic connection on E ex relative to V (it is integrable because dim(U) = 1 implies ~ x v/v = 0). Suppose e 0 e H~ 0 } � S', E I{o} x s'), and choose an extension to e~ x e H~ 0 } � V, E I{0} � v). By the result for smooth base spaces, there exists e eXeH~ X V, Eex) vex with v(e "*) =e~ x. Putting e=W[v� gives a section with V(e) = 0 and v(e) = e 0. This proves that v is an isomorphism in the case of relative dimension 1. Now proceed by induction on the relative dimension k, assuming that the theorem is known for relative dimension k -- 1. Let U 1 denote the disc of dimension k -- 1 obtained by intersecting U with one of the coordinate planes. By the inductive hypothesis, there exists a section e i in H~ � S', E Ivl x s;,) v with e i restricting to e 0 on { 0 ) � S'. Let h : U � S' -+ U i � S' denote the vertical projection. Let V t denote the projection of V into a relative connection for the map h. The map h is smooth of relative dimension 1, so by the previous result, there exists a section e in H~ � S', E) v~ restricting to e i on Ui x S'. In order to show that V(e) = 0 we use the infinitesimal neighborhoods S~. There exist sections e" in H~ � S~,Elvxs~,) v such that e~l,0~� xs~. By the uniqueness result for U~ � S~ over S~, e m is equal to e~ on the subspace U1 x S~. By the uniqueness for U X S~ over Ui � S~, ely� = era" But V is Os,-linear , so V(e) Iv x Sm= V(e,,) = 0. This is true for any infinitesimal neighborhood, so V(e) = 0. This shows that ~ is surjective in relative dimension k. [] Keep the same hypotheses as in this lemma. Suppose E is a trivial bundle of rank n. We can choose n sections el, ..., e, in H~ � S', E) v such that v(e,) form a frame for E I~o} � s'- The lemma implies that H~ X S', E) v ~ H~ ', Os, ) | (Cel Q '' 9 e Ce,). Conversely, if the e~ are a collection of sections such that this formula holds, then the ~(e~) form a frame for E I~0} x s'. We claim that the e~ form a frame for E over U. It suffices to show that over each closed point (u, s) e U � S', the e~ are a basis for the fiber of E. But this follows from the above statement and its converse applied to { u } � S' instead of{O) X S'. Corollary 7.5. -- Suppose el, ..., e, are sections chosen as above. Then the map Ml(r ,) | (eel e ... e Ce.) -+ E v is an isomorphism of sheaves on U � S'. Proof. -- This is injective, because the ei are a frame for the holomorphic bundle E. For surjectivity, suppose e is a section of E v over an open set V C U X S'. Then V can be covered by subsets U' � S" of the form considered above. The restriction of { e~ } to a section { u } x S" is a frame for the restriction of the bundle E, so the previous argument applies. There exist a, e H~ '', 08,, ) with E aiei = e on U'� S". This shows that the map of sheaves is surjective. [] 30 CARLOS T. SIMPSON Corollary 7.6. -- Suppose f: X' -+ S' is a smooth morphism of complex analytic spaces. Suppose (E, V) is a vector bundle with integrable holomorphic connection relative to S'. Let ~- = E v denote the sheaf of sections e ore such that V(e) = 0. Then ~" is a locally free sheaf of f-a(Os,)- modules. Proof. -- Let k denote the relative dimension of X' over S'. We can cover X' by a collection of open subsets of the form U x S" where U C (l k is an open disc, and S"C S' is a subset satisfying the hypotheses of the lemma. By the corollary, ~-[u � s" is free overf-l(Os,). Thus ~- is locally free. [] Lemma 7.7. -- The complex analytic space R~(X/S, ~, n) over S ~ represents the functor which to each morphism S' ~ S ~ of complex analytic spaces associates the set of isomorphism classes of pairs (o~', ~) where o~" is a locally free sheaf of f-x(Os,)-modules of rank n on X' = X ~ xs~, S' , and [3 : ~-1(o ~') ~ O~, is a frame over the section 4. Proof. -- Note that by Lemma 5.7, Part I, the argument of Theorem 6.13, and the analogue of Lemma 2.13, Part I, for the complex analytic case, R~a(X/S , 4, n) represents the functor which to each morphism S' -+ S "~" of complex analytic spaces, associates the set of isomorphism classes of triples (E, V, a) where E is a holomorphic vector bundle over X' = X ~ X San S', V is a holomorphic integrable connection on E relative to S', and ~ : ~'(E) ~ 0~, is a frame. We have to identify this functor with the functor given in the lemma. First, note that the trivial bundle ~x' has a natural connection dx,/s, , the exterior derivative with values projected into DAx,/s,. This connection is f-l(Os,)-linear. If ~" is a locally free sheaf off- l(Os,)-modules on X', then is a locally free sheaf of Ox,-modules , and it has a relative holomorphic integrable connection V = 1 | A frame ~ : ~-1(5) --- @~, yields 0c: ~'(E) ~ ~,. This gives a map from the set R~(X/S, 4, n)(S') to the set R~(X/S, 4, n)(S'). Suppose (E, V) is a holomorphic vector bundle with integrable connection on X' over S'. By Corollary 7.6, the sheaf ~ = E v is a locally free sheaf off- l(Os,)-modules. Suppose , is a frame for E along 4. Let ~ denote the composed map ~-~(E ~) ~ ~-~(E) -+ ~'(E) -~ ~,. The arguments from the proof of Lemma 7.4 show that this is an isomorphism. This completes the construction of the inverse to our previous map, so we obtain an iso- morphism between the set of (E, V, 0c) and the set of (~', 9). [] Proof of Theorem 7.1. -- Lemmas 7.3 and 7.7 show that the spaces R~'~I(X/S, 4, n) and R~a(X/S , 4, n) both represent the same functor. Thus they are naturally isomorphic. The isomorphism between functors is compatible with the group action, so the iso- morphism between spaces is too. [] MODULI OF REPRESENTATIONS. II 31 Proposition 7.8. -- (The Riemann-Hilbert correspondence.) Suppose f: X -7 S is a smooth projective morphism. Then there is a natural isomorphism M~='(X/S, n) - M~(X/S, n). If f has connected fibers and ~ is a section, then this isomorphism is compatible with the isomorphism given by Theorem 7.1. Proof. -- Suppose first of all thatfhas connected fibers and a section ~ exists. Then MD~(X/S, n) is a good quotient of RDR(X/S , ~, n) by the action of Gl(n, C). Propo- sition 5.5, Part I, implies that M~aR(X/S, n) is a universal categorical quotient of R~R(X/S , ~, n) in the category of complex analytic spaces over S "n. On the other hand, M~(X,, n) is the good quotient of R~(X,, ~(s), n) by the action of Gl(n, C), so again by Proposition 5.5, M~(X,, n) is a universal categorical quotient of R ~"(Y~ ~.~,,, ~(s), n). The space M~I(X/S, n) is, locally over S ", of the form M~(X~, n) X S '~. The property of being a universal categorical quotient is preserved under taking the product with another space, as well as localization in the quotient space (hence by localization in S'~), so M]3~(X,, n) � S "~ is a universal categorical quotient of R]3"(X,, ~(s), n) � S '~. The isomorphism of Theorem 7.1 is compatible with the action of Gl(n, C), so it induces an isomorphism of universal categorical quotients M~I(X/S, n) _-_ M~R(X/S , n). Suppose that X is a disjoint union of components, each of which has connected fibers over S and admits a section. The resulting moduli spaces M~(X/S, n) and M~a(X/S, n) are then products of spaces obtained by taking quotients of representation spaces. The above isomorphism for each factor gives the desired isomorphism. In general, we can make a surjective dtale base change S' -~ S such that X'/S' satisfies the hypotheses of the previous paragraph. The isomorphism ~n t t ', n) = descends to give the desired isomorphism. [] The homeomorphlsm between the de Rham and Dolbeault spaces Recall some facts from [Si5]. These results are based on non-linear partial diffe- rential equations, and in particular on the works iNS] [Co] iDol] [Do2] [Do3] [Hil] [Si2] [UY]. There is a notion of harmonic metric for a vector bundle with integrable connection (flat bundle) or a Higgs bundle on a smooth projective variety X. Given a flat bundle and a harmonic metric, one obtains a Higgs bundle, and vice versa. The structures of Higgs or flat bundles obtained from the harmonic metric do not depend on the choice of harmonic metric. The conditions for the existence of a harmonic metric are as follows. A flat bundle has a harmonic metric if and only if it is semisimple [Co] [Do3]. A Higgs bundle has a harmonic metric if and only if it is a direct sum of ~t-stable 32 CARLOS T. SIMPSON Higgs bundles with vanishing rational Chern classes [Hil] [Si2]. A harmonic bundle consists of a flat bundle and a Higgs bundle related by a C oo isomorphism such that there exists a common harmonic metric relating the structures. The set of isomorphism classes of harmonic bundles is exactly the same as the set of flat bundles parametrized by points of the moduli space MD~(X , n). It is also the same as the set of Higgs bundles para- metrized by points of the moduli space MDo~(X , n). We obtain an isomorphism of sets of closed points between these two moduli spaces [Si5]. If X ~ S is a smooth projective morphism, we can take the isomorphisms of sets given in each fiber all together to get an isomorphism between the sets of closed points of MDR(X/S, n) and Mr, o~(X/S, n). Recall that the superscript 5,I t~ denotes the topo- logical space underlying the complex analytic space M". We will show that our isomor- phism of sets gives a homeomorphism of topological spaces MDR(X/S, top n) ~ MDol(X/S to. , n). Wc recall a weak compactness property for harmonic bundles, following the notation of [Si5] (except that the Higgs field which was denoted by 0 there is denoted by ~ here, to conform with Hitchin's original notation). Suppose X -+ S is smooth with connected fibers, and suppose ~ : S ~ X is a section. Suppose { s i } is a sequence of points converging to t in S. Choose a standardized sequence of diffeomorphisms ~F~ : X~. -~ Xt, such that ~'~(~.(s0) = ~(t). Choose a family of metrics on X,, which, when transported via ~F~, are uniformly bounded in any norm with respect to a metric on X t. Use these metrics to measure forms on X~. Proposition 7.9. -- Fix q > 1. Suppose V~ is a harmonic bundle on X,i with harmonic metric K, for each i, such that the coefficients of the characteristic polynomials of the Ifiggs fields % are uniformly bounded in L 1 norm. Then there is a harmonic bundle V, a subsequence { i' }, and isomorphisms r~,, : ~F~,,.(V~,) ~ V of Coo bundles satisfying the following properties. There is a harmonic metric K for V with ~,,(K~) = K, and ifO represents any of the operators O, -0, ~, or combinations thereof, the differences dif(O, i') def ~i', .(Oi') -- O converge to zero strongly in the operator norm for operators from L~ to L ~ Proof. -- This is essentially the same as Lemma 2.8 of [Si5], which is based in turn on Uhlenbeck's weak compactness property [Uh] and the properness of Hitchin's map [Hil]. There are a few new twists. The main difference is that the underlying spaces X,/are varying. In particular, the differences dif(O, i) are differential operators, so they must be measured with respect to operator norms. We recall the proof with this in mind. First of all, the hypothesis that the characteristic polynomials are bounded in L 1 norm implies that they are bounded in C o norm, since the coefficients are holomorphic sections of certain bundles on X~. Fix p large. The bound for the coefficients of the characteristic polynomials implies that I q~i [Ki are uniformly bounded [Si5], Lemma 2.7. The curvatures MODULI OF REPRESENTATIONS. II 33 are therefore uniformly bounded in C ~ Uhlenbeck's weak compactness theorem [Uh] gives a unitary bundle (V, K) with unitary connection 0 + 0 (oftype L[) and a sequence of unitary isomorphisms ~: tF~,.(V~) ~ V such that dif(a + i) = + - a -- converges to zero weakly in L[. In particular, for p big enough this converges strongly to zero in C O (note that dif(O + O, i) is a O-th order operator), and dif(O + O, i) ~ 0 strongly in the operator norm for Hom(L~, L~). We have (o, + = o so (0 + ~) (~,. W) = dif(0 + 0, i) (~,,. q~). Since ~ are unitary isomorphisms, l~q,..q~]~ are uniformly bounded. Thus dif(a+0, i) (~,.q~) ~0 strongly in C o , so (0+0) (:q~,.qh) ~0 strongly in C o . We have to be slightly careful, since ~,. ~ are one-forms--this doesn't constitute an estimate for the full covariant derivatives. The ~.. ~ are of type (1, 0) but each for a different complex structure. More precisely, let T~'~ Tr denote the subbundle of forms of type (1, 0) with respect to the complex structure of X,i as transported to X t by ~,. Then B,,. q~, e H~ End(V) | T~'o) 9 We can choose an open subset U C X~ and a sequence of open immersions ~, : U -+ X t which converge to the identity in any norm, but such that ~;(T~ '~ = T 1'~ U is the subbundle of forms on U of type (1, 0) with respect to the holomorphic structure of X t. We may also choose a sequence of unitary isomorphisms ~ :~(V) ~ V converging to the identity in any norm, so for example ~(0 q- 0) -- 0 -- 0 converges to zero in L]'. Then ~,q)~ ~,,. q~, are End(V)-valued (1, 0)-forms on U with (a + uniformly bounded in (3 ~ Now we can conclude (from the elliptic estimates for 0) that ~, ~ ~q~,. q~ are uniformly bounded in L~ on any relatively compact subset of U. This argument, done for a collection of open sets U covering X~, implies that ~,.(qh) are uniformly bounded in L~. By going to a subsequence, we may suppose that ~,. qh approach a limit q~ weakly in L~ (hence strongly in (30). The limit satisfies (0 + 0) (q~) = 0 and ~ A q~ = 0. Finally, in the conclusion of Uhlenbeck's weak compactness theorem, ~..(F~i+~i) approach Fe+~ weakly in LL In particular (even taking into consideration the change of complex structure), the component of type (0, 2) is the weak limit of the components of type (0, 2), which are zero. Therefore 0* = 0, so (V, 0, ~) is a Higgs bundle. The B~,.(~) approach the K-complex conjugate ~ weakly in L~, and the operator 0 is the one associated to 0 by the metric K. If we set D = 0 + 0 q- ~ q- then the differences dif(D, i) = aq,, .(D,) -- D 5 34 CARLOS T. SIMPSON are 0-th order operators converging to zero weakly in L~. In particular, they converge to zero strongly in C O and hence they converge to zero in the operator norm for operators from L~ to L ~. The weak convergence in L~ implies that the curvature D ~ is the weak D z = 0, proving that V together limit of the curvatures ~i,.(i), which are zero. Thus D z with all of its operators and its metric, is a harmonic bundle. We know from above that dis i) -+ 0 strongly in C O for O = 9 and O = 0 + 0. The same argument as for 9 works for O = ~. We have to extract the cases of O = 0 and O = 0 from the case O = O + 0. Let p~,o denote the projection onto T~ '~ and let pa, o denote the projection for the complex structure of X,. Then pl, o + px, o in any norm. We have ~,,.(~,) = p~,~ ,~, .(~, + 0,). Thus dif(0, i) = p~,0 dif(0 + 0, i) + (p~,o _ p,.o) (0 + 3). Since dif(0 + ~, i) -+ 0 in C O and the PI' 0 are bounded, the first term converges to zero in the operator norm. Now 0 + 0 is a bounded operator from L~ to L ~, and P~' 0 _ px, 0 converges to zero in C C, hence in the operator norm of Hom(L q, Lq). Therefore their composition, the second term, converges to zero in the operator norm of Hom(L~, L~). The same argument works for dif(0, i). This proves the proposition. [3 Let J denote the standard unitary metric on C". Let n) c R o,(x/s, 4, n) denote the subset consisting of triples (s, E, 6) where s e S, E is a Higgs bundle of harmonic type on X,, and [3 : E~,) ~ C ~ is a frame, such that there exists a harmonic metric K for E with ~(K~cs~ ) = J. Note that the harmonic metric K is uniquely deter- mined once it is fixed at one point ~(s) [Si2]--we call this K the chosen harmonic metric. Endow R~o~(X/S, ~, n) with the topology induced by the analytic topology of R Ax/s, n). Suppose s~ is a sequence of points approaching t in S. Choose a standardized sequence of diffeomorphisms ~:X~ ~-X~ such that ~,(~(si)) = ~(t). Corollary 7.t0. -- Suppose (E~, ~) are points in R~ot(X,~, ~(s~), n) which remain inside the inverse image of a compact subset 0fM~ol(X/S, 4, n). Then after going to a subsequence, there is a point (E, 6) in RDaol(X~, ~(t), n) and a sequence of bundle isomorphisms ~ : W~, .(E~) ~ E Suck that: the ~ preserve the chosen harmon# metrics; the operators ~i,.(-O~) and ~,,.(9~) converge to-O and ~ in the operator norm for operators from L~ to Lq; and finally, the frames ~(~) converge to 6. Proof. ~ Since the points remain in the inverse image of a compact subset of M~o~(X/S , ~, n), the eigenforms of the Higgs fields q0 i for Ei are uniformly bounded (this is because of the existence of the map ~ sending (E,, %) to the characteristic polynomial of ~,--see the discussion above Lemma 6.10). From the previous proposition, we can go to a subsequence and obtain a Higgs bundle E with harmonic metric K and MODULI OF REPRESENTATIONS. II 35 a sequence of bundle isomorphisms ~ with the desired convergence properties. Since the unitary group is compact, we may, by going to a further subsequence, assume that the frames ~(~i) converge to a unitary frame ~. Then (E, ~) is a point in n). [] Corollary 7.11. -- The subset R~0~(X/S , ~, n) C R ~"gm.=~ t'q 4, n) is closed. Proof. -- Suppose (s,, E~, ~) is a sequence of points in Rgol(X/S, ~, n), converging to a point (t, E', ~) in R~o~(X,, ~(s~), n). The images in M~o~(X~, ~(s~), n) converge, so they lie in a compact set. Apply the previous corollary to obtain a point (t, E, ~) in R~o~(X/S, 4, n) and a sequence of bundle isomorphisms ~. By Theorem 5.12, Part I, for the case of A ~~ the points (s~, E~, ~) converge to (t, E, ~3). Since R~(X/S, ~, n) is separated, (t, E', ~') = (t, 2, ~). Thus the limit is in ll~(X/S, 4, n). [] Corollary 7. lB. ~ The subset R~ot(X/S, ~, n) is proper over M~oa(X/S , n). Proof. -- Suppose (s~, E,, ~) is a sequence of points in R~ol(XIS, 4, n) lying over a compact subset of M~o1(X]S , ~, n). First, we may choose a subsequence so that the points s~ converge to a point t. Then we can apply Corollary 7.10 and Theorem 5.12, Part I, for the case of A D~ to obtaixt a subsequence which has a limit (t, E, ~3) in RL,(x/s, 4, n). [] We do the same thing for the de Rham spaces. Let n) c R.dX/S, n) denote the subset consisting of triples (s, E, ~) where s e S, 2 is a semisimple vector bundle with integrable connection on X,, and ~ : EL(,) ~ C" is a frame, such that there exists a harmonic metric K for E with ~(K~(~)) = J. The harmonic metric K is uniquely determined once it is fixed at the point ~(s) [Co], and we again call K the chosen harmonic metric. Endow R~R(X/S , ~, n) with the topology induced by the analytic topology of 4, n). Suppose s~ is a sequence of points approaching t in S. Choose a standardized sequence of diffeomorphisms tF, : X,i ~ X t such that tF,(~(s~)) = ~(t). Lemma 7.13. -- Suppose (E~, ~) are points in RJgR(X,;, ~(s,), n) which remain inside the inverse image of a compact subset of M~R(X/S , 4, n). Then after going to a subsequence, there is a point (2, ~) E RJvx(Xt, ~(s,), n), and a sequence of bundle isomorphisms ~q, " ~F,..(2~) ~ 2 such that: the ~ preserve the chosen harmonic metrics; the operators ~q~. .('O~) and ~,..(V~) converge to -0 and V in the operator norm for operators from L~ to Lq; and the frames ~q~(~) converge to ~. Proof. ~ This is die same as for Corollary 7. I0, except that we have to show that the characteristic polynomials of the Higgs fields ~0~ of the harmonic bundles corres- ponding to E~, are bounded. We follow the argument of ([Si4], Lemmas 3 and 5). 36 CARLOS T. SIMPSON Let F = 7rx(Xt, ~(t)), which is also equal to r.l(X,i , ~(s,)) via the diffeomor- phisms tF~. The condition that the points lie over a compact subset of M])~(X/S, 4, n) implies that the monodromy representations of tFi.o(E,, V~) lie over a compact subset of M(F, n) (by Theorem 7.1). The first thing we note is that it is possible to choose frames ~3~ for E, such that the monodromy representations of (E~, [3~) lie in a compact subset of R(F, n). The argument (from [Si4], Lemma 3) is that the subset of zeros of the moment map in R(F, n) is proper over M(P, n) [Ki] [KN] [GS]; our monodromy representations come from harmonic bundles, so they are semisimple--lying in the closed orbits--thus by appropriate choice of frames we can assume they correspond to points in the set of zeros of the moment map. Let p~ denote the monodromy reprcscntations corresponding to (E,, ~). Since they are bounded, it is possible to choose initial p~-equivariant maps from the universal covers X, to Gl(n, C)/U(n), which have uniformly bounded energy (note that the diffeomorphisms ~ are uniformly bounded in any norm). See [Si4], Lemma 5, for a description of how to do this (the process described there works the same way for any rank). Finally, the harmonic equivariant map has lower energy, and the energy is equal to the L z norm of q~,. Thus [1 ~o, Ilr~2,x,i , are uniformly bounded. This implies that the eigenforms of q~i are uniformly bounded in L ~ norm. The eigenforms of % are multi- valued holomorphic sections of f~ which do not depend on our choices of frame ~. Xs i The maximum norm of an eigenvalue of a holomorphic matrix is a subharmonic function, so the eigenforms of % are uniformly bounded in C ~ Thus the characteristic polynomials of the Higgs fields q~ are uniformly bounded in C ~ The rest of the proof is the same as that of Corollary 7.10. [] Corollary 7.14. -- The subset liaR(X/S, 4, n) C R~(X/S, 4, n) is closed. Proof. --- The same as for Corollary 7.11, but using Theorem 5.12, Part I, for the case of A DR. [] Corollary 7.15. -- The subset RaDR(X/S, 4, n) is proper over M~(X/S, n). Proof. -- The same as for Corollary 7.12, but using Theorem 5.12, Part I, for the case of A ua. [] There is an isomorphism of sets Rvol(X/S, 4, n)---RDR(X/S, ~, n). Over each fiber X,, this comes from the equivalence between the category of semistable Higgs bundles with vanishing Chern classes, and the category of flat bundles, constructed in [Si5]. This equivalence of categories is compatible with pullback to a point ~(s) ~ Xo, so it gives an isomorphism between the sets of isomorphism classes of framed objects in the two categories. In other words we get an isomorphism between the set of points in RDol(X,, ~(s), n) and RDR(X,, ~(S), n). Putting these together for all s we obtain the isomorphism of sets stated above. MODUI,I OF REPRESENTATIONS. II 37 Lemma 7.16. -- This isomorphism of sets induces a homeomorphism between the subsets R~(X/S, ~, n) and R~dX/S, ~, n). Proof. -- We prove continuity of the map from the Dolbeault space to the de Rham space. Suppose s~ is a sequence of points approaching t in S. Choose a standardized sequence of diffeomorphisms ~,:Xs/~ X~ such that ~,(~(s~))--~(t). Suppose (si, E~, [~) are points in RDol(Xsi,~(si),n), a converging to a point (t,E, [5') in , , ~l~--~, ~(si), n). Choose any R~ol(X t ~(s,) n). The points lie over a compact set in M s" t3~ subsequence. Apply Proposition 7.9 to obtain a harmonic bundle V over X, and (after going to a further subsequence) a sequence of bundle isomorphisms ~ : ~..(Et) ~- V, such that the transported structures of harmonic bundle on E, converge to the structure of harmonic bundle on V. Then the convergence statements of Lemma 7.13 hold for the operators d" and V giving the structures of vector bundle with integrable connection: the ~..(d~') and ~,.(V~) converge to d" and V in the operator norm for operators from L~ to L ~, and the frames ~i(~) converge to a frame ~. By Theorem 5.12, Part I, for the case A Ira, the points (s,, (E,, d[', V~), ~,) converge in R~t(X/S, ~, n) to the point (t, (V, d", V), ~). Similarly, (s~, (El, 0,, q~), ~) converge to (t, (V, 0, q~), ~) in RaDo,(X/S, ~, n). But this implies that (t, (V, 0, q~), ~) -- (t, E', ~'), so (t, (V, d", V), fS) is the point in Ravrt(X/S, ~, n) corresponding to (t, E', ~'). We have shown that every subsequence has a further subsequence where the corresponding points converge to the correct limit. This proves that the sequence of points in R~a(X/S, ~, n) corresponding to the original sequence of points (s,, E,, ~,) converges to the point corresponding to (t, E', ~). Thus the map from the Dolbeault space to the de Rham space is continuous. The proof of continuity of the map from the de Rham space to the Dolbeault space is exactly the same. n Note that the unitary group U(n) = Aut(C", J) acts on the representation spaces, and preserves the subsets R~o,(X/S, ~, n) and R~It(X/S, ~, n). This action is continuous in the analytic topology. Lemma 7.17. -- The moduli spaces M ~~ vot~,,l~,, ~, n) amt M~(X/S, ~, n) are the topological quotients of the representation spaces R~o,(X/S, ~, n) and R~(X/S, ~, n) by the action of U(n). Proof. -- Let N = R~o,(X/S, ~, n)/U(n) denote the topological quotient. Since U(n) is compact, N is separated (Hausdorff). Two points in R~,(X/S, ~, n) map to the same point in M~o,(X/S, ~, n) if and only if the underlying harmonic bundles are isomorphic, thus if and only if the points are related by a unitary change of frame. Thus the map f: N ~ Mvo,(X/S , 4, n) is one-to-one. The map from the representation space to the moduli space is continuous and proper, and since N is the topological quotient, this implies that the map f is continuous and 38 CARLOS T. SIMPSON proper. Therfore f is a homeomorphism identifying the moduli space with the quotient. The proof for the de Rham spaces is the same. [] We state our next corollary as a theorem. Theorem 7.18. -- The isomorphism of sets induced by the equivalence of categories given in [Si5] is a homeomorphism top MDo~(X/S , n) = to, ~ MD~(X/S , n) of the topological spaces underlying the usual analytic spaces. Proof. -- It is easy to reduce to the case where X --> S has connected fibers and admits a section. Then we may refer to the previous discussion. The modull spaces are identified, in the previous lemma, as topological quotients of R~(X/S, ~, n) and R~(X/S, ~, n). But Lemma 7.16 says that the identification between the representation spaces given by the equivalence of categories of [Si5] is a homeomorphism. This gives a homeomorphism between the quotients. [] Remark. -- Combining this with Proposition 7.8, we obtain a homeomorphism M~I(X/S , n) ~ M~~ n) where the right hand side denotes the topological space underlying M~n~(X/S, n). Corollary 7.19. -- If X is a smooth projective varie~, then any representation of the funda- mental group of X can be deformed to a representation which comes from a complex variation of Hodge structure. Proof. -- By Corollary 6.12, any point in M,o,(X , n) can be deformed to a fixed point of the action of C*. These fixed points correspond to representations which come from complex variations of Hodge structure [Si5]. By the continuity result of the theorem, any connected component of M~(X, n) contains a point parametrizing a complex variation of Hodge structure. But the inverse image of a connected component in Ms(X , n), is connected in RB(X , n), since M~(X, n) is a universal categorical quotient of RB(X, n) by a connected group. A point in the closed orbit over a fixed point of C* comes from a complex variation of Hodge structure. Thus in any connected component of the space of representations, there is a representation which comes from a complex variation of Hodge structure. [] Remark. -- This has topological consequences that were explained in [Si5]. The corresponding result is also true for principal bundles (cf. w 9 below). Counterexample We show that the isomorphism of sets R,g(X, x, n) ~ Rvot(X, x, n) given by the equivalence of categories constructed in ([Si5] Lemma 3.11) is not, in general, continuous. MODULI OF REPRESENTATIONS. II 39 In fact, the isomorphism on the open subset of stable points (which is continuous), has no continuous extension over the whole representation space. Let X be an elliptic curve, with nonvanishing differential dz. Let E = 0 x | 0 x be the trivial bundle of rank 2, with the canonical identification ~ : E, - C ~. Let 0,__ (0 ~ at dz Suppose t is real and approaches 0. Then the point (E, 0t, ~) approaches the point (E, 0o, ~). This limit is independent of the choice of a. However, we will see that the associated representations approach a limit that depends on a. Let gt = 9 Then , (; 0) Ot = gi- 10t g~ = at d Now the metric for (E, 0~) is the usual constant metric, and the associated flat connection is given by the matrix 0 ~ d~)" (;atdz+ Thus the flat connection associated to (E, 0~) is given by the conjugate of this matrix by gt: at ds dz + a~ gt at dz + ~ d~. g[ 1 = (: ~ (i at dz + ~ d~. ~ d~ (note that the entries ofg t are constant so there is no need to differentiate in conjugating the connection). Since we assumed that t was real, this connection matrix approaches (0 ~ as t -+ 0. The limit depends on arg(a). Thus the map between the space of representations and the space of (E, ~) cannot be continuous. It might still be the case that there is a homeomorphism between the topological quotient spaces RDR(X/S, too ~, n)[Gl(n) and top RDoI(X/S , ~, n)/Gl(n), which are non- Hausdorff spaces. Philosophically it would be important because of the interpretation of the topological quotient spaces as non abelian first cohomology spaces. This is an interesting problem for further study 9 40 CARLOS T. SIMPSON 8. The Gauss-Manln connection Suppose f: X ~ S is a smooth projective morphism. We have constructed the relative de Rham moduli spaces MDR(X/S , n). On the other hand, the relative Betd space is in fact a local system of schemes M~(X/S, n). The associated analytic total space M~(X/S, n) has a connection, namely a compatible system of trivializations over artinian subspaces of S ~''. The isomorphism of Theorem 7.1 gives a connection on M~R(X/S , n). We will show that this comes from an algebraic connection on MDI~(X/S , n). We will call this connection the Gauss-Manin connection because it is the analogue for nonabelian cohomology of the usual Gauss-Manin connection on the relative abelian de Rham cohomology. For constructing the algebraic connection, we follow the ideas of Grothendieck's construction for the case of abelian cohomology. Crystalline interpretation of integrable connections The first step is to give an interpretation of vector bundles with integrable connection on X/S as crystals. The advantage of this is that if S' is an S-scheme which contains a closed subscheme S o defined by a nilpotent ideal, and we set X~-----X' X s, So, then a crystal on X'/S' is canonically the same thing as a crystal on X'0/S'. The set of crystals on X'0/S' depends only on the restricted map S' 0 -~ S, so the functor M~R(X/S, n) is itself a crystal on S. The resulting stratifications for MD~(X/S , n) and RDR(X/S, ~, n) provide the Gauss-Manin connections on these schemes over S. This argument shows that the notion of a crystal can be useful in characteristic zero too. We will begin with an intermediate interpretation of vector bundles with connection on a smooth X/S, then proceed to describe what is meant by a crystal (in the present simple case). The contents of this discussion are based on the ideas of Grothendieck [Gr3], by now well known. We present them here for the convenience of the reader, since most of the literature on crystals has concentrated on characteristic p. Our terminology may not be completely standard. Suppose as usual, that X/S is smooth and projective. Denote by (X � X) ^ and (X � X � X) ^ the formal neighborhoods of the diagonals in products of X. We have projections denotedp~ or p~j in an obvious manner. Lemma 8.1. -- Suppose E is a vector bundle on X. Then an integrable connection V is the same thing as an isomorphism ~ : p'~ E-~ p~ E on (X � s X) ^, such that the restriction of to the diagonal is the identity, and on(Xx.~X xsX) ^. MODULI OF REPRESENTATIONS. II 41 Proof.-- Given such an identification 9, we obtain a connection V as follows. Let J denote the ideal of the diagonal in X � s X. If e is a section of E, then set V(e) = p*2(e) -- ~(p~(e)) (modJ2). It is an element ofp~ E | (j/j2), and considered as a module on the diagonal X, j/jz is (by definition) equal to the module of relative differentials ~lis. Note that p~ E/J = E on the diagonal, so V(e) is an element of E | ~x/s. From the discussion below, it will be clear that V is an integrable connection. We would like to see that this construction gives a correspondence between q~ and V. This statement does not depend on the fact that X is projective. We can cover X by open sets V which are finite ~tale covers of open sets U in affine space A~, and it suffices (by considering the direct image from V to U, and the 0v-module structure over U) to verify the lemma for vector bundles on U. We may further assume that E is a trivial bundle, E ~ d~. The isomorphism q~ is then given by a function g(x,y) with values in Gl(n), defined for x,y e U with y infinitesimally close to x (more precisely it is defined on the formal scheme (U � U)^). The conditions on g are that g(x, x) = I, and that g(y, z) g(x,y) = g(x, z). Given such a function g, we can write g(x,y) = 1 + A(x) (x --y) + O((x _y)2), where A(x) is an n � n matrix-valued one-form on U. Then V(e) (x --y) = e(y) --g(x,y) e(x) = e(y) -- e(x) -- A(x) e(x) (x -- y), in other words V = d -- A. This shows that V is a connection. The cocycle condition for g, taken when (y -- z) is a first order infinitesimal, becomes a differential equation: g(x,y) + a(y) (y -- z) g(x,y) = g(x, z) g(x, z) -- g(x,y) = A(y) g(x,y) (y -- z) or d~g(x,y) = Av(y)g(x,y). The subscripts indicate that the differentials are of the form dy. This equation uniquely determines g given the initial conditions g(x, x) = 1, so V determines 9 uniquely. To complete the proof we have to show that ? or g exists if and only if~7 is integrable. Note that if we can solve the equation d~ g(x,y) ---- A~(y)g(x,y) with initial conditions g(x, x) = 1, then the solution will satisfy the cocycle condition. This is because both g(y, z) g(x,y) and g(x, z) satisfy the same differential equation in z, and they are equal when z = y, so they are equal for all values of z. Change variables by setting t =y -- x. Set A(x + t) = ~ A~(x, t) dt,. This is a formal power series in t with coefficients which are regular functions of x ~ U. The differential equation (really a system because there are several tl, ..., tin) becomes Og(x, t) -- = A~(x, t) g(x, t). 6 42 CARLOS T. SIMPSON This is an ordinary differential equation for g(x, t) which is a formal power series in t with coefficients which are regular functions in x E U. The initial conditions are g(x, O) = 1. It has a solution if and only if it satisfies the integrability condition 0A~ 0A~ --+A~Aj-- +A iA~. Ot~ ~t~ The solution may be constructed inductively to higher and higher order in t. This integrability condition is equivalent to (d -- A) ~' = 0, so the function g exists if and only if V is integrable. This completes the proof of the lemma. [] Crystals of schemes Suppose that X is a scheme of finite type over S, not necessarily smooth. Define a category Inf(X/S) as follows. Its objects are pairs (U C V) consisting of an X-scheme U --+ X and an S-scheme V, with an inclusion U ~ V over S, which makes U into a closed subscheme defined by a nilpotent sheaf of ideals (by this we mean a sheaf of ideals I such that I k = 0 for some k). Such a nilpotent inclusion is sometimes referred to as an infinitesimal thickening. A morphism f: (UC V) ~ (U'C V') consists of a morphism f: V -+ V' of S-schemes, such that the restriction f: U ~ U' is a morphism of X-schemes. Let Inf'(X/S) denote the full subcategory of Inf(X/S) consisting of objects (U C V) such that there exists a morphism V -+ X compatible with the map from U. This morphism is not, however, considered part of the data of (U C V). Remark. -- If X/S is smooth, then any object of Inf(X/S) is, locally in the Zariski topology, isomorphic to an object of Inf'(X/S). This is because the infinitesimal tiffing property for smooth morphisms guarantees the local existence of V ~ X. A crystal of schemes F on X/S is a specification, for each (U C V) in Inf(X/S), of a V-scheme F(UCV) ~V; and for each morphism f:(UCV)-+(U'CV'), an isomorphism +(f) : r(v c V) c V')); such that d~(gf) =f'(+(g)) +(f). A crystal of vector bundles, or just crystal for short, is a crystal of schemes F with structures of vector bundles for F(U C V), such that the +(f) are bundle maps. Equivalently, it is a specification of locally free sheaves F(U C V) on V, with isomorphisms of locally free sheaves ~(f) 9 F(U C V) -~f* F(U' C V'). A stratification of schemes F on X/S is the same sort of thing as a crystal of schemes, but with F(UC V) defined only for (UC V) in the restricted category Inf'(X/S). Similarly for a stratification of vector bundles. According to the above remark, if X/S is MODULI OF REPRESENTATIONS. II smooth then stratifications are the same as crystals. In general, a crystal gives a strati- fication but not necessarily vice versa. We will also use the following terminology. If F ~ X is a morphism of schemes, a relative integrable connection for F on X over S is a stratification of schemes with F as the value over X. The corresponding notion for vector bundles is the same as the usual notion of vector bundle with relative integrable connection. This follows from Lemmas 8.1 above and 8.2 below. Suppose F and G are crystals or stratifications of schemes on X/S. A morphism u : F ~ G consists of a specification of morphisms of schemes u : F(U C V) -+ G(U C V), compatible with morphismsfin Inf(X/S) in the sense that +(f) u = u~b(f). A morphism of crystals or stratifications of vector bundles is the same, with the condition that the u should be morphisms of vector bundles, in other words linear. Lemma 8.2. -- A stratification of schemes on X/S is the same thing as a scheme F(X) ~ X, together with an isomorphism x , satisfying a cocycle condition. This condition says that the two resulting isomorphisms P~3(~) P~(~?) and P~s(~) between the restrictions of F(X) � s X � s X and X � s X � s F(X), are equal. A stratification of vector bundles on X/S is the same as above but where F(X) has a structure of vector bundle over X and e? is an isomorphism of vector bundles. Proof. -- Suppose F is a stratification of schemes on X. This gives a scheme F(X) over X. Let (X x s X) c") denote the n-th infinitesimal neighborhood of the diagonal in X � s X, and similarly in triple products. These are objects in the category Inf'(X/S). The maps PI.~,I and P2.~,~ from (X � s X) ~"~ to X give, by definition, isomorphisms F((X X)'"') p;,,.,(F(X)) and F((X � X)'"') ~ p~,,,,(F(X)). Composing these, we get isomorphisms 9 . ~ s, ~,. p,,,,,(r(x)) = p2,,.,(r(x)). Since the pullback isomorphisms defining the stratification F are functorial and satisfy an associativity, we have r xaX~ ~") ~ ~n for n ~< m, so these isomorphisms fit together into an isomorphism cp between the two pullbacks to the formal scheme (X � s X) A. This provides the desired ~. The associativity rule for the pullback maps implies that on (X � s X � s X) ^ the two isomorphisms p; F(X) x x) ^) and p; F(X) ~ pl~ V((X xs X) ^) ~ F((X � X x 8 X) ^) 44 CARLOS T. SIMPSON are equal. Similarly in other combinations. Thus, all of the resulting isomorphisms between p~ F(X) and p; F(X), are equal. This provides the cocycle condition. Suppose given, on the other hand, an isomorphism ~ satisfying a cocycle condition as described in the hypotheses. For every object U C V in the category Inf'(X/S), choose a map i v : V -~ X compatible with the map U ~ X. Define F(UC V) = iV(F(X)). Suppose f: V -+V'. Then iv, fl U is equal to i v ]u, although they may not be equal on V. Since U C V is defined by a nilpotent ideal, the pair (iv, iv, f) maps V into (X xs X) ^. Note that (iv, iv, f)'p'~ F(X) = F(V). while (iv, i v, f)* p~ F(X) =f* F(V'). Our hypothesis gives q~ :p~ F(X) ~ p~ F(X). Thus we may define q~(f) = (iv, iv, f)" (~), to obtain ~p(f) : F(V) ~-f* F(V'). Given f: V ~ V' and g : V' -~ V", we obtain a map (iv, iv, f, iv,,gf) :V -+ (X � X � X) ^. The cocycle condition for ~ implies that the two possible maps i v F(X) -~ (g f)" iv,, F(X) are equal. In other words, f'(~(g)) ~(f) = ~(gf). This shows that we have defined a stratification of schemes. These two constructions are essential inverses, so we get an equivalence of categories. [] Corollary 8.8. -- Suppose X/S is smooth. A vector bundle with integrable connection on X/S is the same thing as a crystal of vector bundles on X/S. Proof. -This follows immediately from the previous two lemmas, and the contention that crystals and stratifications are the same if X/S is smooth. This contention follows from the remark several paragraphs ago, that any object of Inf(X/S) is locally in Infr(X/S). In order to define F(U C V) for (U C V) e lnf(X/S), cover V by Zariski open sets V, which are in Inf'(X/S). Then use the isomorphisms which are provided on overlaps V~, to glue together the objects F(V~), forming F(V). [] Remark. -- Suppose Z/S is another S-scheme, and j:Z-~ X is a morphism of S-schemes. We obtain a functor j:lnf(Z/S) ~ Inf(X/S) in an obvious way. In fact, Inf(Z/S) is a subcategory of Inf(X/S). If F is a crystal of schemes or vector bundles on X, then the restriction is a crystal of schemes or vector bundles j" F on Z/S. The equivalences of categories given by the preceding lemmas and corollary are compatible with pullbacks. The following proposition was Grothendieck's main observation. MODUL1 OF REPRESENTATIONS. II Proposition 8.4. -- Suppose S o C S is a closed subscheme defined by a nilpotent sheaf of ideals. Suppose X is an S-scheme. Let X 0 = X � s So, still considered as an S-scheme. Let j : X 0 ~ X denote the inclusion. Then the pullback functor F ~j" F is an equivalence from the category of crystals of schemes on X/S to the category of crystals of schemes on X0[S. The same is true for crystals of vector bundles. Proof. -- We have a functor a : Inf(X0/S ) ~ Inf(X/S) defined by a(u c v) = (u c v), and a functor b : Inf(X/S) + Inf(X0[S ) defined by b(U C V) -=- (U o C V). The compo- sition ba is equal to the identity. On the other hand, if (U C V) ~ Inf(X/S) then there is a natural map (UoC V) ~ (UC V), so we get a natural morphism ab--+I. The functors a and b (and this natural morphism) preserw; the schemes V. We obtain functors a', from the category of crystals of schemes on X/S to the category of crystals of schemes on X0]S , and b', from the category of crystals of schemes on X0]S to the category of crystals of schemes on X/S. We have a" b'----I, and there is a natural morphism from b" a" to the identity. Note that (b' a" F)(UC V)= F(UoC V). The natural morphism is given by the pullback (using j: (UoC V) ~ (UC V)), ~?(j) : (F(UoC V) -7 F(U C V). But ~(j) is an isomorphism of schemes. A morphism of crystals of schemes which is an isomorphism over each element of Inf(X/S), is an isomorphism of crystals of schemes--the inverse will also be a morphism. Hence b" a" F ~ F. Thus a and b give an equivalence of categories. [] Representability One can define, in exactly the same way as before, the notions of crystal offunctors or stratification offunctors. These mean that for any object (U C V), F(U C V) is a functor of schemes Y -~ V. The set of such functors forms a pre-stack. In fact, given any stack or pre-stack ~ over the category of schemes, one can define a notion of crystal of r The above lemmas, done tor the stacks of schemes or vector bundles, remain valid. (Any comments about glueing are valid only for stacks, not pre-stacks.) Lemma 8.6. -- Suppose F ~ is a stratification of functors on X]S. Suppose F~(X) is repre- sented or universally co-represented by a scheme F(X). Then we obtain a stratification of schemes F, such that for any V E Inf'(X/S), F(V) represents or universally co-represents F~(V). Proof -- Use the characterization of Lemma 8.2. Note that p~ F(X) represents or universally co-represents the functor p] F~(X), and so forth. Hence the isomorphism of functors on (X � s X) ^ translates into an isomorphism between the pullback schemes. The cocycle condition for the isomorphisms of functors implies the cocycle condition for isomorphisms of schemes. [] Remark. -- It is in this lemma that we are forced to go from crystals to stratifications. 46 CARLOS T. SIMPSON The Gauss-Manln connections Suppose S is a scheme over C, and X/S is a smooth projective family. Suppose : S -+ X is a section. Define crystals of functors M~ and R~,~, on S/C as follows. For (S oC S') in Inf(S/C), define M~f~(S' 0C S') to be equal to the set of isomorphism classes of crystals of vector bundles of rank n, on X'0/S'. Define Rc~r,(S 0 C S') to be equal to the set of isomorphism classes of pairs (E, ~) where E is a crystal of vector bundles of rank n, on X0/S' , and Here 1 is the trivial crystal on X'o/S' ~ S0/S'. These crystals of functors restrict to strati- fications of functors. By Lemmas 8.1 and 8.2, we have M~r,s---M~oR(X/S,n), and R~r~ ~ R~R(X/S, 4, n). The first is universally co-represented by MD~(X/S, n), and the sccond is rcprescntcd by RDR(X/S, ~, n). By the previous lemma, we obtain stratifications of schemes M~t~t(X/S, n) and l~t,~t(X/S, ~, n) on the stratifying site Inf'(S/C). By Lemma 8.2, these data are equivalent to the data of isomorphisms q~ :p~ MDR(X/S, n) ~ p~ MDR(X/S, n) and q~ :p~ RDr~(X/S , ~, n) ~ p~ RDR(X/S, ~, n) on (S xeS)^, satisfying the cocycle condition on (S xcS xeS)^. These are the Gauss- Manin connections. We can make the same definitions as above for the category of complex analytic spaces. The algebraic connections induce analytic connections on R~R(X/S , 4, n) and M~(X/S, n). We would like to show that these agree with the connections coming from the Betti realizations. Recall that the Betti objects Rs(X/S, 4, n) and MB(X/S, n) are local systems of schemes over S ~. The associated spaces R~"'(X/S, ~, n) and M~]n'(X/S, n) are, by definition, products locally over S. In other words, if s e S then there exists a neighbor- hood U of s such that (with the subscript U denoting the inverse image of U) S'~~ 4, n)~ = U x rt~~ ~(s), ,,) and M~"~(X/S, n)v = U � M]3"(X ,, n). A product space of the form U � Z has an analytic relative integrable connection, given by the natural equalities of objects over U � U p;(U � xz)=u � � Thus R~~ ~, n)u and M'8"'(X/S, n)v have analytic relative integrable connections. The local product structures over open sets U and V agree over connected components of U n V, so the connections agree over U c~ V. These then glue together to give analytic relative integrable connections on R~'~'(X/S, ~, n) and M~"~(X]S, n). MODULI OF REPRESENTATIONS. II 47 Theorem 8.6. -- The isomorphisms R~'~'(X/S, ~, n) ~ R~a(X/S, ~, n), M~'(XIS, n) ~ M~R(X/S, n) identify the connections coming from the locally constant structure of the Betti objects, with the Gauss-Manin connections constructed above for the de Rham objects. Proof. -- It suffices to treat the case where S = Spec(A) with A an artinian local (]-algebra of finite type, and X/S is smooth, connected and has a section ~. Let s ~ S denote the closed point. The Gauss-Manin connections are equivalent to trivializations RDR(X/S, ~, n) _---- S x RDR(X0, ~(s), n) and MDR(X/S , n) _--__ S � MDR(X., n). In order to show that the associated trivializations of analytic spaces agree with the trivializations R~'(X/S, ~, n) = S x R~(X., ~(s), n) and M~='(X/S, n) = S x M~"(X,, n), it suffices to treat the cases of the representation spaces, since the maps R(X/S, ~, n) ---> M(X/S, n) are universally submersive. For the representation spaces, it suffices to show that if f: S' ~ RDR(X]S, ~, n) is a point with values in an artinian scheme S' = Spec(A'), which has constant projection on the second factor in the above product decomposition, then the resulting monodromy representation =l(Xs, ~(s)) ---> Gl(n, A') takes values in Gl(n, C). Tile point f corresponds to a vector bundle with integrable relative connection (E,V) on X'----X � and frame ~:E]~s,~ d)~,. The fact that the projection on the second factor of the product decomposition given by the stratification is trivial, implies that there is an open set U C X' (containing the image of ~) and trivializations -~ : U ~ S' � U s with ~(~(S')) - S' � ~(s) and (E, v, p)1~, ~ (p~ ~.)'((~:s, v, p,)]~,.). The local system of relatively constant sections of (P9 x)'((Es, V,)Iu,) is just the tensor product of the local system of constant sections of (E,, V,) 1u, with A'. Thus the mono- dromy representation of (P2 x)*((E,, V,, ~s)Iu,) takcs values ~(Us, ~(s)) -> Gl(n, C) C Gl(n, A'). Note that the map on fundamental groups is a surjecdon ~dus, r ~ ~(x,, ~(s)) ~ 1 The trivialization of (E, V, [~)Iu implies that the monodromy representation takes values in Gl(n, s [] 48 CARLOS T. SIMPSON Remark. -- The above proof gives the following criterion: an artinian scheme- valued point f." S' ~RDR(X/S, 4, n) has constant projection on the second factor RDR(X,, ~(s), n) if and only if there exists an open set I~C X' (containing the image of 4) and trivializations .r : U -~ S' � U 8 with "r(~(S')) = S' � ~(s) and For, if such an open set and trivializations exist, then the monodromy representation takes values ill Gl(n, C). Thus the pointf kas constant projection on the second factor for the stratification of the Betti spaces. But since the de Rham and Betti spaces are analytically isomorphic, and this isomorphism is compatible with the stratifications, the pointfhas constant projection on the second factor for the stratification of R~R(X/S , 4, n). Hence it has constant projection for the algebraic stratification. 9" Principal objects Suppose X is a scheme of finite type over C. In what follows, we will use the term tensor category to denote an associative commutative C-linear tensor category with unit object. A tensor functor is a functor together with natural isomorphisms of preservation of the tensor product, compatible with the associative and commutative structures [Sa] [DM]. Suppose G is a complex linear algebraic group. Let Rep(G) denote the tensor category of complex linear representations of G. Let Vect(X) denote the tensor category of vector bundles (considered as locally free sheaves) over X. A morphism u : E ~ F of objects in Vect(X) is strict if coker(u) is a locally free sheaf. In this case, the kernel and image of u are locally free sheaves. A principal right G-bundle over X is a morphism P ~ X together with a right action of G on P such that there exists a surjective dtale morphismf: X' ~ X and a G-equivariant isomorphism P � X' � G. If P is a principal right G-bundle over X, let P � V be the locally free sheaf in the Zariski topology obtained by descending the sheaf {(p, v) eP(Y) � (V| } y eXet ~ (pg, v) ,-~ (p, gv) for g ~ G(Y) from the ~tale topology X et to the Zariski topology. We obtain a functor Pl, : Rep(G) ~ Vect(X) by setting pp(V) = P � V. This has the following properties: that 0P is strict, in other words ifu : V ~ W is a morphism in Rep(G) then Op(u) is a strict morphism in Vect(X) ; MODULI OF REPRESENTATIONS. II 49 that 0i, is exact, that is pp(ker(u)) = ker(pp(u)) and pp(coker(u)) = coker(pe(u)) ; and that pp is faithful. Furthermore, for any closed point x e X the functor V ~ 9e(V)~ is a fiber functor [Sa] [DM]. Nori has proved the following converse: Proposition 9.1. -- Suppose p : Rep(G) ~ Vect(X) is a strict exact and faithful tensor functor. Then there exists a principal right G-bundle P over X and an isomorphism of tensor functors p ~- 9P; and P is unique up to unique isomorphism. Proof. -- [No]. [] Principal Higgs bundles Suppose X -+ S is a smooth projective morphism to a scheme of finite type over C. Let g denote the Lie algebra of G with G acting by the adjoint representation. A principal Itiggs bundle on X over S, for the group G, is a principal right G-bundle P ~ X together with a section 0 of (P X e' g) | f2~x/s such that [0, 0] = 0 in (P � g) | f2x/s 9 This is the relative version of one of the definitions given in [Si5]. Given such an object and a representation V of G, we get a relative Higgs bundle p~(V) = P � V. Say that P is of semiharmonic type if the Chern classes of the restrictions of P to fibers X, are zero in rational cohomology, and if there exists a faithfifl representation V such that pp(V) restricts to semistable Higgs bundles on the fibers. In this case, the same is true for any other representation (cf. [Si5], remarks after Lemma 6.13). The category of semistable Higgs bundles with vanishing Chern classes (Higgs bundles of semiharmonic type) has a natural structure of tensor category--the tensor product of two semistable Higgs bundles is again semistable [Si5]. Lemma 9.8. -- The construction P ~ pp provides an equivalence between the categories of principal Higgs bundles of semiharmonic type for the group G, and strict exact faithful tensor functors p from Rep(G) to the category of Higgs bundles of semiharmonic type on X over S. Proof. -- This follows from the previous proposition--see [Si5], remarks after Lemma 6.13. [] Lemma 9.3. ~ Suppose E is a Higgs bundle of semiharmonic type on X over S. Fix a number k. There is a projective S-scheme N(E, k) ~ S representing the functor which associates to each S-scheme f: S' ~ S the set of quotient Higgs bundles f*(E) -+ F ~ 0 of rank k such that the Chern classes of F vanish on fibers of X' ~ S' (note that any such F is a semistable Higgs bundle on X' over S', hence of semiharmonic type). Suppose that the fibers X~ are connected, and : S --> X is a section. Then the morphism N(E, k) -+ Grasss(~*(E), k) is a closed embedding. Proof. -- Let P0 denote the Hilbert polynomial of d~ x over S. Let Hilb(E, kPo ) denote the Hilbert scheme parametrizing quotient sheaves E-+ F ~ 0 fiat over S, with Hilbert polynomial kpo. Denote the kernel by O ~ K ~E ~ F ~0. Let 50 CARLOS T. SIMPSON N(E, k)C Hilb~ kpo ) denote the closed subscheme representing the condition that the map 0 : K -+ F |162 f2~'s is zero (see the first paragraph of the proof of Theorem 3.8, Part I). The points of N(E, k) with values in f: S' -+ S correspond to quotient Higgs sheavesf'(E) -+ F -+ 0 on X' over S', such that F is fiat over S' with Hilbert polynomial P. If F is such a quotient, then for any s E S' the fiber F s = F Ix; is a quotient Higgs sheaf of E 8 = E Ixs with normalizcd Hilbert polynomial equal to that of E s. Let K8 denote the kernel of E 8 -+ F,. Then K s is a sub-Higgs sheaf of E~ with the same normalized Hilbert polynomial. By Proposition 6.6, K~ is a strict subbundle with vanishing Chern classes, hence F s is locally free and has vanishing Chern classes. By Lemma 1.27, Part I, this implies that F is locally free. Thus the points of N(E, k) correspond to quotient Higgs bundles F which are locally free of rank k and have Chern classes restricting to zero on the fibers. This is the desired parametrizing space. Note that Hilb(E, kpo ) is projective over S and N(E, k) is a closed subset, hence it is also projective. Suppose X has connected fibers over S and ~:S -)-X is a section. Associating, for each quotient E -+ F ~ 0, the quotient vector bundle ~'(E) -+ ~'(F) --~ 0, gives a morphism N(E, k) ~ Grass(~*(E), k). It is proper, since N(E, k) is proper over S. Suppose F 1 and F, the quotients given by points of N(E, k) (S), such that ~*(F1) = ~*(F,) as quotients of ~*(E). Let K 1 denote the kernel of E --~ F t. Then K 1 and F 2 are Higgs bundles on X over S with the same normalized Hilbert polynomials. By Proposition 6.6, they both satisfy condition LF(X). The morphism +:K1 ~ F2 has ~*(+) = 0, so by Lemma 4.9, Part I, for the case ofA ~'gg~, + = 0. Thus F 2 is a quotient of FI; similarly in the other direction, F 1 is a quotient of F 2 so F1 = 1: 8. This shows that the map N(E, k) (S) -+ Grass(~'(E), k) (S) is injective. The same is true for points with values in any S-scheme S'. A morphism which is proper and injective on the level of points is a closed embedding. [] Suppose that the fibers X s are connected, and ~ : S -+ X is a section. Suppose G C H is a subgroup. Suppose P is a principal Higgs bundle for the group H which is semistable with vanishing Chern classes, on X over S. Suppose b:S-~ ~'(P) is an S-valued point. We say that the monodromy of (P, b) is contained in G if the following condition holds: for every linear representation V of H, and every subspace W C V preserved by G, there exists a strict sub-Higgs bundle of semiharmonic type FC P � such that ~'(F) ={b} � WC ~'(V � If S is a point, we define the monodromy group Mono(P, b) to be the intersection of all algebraic subgroups G C H such that the monodromy of (P, b) is contained in G. Note that the monodromy group jumps down under specialization. Lemma 9.4. -- Suppose O C H. Suppose P' is a principal Higgs bundle of semiharmonic type on X over S, for the group G. Then the principal Higgs bundle P = P' � c H obtained by MODULI OF REPRESENTATIONS. II extending the structure group to H is also of semiharmonic type. This construction gives an identi- fication between: (1) the set of isomorphism classes of pairs (P', b') where P' is a principal Higgs bundle of semiharmonic type for the group G and b' is an S-valued point of 4*(P'); and (2) the set of isomorphism classes of pairs (P, b) where P is a principal Higgs bundle of semiharmonic type for the group H and b is an S-valued point of 4*(P'), such that the monodromy of (P, b) is contained in G. Proof. -- The Chern classes of P are induced by those of P', hence they vanish. To check sernistability of P, choose a faithful representation V of H. This restricts to a faithful representation of G, and we have P x ~ V = P' x ~ V. By the assumption of semistability of P', this is semistable, so P is semistable. Our construction gives a functor from the category of objects (1) to the category of objects (2). To go in the opposite direction, let Rep(G, H) denote the category whose objects are pairs (V, W) where W is a representation of H and V is a G-invariant subspace; and whose morphisms are the G-equivariant morphisms between the subspaces V. Forgetting W gives an equivalence of categories Rep(G, H) -~ Rep(G). On the other hand, suppose we have a principal H-bundle P with a point b e 4*(P) (S), such that the monodromy is contained in G. By definition, for any (V, W) E Rep(G, H) there is a unique sub-Higgs bundle F(V, W) C P x ~ W of semiharmonic type with 4*(F(V, W)) = { b } X V. Given (V, W) and (V', W') and a G-equivariant morphism f: V-+V' we obtain a G-invariant subspace L C W @ W' giving the graph of the map f. The hypothesis of monodromy in G implies that there exists a sub-Higgs bundle L(f) C F(V, W) | F(V', W') which restricts to L on the section 4- This gives the graph of a morphism F(V, W) --+ F(V', W') restricting to f over the section 4 (and the morphism is unique by Lemma 4.9, Part I). We obtain a functor from Rep(G, H) to the category of Higgs bundles of semiharmonic type on X over S, commuting with the functor of taking the fiber along 4. Composing with the inverse of the above equivalence of categories gives a functor from Rep(G). This has a natural structure of neutral tensor functor (one can define a tensor operation (V1, W1) | (V~, W,) = V~ | V,, W~ @ W2) on Rep(G, H) as an intermediate in the definition of the tensor structure). By Lemma 9.2, this gives a principal G-bundle P' as desired. [] Lemma 9.5. -- Suppose E is a Higgs bundle of semiharmonic type, of rank n on X over S. Then the frame bundle P of E has a natural structure of principal Higgs bundle of semiharmonic type for the group Gl(n, C) on X over S. The Higgs bundle is recovered as E = P x alC"'c~ C". This construction provides an identification between the sets of isomorphism classes of (E, ~) and (P, b). Proof. -- Define a category Rep(Gl(n, C), std) whose objects are pairs (V, T a' ~(C")) where the second element refers to the tensor product (C")|174 | and V C T "' b(C") is a Gl(n, C)-invariant subspace. The morphisms are equivariant morphisms of the subspaces V. This category is equivalent to Rep(Gl(n, C)) (and it even has a tensor operation compatible with the tensor product on Rep(Gl(n, C))). Suppose 52 CARLOS T. SIMPSON V CTa, b(C ") is a Gl(n, C)-invariant subspace. Then for any n-dimensional vector space U we obtain a subspace V s T"'b(U) which does not depend on the choice of basis. The same construction holds for vector bundles, so we get a subbundle F C T ~' b(E) with ~(F)= V. The construction of V is also compatible with infinitesimal auto- morphisms, so the subbundle F is preserved by 0. There is a complementary subspace V  and a corresponding complementary subbundle F l. The tensor product T~ is also of semiharmonic type [Si5], so any direct factor such as F is of semi_harmonic type. Morphisms of representations V give rise to morphisms of the Higgs bundles F, and it is compatible with tensor product, so we obtain a tensor functor from Rep(Gl(n, C), std) to the category of Higgs bundles of harmonic type on X. Lemma 9.2 gives the desired principal bundle P. [] Suppose G C Gl(n, C). Suppose E is a Higgs bundle of semiharmonic type on X over S, of rank n, and suppose ~: ~'(E) ~ ~. Let P denote the frame bundle of E, and b the point corresponding to ~. We say that the monodromy of (E, ~) is contained in G if the monodromy of (P, b) is contained in G in the sense defined above. If S is a point, the monodromy group Mono(E, ~) is again the intersection of all subgroups G C Gl(n, C) such that the monodromy of (E, ~) is contained in G. Theorem 9.6. -- Suppose ~:S ~ X is a section. There is a scheme Rr~I(X/S, ~, G) over S representing the functor which associates to any S-scheme S' the set of pairs (P, b) where P is aprincipal Higgs bundle for the group G on X' = X x s S' over S', semistable with vanishing Ghern classes, and b : S' ~ ~'(P) is a section over ~. If f: G ~-~ H is a closed embedding, then f induces a closed embedding ]R~I(X/S , ~, G) ~ ]R~I(X/S, ~, H). Proof. ~ By Lemma 9.5, RI~,(X/S, ~, Gl(n, C)) d_a R~,(X/S, ~, n) does the job for the group Gl(n, C). Suppose now that the existence of RDoI(H ) = RDoI(X/S, ~, H) is known, and that G C H is an algebraic subgroup. Suppose that V is a representation of H and W is a subspace preserved by G. Let (P~', b my) denote the universal principal object on X x s RDo~(H), and let E'~l" = p,~sv � V denote the universal Higgs bundle asso- ciated to the representation V. Let Y"= V|162 ORDoI~r~ ~ and let ~i':~*(Em')- denote the frame given by the point b ~. Let k =dim(V)--dim(W), let YW = W | dYR~(~) denote the corresponding subobject of ~, and let o~-lu~ : R~I(H ) ~ GraSSRDolirn(~r k) denote the section corresponding to the quotient ,v'/~. Let N(E ~j*, k) C Grasssr, l~Hj(~ , k) denote the closed subscheme given by Lemma 9.3 and transported by the frame ~*. Define the closed subscheme C(V, W) d~ a~/~(y(E,~,, k)) C RDo,(H). MODULI OF REPRESENTATIONS. II By Lemma 9.3, this subscheme represents the condition on points g:S'-+Roo~(H), that there exists a quotient Higgs bundle F' of harmonic type of g*(E"~*), with g.(~t~) (~*(F')) = g*($f/W'). This is the same as the condition that there exists a strict sub-Higgs bundle F of harmonic type with g.(~l,) (4*(F')) = g*(W'). Set R.o,(X/S, 4, G) = fl C(V, W) (V, W) where the intersection is taken over all representations V of H and subspaces W preserved by G. It is a closed subscheme of RDo~(H ) which represents the functor associating to an S-scheme S' the set of (P, b) where P is a principal Higgs bundle of semiharmonic type for the group H on X' over S', and b is a point, such that the monodromy of (P, b) is contained in G. By Lemma 9.4, RDoI(X/S , 4, G) also represents the functor associating to S' the set of (P', b') where P' is a principal Higgs bundle of semiharmonic type for the group G and b' is a point. Every linear algebraic group G is a subgroup of Gl(n, C) for some n, so we obtain all of the required spaces Rvo~(X/S, 4, G). The last statement is immediate from this construction. [] Remark. -- The last statement of the theorem, applied to G CGI(n, C), gives RDoa~X;S, ~, G) C Rr~,(X]S, 4, n), because Rvo~(X/S, 4, Gl(n, C)) = R,o~(X/S, 4, n). Our next task is to study the universal categorical quotients of these representation spaces. Assume from now on that G is reductive. Note that G acts algebraically on R~(X/S, 4, G). IfG C Gl(n, C) is a faithful representation, then G acts on Rr~(X/S , 4, n) through its inclusion in Gl(n, C), and this induces the natural action on the subscheme RDoI(X/S, 4, G). Choose a Gl(n, C)-linearized line bundle .Lf on RDoI(X/S, 4, n), such that every point is semistable for the action of Gl(n, C) (cf. Theorem 4.10, Part I). By Mumford's criterion involving one parameter subgroups [Mu], every point is also semistable for the action of G. Thus every point of the closed subset RDoI(X/S, ~, G) is semistable for the action of G with respect to the linearized line bundle .2 ~ By [Mu], we may form the universal categorical quotient MDoI(X/S, 4, G) d~ Rml(X/S, 4, G)/G. Proposition 9.7. -- Suppose that X -+ S is smooth and projective. There exists a space MDo,(X/S, G) which universally co-represents the functor associating to S' ~ S the set of iso- morphism classes of principal tIiggs bundles P of harmonic type on X' over S' for the group G. If the fibers X~ are connected and 4 : S --~ X is a section, then there is a natural isomorphism between MDol(X/S , G) and the universal categorical quotient MDol(X/S , 4, G) constructed above. In this case, the points 0f MDo~(X ~, G) parametrize the closed G-orbits in RDo~(Xs, 4(s), G). Proof. -- Choose an dtale morphism S' -~ S with S' connected, such that each connected component X~ of X' admits a section 4~. Then MDo,(X'/S' , G) ---- I-IMDo,(X/[S', ~,, G) iRDol~X/U,~,O). 54 CARLOS T. SIMPSON universally co-represents the appropriate functor. If{ S'~ } is a collection of 6tale S-schemes covering S, then the collection of spaces M,o~(X'JS'~, G) constructed in this way is provided with descent data (since the functors they co-represent are provided with the corresponding descent data). They descend to give M,o~(X/S , G) which co-represents the desired functor. [] Theorem 9.8. -- Suppose G is a reductive group and f:G---~ Gl(n, C) is a faithful representation. Suppose (P, b) ~ RDo,(X,, ~(s), G) maps to (E, ~) ~ RDo1(X,, ~(s), n). Then (P, b) is in a closed G-orbit in RDo~(X~, ~(s), G) /f and only if (E, ~) in a closed Gl(n, C)-orbit in RDo~(X,, ~(s), n), or equivalently the monodromy group of E is reductive, or equivalently E is semisimple. Proof. -- The subobjects of (E, ~) correspond to the subspaces of C ~ preserved by the monodromy group of E. The monodromy group is reductive if and only if the representation C ~ is completely reducible, thus if and only if E is semisimple. The statement of the theorem is true if G = Gl(n, C): in Theorem 4.10, Part I, as applied in w 6, we have identified the closed Gl(n, (])-orbits in RDol(Xs, ~(s), n) as corresponding to the semisimple representations. Certainly if E is semisimple, then its G-orbit is closed, since the G orbit is a closed subset of the Gl(n, C)-orbit. Suppose E is not semisimple, so H = Mono(E, ~) is not reductive. Let U be the unipotent radical of H. There exists a one parameter subgroup C* -+ G and a family of morphisms ft : H --~ G for t ~ C such that f,(g) = tgt -1 for t ~ C*, and such that the imagef0(H) is not conjugate to H. To see this, we apply the theorem ofMorozov [Mo] --see also [BT]--to conclude that since H is not reductive, it is contained in a proper parabolic subgroup Q., and its radical U intersects the unipotent radical of Q. Now we may assume that O is defined by a torus C* --> G. The Levi component of Q. is the centralizer of this torus, and the torus acts with positive weights on the unipotent radical of Q. In particular, if q ~ Q. then the limit lim,_0 tq t-1 exists. These limits give the map f0, which completes the family ft(q) = tq t-1 defined for t ~e 0. The limits of the elements of the unipotent radical of Q. are the identity element, so there is u ~ H whose limit is the identity. In particular f0(H) has dimension smaller than that of H. This is the required family. The construction of the previous paragraph gives a morphism of group schemes over A 1, f: H � fik I ~ G � A 1. Let (P', b') be the principal Higgs bundle for the group H with P = P' � ~ G. The map f gives an associated relative principal Higgs bundle Pf = P' � on X x A 1 over A 1, with an Al-valued point b I. For t ~ Ak 1, Mono(P/, bf) =f,(H), and furthermore (Pf, b[) ~ (P, tb) for t + 0. Thus (P[, b{) are points in the G-orbit of (P, b), which approach the limit (P0 I, b0 f) as t ~ 0. This limit is not in the same orbit, since its monodromy group, f0(H), is not conjugate to H. Thus the G-orbit is not closed. [] MODULI OF REPRESENTATIONS. II 55 The de Rham spaces We can define spaces RDlt(X/S, ~, G) and MDR(X/S , G) in the same way as above, and obtain the same results. Suppose X + S is a smooth projective morphism to a scheme of finite type ovre C. Let g denote the Lie algebra of G with G acting by the adjoint representation. A principal bundle with integrable relative connection on X over S, for the group G, is a principal right G-bundle P ~ X together with an integrable connection V. For purposes of brevity, we can define an integrable connection as being a G-invariant structure of stratitication of schemes for P on X/S in the sense of w 8. Given (P, V) and a representation V of G, we get a vector bundle with integrable relative connection pp(V) = P � The construction P ~pp provides an equivalence between the categories of principal bundles with integrable relative connection, and strict exact faithful tensor functors p from Rep(G) to the category of vector bundles with relative integrable connection. Lemma 9.9. -- Suppose E is a vector bundle with relative integrable connection on X over S. Fix a number k. There is a projective scheme N(E, k) ~ S representing the functor which associates to each S-scheme f: S' -+ S the set of quotients f*(E) ~ F ~ 0 compatible with the connection. Suppose that the fibers X, are connected, and ~:S ~ X is a section. Then the morphism N(E, k) ~ Grasss(~*(E ), k) is a closed embedding. Proof. -- Let A ---- A D~ be the sheaf of rings of all relative differential operators on X over S. We may consider 1" as a A-module. Let P0 denote the Hilbert polynomial of 9 x over S. The Hilbert scheme Hilb(E, kPo ) parametrizes quotient sheaves E -+ F -+ 0 fiat over S with Hilbert polynomial kpo. Let E~lV~ Fun~v ~0 denote the universal quotient on Xun~v = X � kP0), and let K~'vC E un'v denote the kernel. We get a map A~ "iv | K unlv ~ F univ. Let N(E, k) be the closed subscheme representing the condition that this map pulls back to zero. Then N(E, k) parametrizes quotients E ~ F ~ 0 compatible with the action of A, such that F is flat with Hilbert polynomial kpo over the base. Any such quotient restricts to a sheaf with connection on each fiber, hence to a locally free sheaf. By Lemma 1.27, Part I, F is locally free, so it is a vector bundle with integrable relative connection. From the Hilbert polynomial, it has rank k. Thus N(E, k) represents the desired functor. Furthermore, N(E, k) is projective over S and the natural morphism to Grass(~*(E), k) is injective on the level of S'-valued points, by an application of Lemma 4.9, Part I. Hence the map to the Grassmanian is a closed embedding. [] Suppose that the fibers X, are connected, and ~ : S ~ X is a section. Suppose G C H is a subgroup. Suppose P is a principal bundle with relative integrable connection for the group H on X over S. Suppose b : S -~- ~'(P) is an S-valued point. We say that 56 CARLOS T. SIMPSON the monodromy of (P, b) is contained in G if the following condition holds: for every linear representation V of H, and every subspace W C V preserved by G, there exists a strict subbundle preserved by the connection FC P � such that ~*(F) ={b} x WC ~*(P � If S is a point, we define the monodromy group Mono(P, b) to be the intersection of all algebraic subgroups G C H such that the monodromy of (P, b) is contained in G. We obtain the same result as in Lemma 9.4. Note that the concept of " semi- harmonic type " is not needed, since all Am~-modules are automatically p-semistable with vanishing rational Chern classes. Suppose G C H. Suppose P' is a principal bundle with relative integrable connection on X over S, for the group G. Then P ---- P' � H is a principal bundle with relative integrable connection for the group H. This cons- truction gives an identification between: (1) the set of isomorphism classes of pairs (P', b') where P' is a principal bundle with relative integrable connection for the group G and b' is an S-valued point of 4*(P') ; and (2) the set of isomorphism classes of pairs (P, b) where P is a principal bundle with relative integrable connection for the group H and b is an S-valued point of 4*(P'), such that the monodromy of (P, b) is contained in G. Suppose E is a vector bundle of rank n with integrable relative connection on X over S. Then the frame bundle P of E has a natural structure of principal bundle with integrable relative connection for the group Gl(n, C), and E is recovered as P � o11,. c~ t3". This construction provides an identification between the sets of isomorphism classes of (E, ~) and (P, b). Theorem 9.10. -- Suppose 4 : S --~ X is a section. There is a scheme RIm(X/S, 4, G) over S representing the functor which associates to any S-scheme S' the set of pairs (P, b) where P is a principal bundle with relative integrable connection for the group G on X' -~ X � s S' over S', and b : S' -~ 4*(P) is a section over 4. If f: G ~ H is a closed embedding, then f induces a closed embedding R,R(X/S , 4, G) ~ RD~(X/S, 4, H). Proof -- The same as the proof of Theorem 9.6. [] The analogues of Proposition 9.7 and Theorem 9.8 also hold. Relationship with Bettl spaces Suppose X ~ S is smooth and projective, with connected fibers, and suppose 4 : S ~ X is a section. If G is any linear algebraic group, we obtain a local system of schemes R~(X/S, 4, G) on S ~. These are obtained from the fundamental group I" = nl(Xs, 4(s)) by setting R(P, G) = Horn(F, G) ; the fundamental group nl(S, s) MODULI OF REPRESENTATIONS. II 57 acts on P so it acts on R(F, G), and RB(X/S , {, G) is the corresponding local system of schemes. If G is reductive then define MB(X/S , G) to be the local system of schemes whose fibers are the good quotient R(F, G)/G (which exist because the representation space is affine). If X-+ S is any smooth and projective morphism, we obtain My(X/S, G) by the same dcscent as usual. Recall that the superscript " (~"~ " denotes the analytic total space associated to a local system of schemes. Theorem 9.11. -- We have isomorphisms of complex analytic spaces -- (X/S, G) and, if G is reductive, M~R(X/S, G) _-_ M~'~'(X/S, G). These are compatible with the morphisms of funetoriality induced by homomorphisms of algebraic ,Croups, and they are equal to those given by Theorem 7.1 in the case G -=-- Gl(n, C). The monodromy ,croup corresponding to a point in RDR(X,, ~(S), G) is equal to the Zariski closure in G of the image of the representation parametrized by the corresponding point in RB(X,, ~(s), G). Proof. -- Fix an injective homomorphism G C Gl(n, C). Over an analytic base, a relative vector bundle with integrable connection has monodromy contained in G if and only if the corresponding family of representations has image in the subgroup of points with values in G. Thus the subsets a. R ~" rX'~ " Gl(n, C)) RuR(X/S , ~, G) C DR, /~, ~,, and R~'(X/S, ~, G) C R~'(X/S, ~, Gl(n, C)) represent the same functor of analytic spaces S' -~ S a~. Hence they correspond under the isomorphism of Theorem 7.1. We obtain an isomorphism between the moduli spaces by applying Proposition 5.5, Part I. [] Corollary 9.12. -- If G C Gl(n, C) is a closed embedding then the Gauss-Manin connection preserves the subspace RDR(X/S, ~, G) C RDR(X/S, ~, n). If G is reductive, we obtain a Gauss- Manin connection on the universal categorical quotient 1VIDR(X/S, G). Proof. -- The Gauss-Manin connection on the associated analytic space is the same as the connection given by the local trivializations of the Betti spaces. These trivia- lizations are compatible with the subspaces of representations for the group G. Thus the analytic connection preserves the analytic subspace R~(X/S, ~, G). This implies that the algebraic connection preserves the subspace RD~(X/S, 4, G). The connection descends to the universal categorical quotient by Lemma 8 5. [] 8 58 CARLOS T. SIMPSON Principal harmonic bundles Let J denote the standard metric on C". Recall that we defined in w 7 the space R~oI(X/S , ~, n) whose points over s E S consist of pairs (E, ~) where E is a Higgs bundle on X, and ~ is a frame for E~,) such that there exists a harmonic metric K on E with ~(K~(,)) = J. Similarly, Rarer(X/S, ~, n) was the space whose points over s consist of pairs (E, ~) where E is a vector bundle with integrable connection and ~ is a frame for E~,) such that there exists a harmonic metric K for E with ~(K~,)) = J. Suppose G is a reductive algebraic group. Fix a maximal compact subgroup V C G. Choose an inclusion G ~ Gl(n, C) so that the standard metric J is invariant under V; then V = G c~ U(n). Define Rgo (X/S, 4, = 4, G) ngo,(X/S, 4, n). Endow this space with the topology induced by the usual topology of R~oI(X/S , 4, G). Define 4, O) = 4, G) n 4, n), endowed with the topology induced by the usual topology of R~g(X/S, ~, G). Note that V (with its usual topology) acts continuously on R~I(X/S , 4, G) and R~R(X/S , 4, G). Lemma 9.13. -- The map R~(X]S, 4, G) ~ M~o~(X/S, G) is surjective and proper, and identifies M~(X/S,G) with the topological quotient RJ~(X/S,~G)IV. The map RJ,~(X/S, ~, G) ~ M~(X/S, G) is surjective and proper, and identifies M~(X/S, G) with the topological quotient R JR(X/S, 4, G)/V. Proof. -- We give the proof for the Dolbeault spaces. Recall that R,oI(X/S , 4, G) is a closed subset of RDoI(X/S, 4, n), so R~I(X/S, 4, G) is a closed subset of R~oI(X/S , 4, n). But R~oI(X/S, ~, n) is proper over Mgol(X/S , n), so R~1(X/S, 4, G) is proper over Mgo~(X/S , G). Furthermore, if RJI(X/S, ~, G) ~ Mvo~(X/S, G) is surjective, then M~(X/S, G) is proper over M,o~(X/S , n). To show surjectivity, suppose s ~ S and suppose q is a point of Mvol(X,, G). This can be lifted to a point (E, ~) in a closed G orbit of Rvol(X,, ~(s), G), in which case, by Theorem 9.8, the associated rank n Higgs bundle E is semisimple. Write E = O E~ | A~ where E i are the distinct stable summands of E, and A s are vector spaces. Choose good metrics K~ for E i. Then the good metrics for E are those of the form ]~ K,| i where L i are any metrics on A~. The monodromy group fixes the decomposition of E and acts irreducibly on the components E~. Choose a metric K = ~ K~| for E. Let Mono(E, 1~(8) ) CGI(E~Is)) denote the monodromy group induced by the identity frame lg(,): Er -~ Eg(,). Let W = Mono(E, 1~(,)) r3 U(E~(,,, K~,)). MODULI OF REPRESENTATIONS. II 59 We claim that this is a maximal compact subgroup of Mono(E, 1r162 ). Let ~ denote complex conjugation in GI(E~I,I) with respect to the metric K,. We will prove that fixes Mono(E, l~lsj), and also that every component of Mono(E, l~c,I ) contains a fixed point of ~. Then W, being a compact real form which meets every component, will be maximal compact. Since Mono(E, 1r is reductive, it is equal to the group of elements fixing a subspace of tensors T C Er174 | | Furthermore we may assume that T is the space of all tensors so fixed, and hence there is a decomposition of Higgs bundles E|174 E') | = (T | Ox) | F with F not containing any trivial subobjects. In particular, Mono(E, l~l,~) preserves the subspace F~c,~. Now the harmonic metric K on E induces a harmonic metric on the tensor product, and it tbllows that the direct sum (T | ~x) | F is an orthogonal direct sum of bundles with harmonic metrics. For any g s GI(E~,~), let g" denote the adjoint with respect to the metric K~,~, defined by the formula (ge, f) = (e, g'f) (we will suppress reference to the metric K~,~ in the notation (., .) for the metric on E~,~ or any tensor power thereof). The complex conjugation ~ is given by s(g) = (g.)_h Suppose g e Mono(E, l~c,, ). Then for t s T andfs F~cs~ , we have (g" t,f) = (t, gf) = O, since gfs F~c,~. Similarly if s, t e T then (g" t, s) = (t, gs) = (t, s). Therefore if t s T, g" t = t. In other words, g* e Mono(E, l~l,: ). Thus s(g) = (g')-~ is also in Mono(E, 1 ~,1). This proves that W is a compact real form of Mono(E, 1 ~,~). We still have to prove that it meets every component; this we do by a standard argument. Suppose g s Mono(E, 1 ~c,~). Then gg* is a positive definite self adjoint matrix, so it can be raised to any real power, and we get a real one parameter subgroup of Gl(n) consisting of the self adjoint matrices (gg*)t, t s R. Furthermore, it is easy to see that (gg')~ preserves any tensor preserved by gg', so this one parameter subgroup is in Mono(E, 1~,~). Furthermore, we have a((gg')*)= (gg')-*. Let f(t)=g-'(gg')~; it Ls in Mono(E, 1~,~). Note that f(0) = g-~ and f(1) = g*. On the other hand, a(f(t)) = g'(gg')-' -= g-~(gg') (gg')-' =f(1 -- t). Thus f(1/2) is fixed by a. We have joined the element g-~ to an element of W by a path of elements of Mono(E, 1~1,~). This shows that every component of Mono(E, 1~r contains an element of W, completing the proof that W is a maximal compact subgroup. The group W preserves the metric K~,~ on E~c,~ , and fixes the factors E~, so it preserves the metrics K~ on E~, ~,~. On the other hand, Mono(E, 1~,~) acts irreducibly on E~. ~,~, and W--being a maximal compact subgroup--does too. Therefore K~. ~,; is, up to scalars, the unique metric on E,, ~1,~ preserved by W. Since W is compact, there exists g ~ G such that gWg- ~ C V. Then we may replace the point (E, ~) by (E, g~) ~ Rgo~(X,, ~(s), G), so we may assume W C V. Now J is a W-invariant metric on E, ~= @ E~. ~,1 | A~. But since K~. ~,~ is the unique W-invariant metric on E~. ~r up to scalars, and the E~, ~,~ are distinct irreducible representations fi0 U,A.RLOS T. SIMPSON of W, there exist metrics L~ on A i such that J = Y, K~, z | L,'. Thus our point lies in R~ol(X,, 4(s), n). Set K' =  K~| L[, and ~(K~,,~) ---- J. This proves that R~ol(X/S , ~, G) --> MDol(X/S, G) is surjective. The map is clearly V-invariant, so finally we must prove that two points in Rs~(x/s, ~, G) which map to the same point in MInt(X/S, G) differ by an element of V. Then the properness and surjectivir>" will imply that M~(X]S, G) is the topo- logical quotient space. Again, we may restrict our attention to the fiber over a point RDol(X,, ~(s), G) corresponds to a semisimple object, in other words s E S. Any point in a it is contained in a closed orbit. But the inverse image of a point in Mvol(X,, G) contains exactly one closed orbit. Thus if two points map to the same point in M~(X,, G), we may assume that the two points are (E, ~) and (E, g~). Then there are two harmonic metrics on E, say E K~ | L i and 52 K i| L;, which map to the metric J via ~ and g~ respectively. Note that the stabilizer of E in Gl(n, C) is Stab(E) = l-I GI(A,). There is an element s E Stab(E) such that gs~ takes the metric (Y, K, | Li)~ to J. Thus gs ~ U(n), so g ~ U(n).Stab(E), We have a unique decomposition Stab(E) = (U(n) c~ Stab(E)). (exp(p) n Stab(E)), where gl(n) = u(n) | p is 1he Cartan decomposition. Thus we may write g = up for u ~ U(n) and p e exp(p) n Stab(E). Furthermore, since V = G c~ U(n), we get a Cartan decomposition g =v| c~g), and we may write g: vp' uniquely for vcV and p' ~ exp(p c~ g). It follows that v -- u and p' = p. In particular, p' ~ exp(p) n Stab(E). Thus (E, p' ~) ~ (E, ~) so (E, g~) ~ (E, v~). Thus our two points differ by an element of the maximal compact group V. This completes the proof for the Dolbeault spaces. "['he proof for the de Rham spaces is the same. [3 Lemma 9. la,. -- The equivalence of categories constructed in [Si5] gi~es horr~eomorphisms of topological spaces R~ot(X/S, ~, G) -- R~Da(X/S, ~, G) and M~,(X/S, G) ~ MD,(X/S, G). Proof. -- The equivalence of categories of [Si5] gives an isomorphism of sets R o,(X/S, n) - 4, We have seen in Lemma 7 16 that this restricts to a horneomorphism of subspaces g~t(X/S, ~, n) ~ R~(X/S, 4, n). Furthermore, the equivalence of categories is a tensor functor, so it preserves the mono- dromy groups. Thus it gives an isomorphism of subsets RL,(X/S, 4, G) = R~(X/S, ~, G). MODULI OF REPRESENTATIONS. II 61 Note that Rgol(X/S , 4, G) and Rg.(X/S, 4, G) are respectively closed subsets of Rawol(X/S, 4, n) and Ro.R(X/S, 4, n), endowed with the subspace topologies. Therefore the above isomorphism gives a homeomorphism of topological spaces R~o,(X/S, 4, G) ~ R~a(X/S, 4, G). Furthermore, this is compatible with the action of V. Thus it descends to a homeo- morphism between the quotient spaces which are iDol(X/S, G) and MD~(X/S , G). To finish the proof, note that this homeomorphism is compatible with descent data for going from the case where the fibers are connected and there exists a section, to the general case where the moduli spaces are constructed. [] Corollary 9.15. -- Suppose G and H are reductive algebraic groups, and G ~ H is an injective homomorphism. Then the induced maps between moduli spaces M,ol(X/S, G) ~ MDoI(X/S , H) and MDR(X/S , G) -> MD~(X/S, H) are proper. Proof. -- We may assume that X -> S has a section 4, and that the fibers are connected. Then R~ot(X/S , 4, G) is a closed subset of RaDol(X/S, 4, H). Therefore the map RaDon(X/S, 4, G) -> MDo~(X/S, H) is proper. But this factors through MDo~(X/S, G), and the map li~ol(X/S , 4, G)--*M,ol(X/S , G) is surjective. Therefore the map MDol(X/S , G) ~ MDo~(X/S , H) is proper. The same proof works for the de Rham spaces. [] Surprisingly, we obtain a result about representations of any finitely generated group. Corollary 9.16. -- Suppose Y is a finitely generated group. Suppose G ~ H is an injective homomorphism of reductive algebraic groups. The resulting morphism of moduli spaces M(Y, G) -> M(Y, H) is finite. Pro@ -- Suppose X is a connected smooth projective variety with basepoint x e X. The previous corollary implies that i,a(X, G) -* MDR(X, H) is proper. By Theorem 9.11, this implies that the map Ms(X , G) ->Ms(X, H) is proper. However, the Betti spaces are affine, and an affine proper map is finite. Thus Ms(X , G) -+ MB(X , H) is finite. If Fg denotes the free group on n generators, and if X is a smooth connected projective curve of genus g with basepoint x, then there is a surjection from ~I(X, x) -+ Fg -+ 1 (this is easy to see by drawing a picture of the Riemann surface X ~ as the surface of a solid with g holes). Thus if Y is any group generated by g elements, there is a surjecfion z~I(X , x) -> Y. The additional relations in Y give closed conditions on the representation space, so R(T, G) C RB(X, x, G) and R(Y, H) C RB(X , x, H) are closed equivariant embeddings. Reductivity of the groups G and H implies that the corresponding maps on good quotients M(Y, G) -> M~(X, G) and M(Y, H) -> M~(X, H) are closed embeddings. This implies that the map M(Y, G) -> M(Y, H) is finite. [] 62 CARLOS T. SIMPSON Lemma 9.17. -- Suppose A is a C-algebra of finite type, and N is a finitely generated A-module. Suppose that a reductive algebraic group G acts algebraically on A and N. Then the module of invariants N ~ is finitely generated over A c'. Proof. -- [Mu]. [] Corollary 9.18. -- Suppose Y is a finitely generated group. Suppose G --~ H is a homo- morphism of reductive algebraic groups with finite kernel. Then the resulting morphism of moduli spaces M(Y, G) ---> M(Y, H) is finite. Proof. -- Let G'C H denote the image of G. From Corollary 9.16, the map M(Y, G') ~ M(Y, H) is finite. The map G -+ G' is finite, and the representation spaces are embedded as closed subsets in products of copies of the groups, so the map R(Y, G) ~ R(Y, G') is finite. The map G ~ G' is surjective, so M(Y, G') is a good quotient of R(Y, G') by the action of G. The previous lemma implies that the map M(Y, G) ~ M(Y, G') is finite. Composing these statements gives the corollary. [] This in turn gives finiteness for the maps of Corollary 9.15. Corollary 9.19. -- Suppose G and H are reductive algebraic groups, and G-+ H is a homomorphism with finite kernel. Then the induced maps between moduli spaces MDol(X/S, G) --> MDo,(X/S, H) and M.~(X/S, G) ~ M.a(X/S, H) are finite. Proof. -- The map M~(X/S, G)~M~'(X/S, H) is finite by the previous corollary, and the Dolbeault and de Rham spaces are homeomorphic to these Betti total spaces. Thus M~ot(X/S , G) --~ M~o,(X/S , H) and M~R(X/S , G) --~ M~(X/S, H) are finite. This implies that the corresponding algebraic maps are finite. O Limits of the C" action There is an action of C* on the category of principal Higgs bundles: z e C* sends (P, ~) to (P, zr If G is a reductive group, we obtain an action of C" on MDol(X/S, G). This is compatible with the morphisms of functoriality induced by morphisms of groups, and is equal to the action defined in w 6 in the case G = Gl(n, C). Corollary 9.20. -- For any point y e M~I(X/S, G) the limit lim,_~0 zy exists, and is a fixed point of the action of C*, in M~(X/S, G). Proof. -- Corollary 6.12 gives this statement for the group Gl(n, C). Choose a faithful representation G CGI(n, C). The map C ~ -+M~(X/S, n) extends to a map Ax-~ MDot(X/S, n), and by the properness of the maps in Corollary 9.15, the orbit C* -+ MDo~(X/S, G) extends to a map A 1 -7 Mvo,(X/S , G). The image of the origin is the desired fixed point of C*. [] MODUL! OF REPRESENTATIONS. II 63 Corollary 9.9.1. -- Suppose X is a smooth connected projective variety with basepoint x, and G is a reductive complex algebraic group. Any representation ~I(X, x) -+ G can be deformed to a representation which comes from a complex variation of Hodge structure. Proof. -- The points in the closed orbit of R,ol(X, x, G) lying over fixed points of C* correspond to the representations of the fundamental group which come from complex variations of Hodge structure [Si5]. The same proof as for Corollary 6.12 now works. [] 10. Local structure We will now review the deformation theory of Goldman and Millson (descended from Deligne, Schlessinger and Stasheff) [GM]. The cases of RDn and R~ are identical to [GM], and the case of RDo I is analogous. A differential graded Lie algebra [GM] is a collection A = (A ~ A 1, ... ) of C-vector spaces, with differentials d : A' ~ A ~ + 1 and a bracket [ , ] : A' |162 A t -+ A ~ + ~ such that the following axioms hold: d~= 0; the bracket is graded-anticommutative, [a, b] = (-- 1) '~+1 [b, a] for a EA' and b ~A~; the differential and bracket are compa- tible, d[a, b] ---- Ida, b] q- (-- 1)' [a, db] if a ~ A'; and the Jacobi identity holds with the appropriate signs. Fix a finite dimensional Lie algebra g. A g-deformation diagram is a pair (A, ~) where A ~ is a differential graded Lie algebra and g : A ~ ~ g is a morphism of Lie algebras. Let H' denote the i-th cohomology of the complex (A', d). We say that (A, g) is finite dimensional if the spaces H' are finite dimensional. We say that (A, e) is rigid if the map s : H ~ -~ g is injective. Denote by h the image of H ~ in g, and let h J- denote a subspace transverse to h (for example, if g is semisimple we can take the perpendicular space with respect to the Killing form). The main examples are as follows. Suppose X is a connected smooth projective variety over Spec(C) with a point x ~ X. Let E be a Higgs bundle of semiharmonic type of rank n, with a frame ~:E. ~ C ". Let g = gl(n, C). Then we can define a g-deformation diagram (A,o1(E), e) with A' equal to the space of smooth/-forms with coefficients in End E, the differential d given by the operator D", and the Lie bracket given by the graded commutator of forms. The map ~ is evaluation at x composed with the frame ~. Let G CGI(n) be a complex algebraic subgroup, and suppose (E, ~) satisfies condition Mono(E, [5) C G (in other words (E, ~) represents a point in R,ol(X , x, G)). Put g = Lie(G). Let P be the associated principal Higgs bundle and Ad(P) = P � the adjoint Higgs bundle. We can define a g-deformation diagram (Avot(P), r) with A' equal to the space of smooth/-forms with coefficients in Ad(P). The Lie bracket comes from the Lie bracket of g and the graded commutator of forms, and the augmentation r is given by evaluation at x using P~ --- G. CARLOS T. SIMPSON Similarly, suppose E is a flat bundle (thought of as a representation of the fundamental group, or equivalently as a holomorphic vector bundle with integrable connection), with frame ~:E~-~ C". Let g = gl(n, C). The g-deformation diagram (AB(E), ~) = (A~(E), ~) has A' equal to the space of smooth /-forms with coefficients in End(E), with differential d given by the flat connection D on E, Lie bracket given by graded commutator of forms, and augmentation r given by evaluation at x. If the monodromy group is contained in G CGI(n, t3) (in other words (E, $) represents a point in Rr~X, x, G) or RDj~(X , x, G)), and P denotes the associated flat principal bundle, then we obtain a Lie(G)-deformation diagram (AB(P), ~) = (ADx~(P), e) where A * are the spaces of forms with coefficients in Ad(P), the differential is again given by D, the Lie bracket comes from that of Lie(G), and the augmentation is given by evaluation at x. The deformation theory associated to a deformation diagram We recall the basic elements of the theory of Goldman and Millson--see [GM] for details. Let Art denote the category of artinian local schemes of finite type over Spec(C). An object S a Art is of the form S = Spec(0s) for a local C-algebra Os of finite length. Let m s denote the maximal ideal of O s. Fix a Lie algebra g and let G be an algebraic group with lie(G) = g. Let G~ S) C G(S) denote the set of S-valued points sending the closed point to the identity in G. The group G~ S) depends only on g, not on the choice of G. We have an exponential map from g| to G~ S), denoted u ~ e ", which is an isomorphism of sets. The formulas giving the group structure of G~ S) in terms of the exponential isomorphism of sets are universal, applying also to the case of infinite dimensional Lie algebras. Suppose (A, e) is a deformation diagram. For S e Art, we obtain a group G~ ~ S) with exponenlial map A ~ | ms ~ G~ A~ S). The Lie algebra A ~ acts on the A ~, and this gives an action of the group G~ ~ S) on A ~ | ms- We denote the composition of this action with the exponential map by (u, a) ~ e-" ae". There is also an expression e-"d(e") e A 1 | ms- The formulas for these actions are the same as those that one calculates from the terminologies in the case of a finite dimensional Lie algebra. Given a g-deformation diagram (A, e) and an artinian scheme S cArt, let F(S, A, s) denote the set of pairs (~, g) with ~ e A 1 | m~ and g ~ G~ such that d(~) + ~[~, ~] = 0. The group G~ ~ S) acts on F(S, A, e) by the formula e" : (r~, g) ~ (Ad(e -u) ~ q- e-" d(e"), e-"*' g). Let R(S, A, r denote the quotient of the set F(S, A, ~) by the action of G~ ~ S). MODUI.I OF REPRESENTATIONS. II 65 Lemma 10.1. -- Suppose the diagram (A, e) is rigid and finite dimensional. Then the functor S ~ R(S, A, e) is pro-represented by a formal scheme R(A, ~). Proof. -- [GM]. [] Lemma 10.2. -- Fix a linear algebraic group G and put g = Lie(G). If (P', b') is a principal Higgs bundle of harmonic type (resp. a principal flat bundle) for the group G, with a point b ' e P'~ , then (ADoI(P), r ( resp. (ADR(E), e)) is a rigid and finite dimensional g-deformation diagram, and the formal scheme R(A, e) is naturally isomorphic to the formal completion of the representation space RD,I(X, x, G) (resp. RD~(X , x, G) or R~(X, x, G)) at the point corres- ponding to (P', b'). Proof. -- This is a simple variant of one of the theorems of Goldman and Millson [GM]--theirs is the statement for the space RB(X, x, G). Note that the formal completions of RB(X, x, G) and RDR(X , X, G) at corresponding points are isomorphic, by the analytic isomorphism given in Theorem 7.1. This isomorphism is compatible with the equality of deformation diagrams. We may thus restrict our attention to the case of A~ and RDo ,. Suppose H is a linear algebraic group and N C H is a normal unipotent subgroup. Fix a principal Higgs bundle ofsemiharmonic type with frame (P', b') for the group H/N. Let Z denote the set of triples (P, b, ~) where P is a principal Higgs bundle of semi- harmonic type for H, b is a frame, and a : (P, b) � a (H/N) ~ (P', b') is an isomorphism. Choose (Po, b0, ~o) e Z. Since H acts on Lie(N), we obtain a Higgs bundle with Lie algebra structure P0 � n Lie(N). Let 00 be the operator giving the holomorphic structure of P0, and let q~o be the Higgs fie.ld. Let Y~' denote the set of pairs (u, ~) with u e N and aq e AI(X, Po � Lie(N)), + + = o, up to equivalence under the action of A~ P0 � Lie(N)) given by the same formula as above. Then there is a natural isomorphism between Z' and Z. The principal Higgs bundle corresponding to (u, ~) is P~ = (P0, 0o + ~o,1, q~0 + ~t,o), and the frame is b~ = b o u. The isomorphism stays the same, 0% = ~o. These constructions are functorial in terms of the pair (H, N). Suppose G is a linear algebraic group and S is an artinian local scheme of finite type over Spec(C). Then we obtain a new group scheme G(S) defined by setting G(S) (T) = Hom(S � T, G). There is a morphism G(S) -+ G, and the kernel G~ is a normal unipotent subgroup. There is a morphism of group schemes over S, : G(S) xS G� equal to the identity in the second factor, and equal to the element ofHom(G(S) � S, G) corresponding to the identity in Hom(G(S), G(S)) in the first factor. 9 66 CARLOS T. SIMPSON Let pt : X x S -+ X be the projection on the first factor. If Pa is a principal object for the group G(S) on X, then P;(Ps) is a principal object for the group G(S) on X x S over S, and @(Ps) a"fp;(Ps) XGts'�162 X S) is a principal object for the group G on X x S over S. There is a quasMnverse: if P is a principal G-bundle on X x S then put P(S) (T) ----Hom(S x T, P); this is a principal G(S)-bundle over X with P = F(P(S)). If P has some extra structure then P(S) is provided with the same extra structure. Thus, the functor r gives an equivalence of categories between principal objects for the group G on X x S, and principal objects for the group G(S) on X. Furthermore, the S-valued points of O(Ps)[~,} � s correspond to the points of (Ps),- This construction works for principal bundles, principal Higgs bundles, and principal bundles with integrable connection. Denote by s o e S the closed point. The restriction of O(Pa) to X X { s o } is naturally identified with Ps x a~sl G (and this identification is compatible with the identification of the S-valued points above { x } x S given above). Consequently, the construction 9 applied to principal Higgs bundles preserves the property of semiharmonic type. Applying the previous construction with H = G(S) and N ---- G~ we obtain a natural identification between: the set of triples (P, b, 0c) where P is a principal Higgs bundle of harmonic type on X x S over S, b is an S-valued point of P[t,}� and ~: (P, b)Ix� t,0} ~ (P', b'); and the set of elements of R(S, ADol, ,-). We obtain an isomorphism of functors of artinian local C-schemes of finite type, giving an isomorphism of formal schemes between R(AI~, ~) and the formal completion of RDo~(X, x, G) at (P', b'). [] Remark. -- Under the isomorphism of functors given above, the G-orbit of (E, [5) goes to the set of elements represented by (0, g). Remark. -- Let H C G be the stabilizer of (E, ~). Then H acts on RDol(G ). Since H d G, it preserves the bundle Ad(P0). Thus H acts by conjugation on the diagram DDoI(E, G) (the action on the Lie algebra g is also by conjugation). Thus H acts on the functor F and the representing formal scheme R(F). Our isomorphism is compatible with these actions of H. A morphismffrom a diagram D 1 to a diagram D~ is a collection off ~ from A~(D1) to A~(D~), such thatf ~ ~ ---- ~ f'-1, such thatf(ab) ----f(a)f(b), and such that ~f~ = ~(s). Given such a morphism we get a map of functors F(S, D~) -+ F(S, D,), and hence a map f: R(D1) ~ R(D2). We say that a morphism f is a quasi-isomorphism of diagrams if fo: H0(D,) HO(D ) f~: H'(Dj.) --% HI(D,,) MODULI OF REPRESENTATIONS. II are isomorphisms, and if f*: H'(I),) is injective. The fundamental step in the theory of deformations of Goldman-Millson-Deligne- Schlessinger-Stasheff is the following statement [GM]. Proposition 10.3. -- If f: D 1 --*D 2 is a quasi-isomorphism of rigid finite-dimensional diagrams then it induces an isomorphism of formal schemes f: R(I)~) -~ R(D,). Proof. -- [GM]. [] We will apply this by using the formality results from ([Si5] w 3). Suppose (A, 5) is a diagram where the differentials 8 are zero. Let C C HI(A ") be the quadratic cone which is the zero set of the map from HI(A ") to H2(A ") given by B ~-* B ^ B. Recall that h I is the perpendicular space of the image of H~ ") in g. Goldman and Millson show that the formal scheme R(A, ~) is equal to the formal completion of (~ � it  at the origin [GM]. Theorem 10.4. -- Let G be a reductive algebraic group. Suppose (P, p) is a point in a closed orbit in RDol(X , G) (resp. RDR(X , G)). Let C be the quadratic cone in HX(Ad P) defined by the map ~ ~ ~q ^ ~ ~ H2(Ad P). Let C denote the cone defined above for the formal deformation diagram (AH, r and, let h a. denote the perpendicular space to the image under ~ of H~ P) in g. Then the formal completion (RI)o,(X,G), (P,p))^ (resp. (RDR(X,G), (P,p))^) /s isomorphic to the formal completion (C � la  0) ^ Proof. -- Suppose (P, b) is a framed principal Higgs bundle of harmonic type. We get a Higgs bundle Ad(P) with Lie algebra structure. The Higgs bundle Ad(P) is a direct sum of stable Higgs bundles with vanishing Chern classes. By the results of [Si5], there is an operator D' on C ~~ Ad(P)-valued forms. Let (A1),(P), 5) be the diagram with A ~ equal to the space of Ad(P)-valued/-forms u such that D'(u) = 0 (in the notation of [Si5]). The map ~ is given by D"= 0 + 9 or equivalently by D = D' + D". Let (Aa(P), *) be the diagram with A ~ equal to the space of harmonic forms, which is equal to Hi(A~ol(P)). Here the maps 8 are zero, in other words (AH(P), 5) is formal. Let (Axm(P), 5) be the deformation diagram for the flat principal bundle (P, D) corresponding to the principal Higgs bundle P by the correspondence of [Si5]. We have natural morphisms (AD (P), ") (AD,(P), 5) --+ (ADa(P), 5) (AD,(P), 5) --+ (An(P), r 68 CARLOS T. SIMPSON By [Si5] Lemma 3.2, these are quasi-isomorphisms. By Proposition 10.3 we get iso- morphisms of formal schemes R(AD,(P), s) ~ R(ADo,(P), a) R(AD,(P), e) -~ R(AD~(P), ~) R(AD,(P), 5) ~ R(Aa(P), s). Finally, the formal scheme R(AH(P), 5) is isomorphic to the formal completion of the cone (C � h 1, 0) A. Now apply Lemma 10.2. [] Remark. -- Following Goldman and Millson, we may apply the Arfin approxi- marion theorem [Ar] to conclude that the isomorphism of formal completions comes from an isomorphism of analytic or ~tale neighborhoods. Remark. -- The stabilizer H of (E, ~) acts on all of the above spaces and diagrams, and in particular, H acts on the cone C and on 11  The quasi-isomorphisms of diagrams are compatible with the action of H. Therefore the isomorphism of formal neighborhoods is compatible with the action of H. The cone C is affine and H acts linearly, so there is a good quotient C/H. Proposition 10.5. -- Suppose P is a principal harmonic bundle. The formal completion of the moduli space MDo,(X , G) (resp. I~IDR(X , G), 1VI~(X, G)) at the point P is isomorphic to the formal completion of the good quotient C/H of the cone C by the action 0flL Proof. - Apply Luna's Etale slice theorem [Lu] to construct an H-stable subscheme Y C R,o~(G ) passing through (E, [~), and such that the map Hi� Y ~li~l(G ) is locally an isomorphism in the analytic or Etale topology. Here H  is an H-stable subspace of G passing through the identity, such that H  � H -~ G is locally an isomorphism. Now we have an isomorphism (H  � Y) ^ ~ (H  � C) ^ of formal schemes, preserving the subscheme (H  � { 0))^. From this we get projections yA ~ CA and C A ~ Y^. Their composition is a map Y^ ~ Y^ such that the scheme theoretic inverse image of the origin is just the origin. An argument of counting dimensions of the local ring modulo powers of the maximal ideal shows that this must be an isomorphism, so we get an isomorphism yA ~ C A, This is compatible with the action of the group H. Let Y/H and C/H denote the good affine quotients. Now H is reductive, since it is the stabilizer of a point in a closed orbit [Lu], and because of this, we have (Y/H) A = yA/H and similarly for C. Thus (Y/H) ^ ~ (C/H)^. But Y/H is equal to the moduli space MDo~(G), locally at E. Thus the formal completion of the moduli space is isomorphic to the formal completion of the affine quotient of the cone C by the action of H. ~3 We get canonical isomorphisms between the formal completions of the spaces llDot(X, x, G) and Rnrt(X , x, G), or Mvol(X, G) and Mo~(X, G), at points corres- ponding to the same harmonic bundle. These isomorphisms are not related to the identification between the sets of points given by the harmonic theory of [Si5]. MODULI OF REPRESENTATIONS. II 69 Theorem 10.6 (Isosingularity). -- For any point y e RDoI(X , x, G) (resp. y~RDR(X ,x,G), y~M,ol(X,x,G), or y ~ MD~(X ,x,G)) there exists a point z E RD~(X , x, G) (resp. y E R,ol(X, x, G), y e MD~(X , x, G), or y ~ MDoI(X , x, G)) and gtale neighborhoods U of y and V of z such that (U,y) ~ (V, z); and the local systems corresponding to y and z have isomorphic semisimplifications. Proof. -- By the Artin approximation theorem [Ar], it suffices to show that the formal completions aty and z are isomorphic. Suppose first of all thaty lies in a closed orbit, so it corresponds to a reductive representation. Then let z be a point in the other space corresponding to the same reductive representation o. Then there are natural isomrphisms of cohomology rings H~oI(X , Ad(0))_-__ H~R(X , Ad(p)) [Si5]. Thus the cones that appear in Theorem 10.4 for y and z are isomorphic. The automorphism groups H are also the same in both cases. The formal completions of the representation spaces are both isomorphic to the formal completion of the cone C � g and the formal completions of the moduli spaces are isomorphic to C/H. Suppose y does not lie in a closed orbit. Lety' denote a point in the closed orbit adhering to the orbit ofy. There exists a point z' in the other representation space, and isomorphic dtale neigh- borhoods U' ofy' and V' of z'. There is a pointyl ~ U', mapping to a point in the orbit ofy. In particular, the formal completion ofU' atyl is isomorphic to the formal completion of the representation space at y. Let z 1 denote the point corresponding to Yl under the isomorphism U'~ V', and let z denote the image in the other representation space of z 1. The formal completion of V' at zx is isomorphic to the formal completion of U' at yx, so the formal completion of the representation spaces at y and z are isomorphic. We may suppose that y' is in the closure of the orbit Hya, so z' is in the closure of the orbit Hz 1. In particular, the closed orbits adhering to the orbits ofy and z correspond to the same reductive representations. [] Remark. -- Ify E R,R(X , G) (resp. y ~ MD~(X , G)) and if z denotes the corres- ponding point in RB(X , G) (resp. MB~X , G)) then there are dtale neighborhoods U ofy and V of z, and isomorphisms (U,y) ~ (V, z). This tbllows directly from the Artin approximation theorem, since the analytic isomorphism of Theorem 7.1 gives an isomorphism of formal neighborhoods. The Zariski tangent space We give a result valid for any representation, not necessarily reductive. Lemma 10.7. -- Suppose (P,p) a RDR(X, G). Then the dimension of the Zariski tangent space to R.R(X, G) at (P,p) is equal to h~R(X, Ad(P)) + dim(g) -- h~ Ad(P)). The same for R,ol(X, G). Proof. -- Let D be the de Rham or Dolbeault deformation diagram corres- ponding to (P, p). The Zariski tangent space of the representation space is equal to 70 CARLOS T. SIMPSON R(Spec(C[t]/t) ~, D). The set F(Spec(C[t]/t~), D) is equal to the set of pairs (6, g) where g e g and ~ e A 1 with d(~) = 0. The action ofG~ ~ Spec(C[t][t~)) amounts to changing (~, g) by adding (d(s), r for s 9 A ~ The quotient by this action is H a | (g/~(H~ [] 11. Representations of the fundamental group of a Rierna.n surface Theorem 11.1. -- If X is a connected smooth projective curve of genus g >1 2, then the moduli spaces M~(X, n), MDR(X , n), Mml(X , n), and the representation spaces R~(X, n), RD~(X, n), and RDoI(X, n), are normal irreducible varieties. Most of the rest of the section is devoted to the proof. First we prove that the schemes are reduced and normal. Note that a normal connected variety is irreducible, so for the second statement it suffices to prove connectedness, which we do afterward. At the end of the section, we give some auxiliary statements about the local structure of the representation space. These were obtained in my original proof of the theorem; they are no longer needed in the present proof but it seemed like a good idea to record them anyway. Nor The idea for this part of the proof was suggested by M. Larsen (cf. Corollary 11.6 below). Suppose X is a connected smooth projective curve of genus g/> 2. Choose a basepoint x e X. Lemma 11.2. -- Every irreducible component of Rs(X, x, n) has dimension greater than or equal to 2gn ~- n* + 1. The Zariski open subset R~(X, x, n) parametrizing irreducible representations is smooth of dimension 2gn 2 -- n 2 + 1. Proof. -- First note that R~(X, x, n) is the subvariety of Gl(n, C) ~ defined by one relation. The relation is a map R : Gl(n, C) ~ ~ Sl(n, C), and RB(X , x, n) : R-l(e). This implies that every irreducible component of Rs(X, x, n) has dimension /> 2gn ~- n~+ I. Suppose 0 is a point in R~(X, x, n). Let V denote the local system corresponding to p and let Ad(p) denote the local system End(V). Then Tr : H'(X, Ad(p)) ~ H'(X, C) are isomorphisms for i = 0 and i = 2. On the other hand, if ~ 9 Hi(X, Ad(p)) then Tr([~, ~]) = 0. Therefore the cone C which appears in Theorem 10.4 is equal to all MODULI OF REPRESENTATIONS. II 71 of Hi(X, Ad(p)). By Theorems 7.1 and 10.4, RB(X , x, n) is smooth at O. Finally, the rank of Ad(p) is n 2 so a calculation of Euler characteristics gives dim(C) = dim(H 1) = (2g -- 2) n ~ + 2. In the notation of the previous section, dim(h 1) = n 2- I, so the dimension of Rs(X,x,n) at p is 2gn ~-n 2+ 1. [] Proposition 11.3. -- The dimension of any irreducible component of RB(X , x, n) is equal to 2gn ~ -- n + 1, all irreducible components are generically smooth, and RB(X , x, n) is a complete intersection. The dimension of the subspace of reducible representations has codimension at least two, except in the case g = 2 and n = 2 when it has codimension at one. Proof. -- We suppose that the proposition is known for any n' < n. We will prove the proposition for representations of rank n. For 1 ,< k < n, let Pk denote the parabolic subgroup of Gl(n, C) consisting of block-upper triangular matrices with 2 blocks, where the first block has size k and the second block has size n -- k. There is an exact sequence 0 ~ C k''-'' ~ Pk -~ Gl(k, C) x Gl(n -- k, C) ~ 1, where the kernel represents the abelian group of block upper triangular matrices with the identity matrix in the diagonal blocks. For each Pk, let G k = Gl(n, C)/P k. It is the Grassmanian of k-planes in C", with dimension k(n -- k). Choose a constructible section q~ : G k ~ Gl(n, C). Let RB(X , x, Pk) denote the space of representations of hi(X) into Pk. We obtain a constructible family of representations of ~h(X, x) into Pk indexed by G k � RB(X, x, Pk), corresponding to the constructible map a : G k � RR(X , x, Pk) --*RB(X,x, Gl(n)) defined by a(y, p) = q~(y) pq~(y)-~. This has the property that a(y, 9) is a representation of ~(X) into the conjugateyP~y-~ (this conjugate doesn't depend on the choice of lifting q~(y)). Let n~r -. Gl(n, C)) denote the space of reducible representations. Since every reducible representation has a fixed subspace and is therefore conjugate to a represen- tation in some Pk, we have U .(G~ � RB(X, x, Pk)) = R~(X, x, Gl(n, C)). l~<k<. In particular, the dimension of R~d(X, x, Gl(n, C)) is bounded by the maximum of the dimensions of G, � RB(X, x, Pk). [_.emma 11.4. -- Suppose that Proposition I1.3 is known for representations of rank n' < n. Then for any I ~ k < n, the dimension ofG k x RB(X, x, P,) is less than or equal to 2gn 2 -- nZ; and if g >t 3 or n >I 3 then the dimension is less than or equal to 2gn 2 -- n ~ -- l. Proof. -- We count dimensions, looking at the morphism RB(X , x, Pk) -* RB(X, x, Gl(k, C)) � R,(X, x, Gl(n -- k, C)) 72 CARLOS T. SIMPSON which associates to a representation p its diagonal parts (Pl, P2). We would like to know the dimension of the space of representations into Pk which have given diagonal part (~, ~). Let 7~, ..., 7~,~ denote the standard generators of the fundamental group of X, and let r(Ta, ..., Yso) denote the relation. If we fill in the diagonal parts of the matrices P(7~) according to the given representations Pl, 02, then to specify the remaining part of the representation we have to choose a vector (Ax, ..., A2~ ) with each A~ in the kernel C k("-k~ of the above exact sequence. Putting the resulting matrices into the relation gives a map r 0 : C k(.-k~ � ... � C k(--k~ _+ Ck(--k~. The kernel of this map is the fiber over (Pl, P2), in other words the space of representations with diagonal parts P1, P2. This is the last part of a complex calculating the group cohomology of rq(X, x) with coefficients in the vector space C ~'"-k~, so the cokernel of the map is H2(rq(X, x), Ck("-k~). The action of ~I(X, x) on the vector space of coefficients comes from the adjoint action on Lie(Pk) using the representation p. This only depends on Pl and P2. More explicitly it can be seen by expressing Ck(,-k~ = Ck| C,-k, with action on C a given by Pl and the action on C "-k given by P2- By Poincar6 duality, the dimension of the H s is the same as the dimension of H~ x), p~ | P2)- Thus the dimension of the fiber over (Pl, 02) is (2g -- 1) k(n -- k) + h~ ?~ | p~). If Pl and 02 are irreducible and not isomorphic, then H~ x), 9] | Ps) = 0. Therefore we can count the dimension of the fiber over (Pl, P,) as (2g -- 1) k(n -- k). By induction, the dimension of the space of choices of (Px, 98) is (2g -- 1) (k s + (n -- k) 2) + 2. The dimension of this part of G, x RB(X, x, Pk) is (2g-- 1) (k s+ (n--k) 2 + k(n - k)) + 2 + k(n -- k) = (2g-- 1) n2+ 1-- ((2g--2) k(n--k) -- 1). In particular, as gt> 2 and n>/ 2, the dimension is at most 2gn 2-n 2. If g/> 3 or n/> 3 then the dimension is at most 2gn 2 -- n 2 -- 2. The set of pairs (Pl, P2) such that both representations are reducible has (by induction) dimension bounded by (2g -- 1) (k 2 + (n -- k)2). For these points we make a coarse counting of the dimension of the fiber over (01, P2): it is less than 2gk(n -- k). The dimension of this part of G k � RB(X, x, Pk) is therefore bounded by (2g-- 1) (k 2+ (n--k) *) + (2g+ 1) k(n--k) = (2g-- 1) n *+ l--((2g--3) k(n--k) + 1). The dimension is at most 2gn 2 -- n 2 -- I for g/> 2 and n t> 2. The set of pairs (01, P~) which are irreducible and isomorphic (hence of rank k = n- k = hi2) has dimension less than or equal to (2g- 1)kS+ 1. The H ~ has MODULI OF REPRESENTATIONS. II dimension 1, so the fiber has dimension (2g- 1)kS-t - 1. The sum of the dimensions is less than or equal to (2g -- 1) n~[2 + 2. For n 1> 2 and g/> 2 this is less than or equal to 2gn 2 - n ~ - 4. The set of pairs (P1, P2) such that one representation is reducible and one repre- sentation is irreducible has dimension bounded by (2g- 1) (kS+ (n- k) ~) + 1. For such a pair, the H ~ discussed above has dimension 0 or I. Therefore the dimension of the fiber over (Pl, P2) is bounded by (2g- 1)k(n- k) + 1. The dimension of this part of the space G k � liB(X, x, Pk) is bounded by (2g-- 1) (k s+ (n--k) 2 § k(n--k)) +2 + k(n--k) = (2g-- 1) n'+ 1 -- ((2g--2) k(n--k)-- 1). In this case, note that n must be at least 3. Therefore, the dimension is at most 2gn ~ -- n 9 -- 2. We have shown, in all the cases, that the dimension of G~ x liB(X, x, Pk) is less than or equal to 2gn ~ -- n 2, and if n i> 3 or g >/ 3 then the dimension is less than or equal to 2gn z- n 2- 1. [] We continue with the proof of the proposition. It follows from the lemma that the dimension of the subspace of reducible representations is less than or equal to (2g -- 1) n ~. From the lower bound of Lemma 11.2, no irreducible component can consist entirely of reducible representations. Therefore the open set of irreducible representations is Zariski dense, so the dimension of each component is equal to (2g -- 1) n ~ + 1. It follows from the equations for liB(X, x, n) given in the proof of Lemma 11.2 that liB(X, x, n) is a complete intersection. Since li~(X, x, n) is smooth, each component is generically smooth. Finally note that, except in the case g = 2 and n = 2, Lemma 11.4 shows that the dimension of the subspace of reducible represen- tations is less than or equal to (2g -- I) n 2 -- 1. This proves the proposition. [] Lemma 11.5. --- The scheme llB(X , x. n) is smooth outside of a subset of codimension >t 2. Proof. -- This follows from the previous proposition except when g = 2 and n = 2. We are reduced to that case, where the space of representations has dimension 13. From the proof of Lemma 11.4, the codimension 1 part of the locus of reducible representations consists of those representations conjugate to an upper triangular representation with distinct diagonal parts. We show that the space of representations is smooth at such points. This statement is invariant under conjugating the representation; so we may fix an upper triangular representation p, with diagonal entries Pl + Oz. The space of semisimple reducible representations has dimension 10, so we may assume that p is not semisimple. For a 2 � 2 representation, this implies that there is a unique sub- representation of rank 1 and a unique quotient. Make the convention that the sub- representation is Pl and the quotient is P2- We claim that there are no nonscalar endomorphisms of the representation p. 10 74 CARLOS T. SIMPSON For iffis an endomorphism of rank 1, then the image off must be the subrepresentation PI, while the coimage must be the quotient p,, contradicting the condition that Pl 4- P2. Thus if ~ is an eigenvalue of an endomorphismf then f-- k = 0 andfis a scalar. Thus, as claimed, H~ Ad(p)) = { 0 } (we say that p is simple). By Lemma 10.7, the dimension of the Zariski tangent space to RB(X , x, n) at is equal to hX(Ad(p)) + n ~ -- h~ Since p is simple, h~ = 1. By Poincar~ duality, h~(Ad(p)) = 1, so hX(Ad(p)) = (2g -- 2) n' + 2 = 10, and the dimension of the Zariski tangent space is 10 + 4 -- 1 = 13. Since this is equal to the dimension of any irreducible component of the space, the local ring is regular and R~(X, x, n) is smooth at p. This completes the proof of the lemma. [] Corollary 11.6. --- The space of representations Rs(X , x, n) # reduced and normal. Proof. -- This was pointed out to me by M. Larsen (he refered me to [Ha], Proposition 11-8.23). By Proposition 11.3, RB(X, x, n) is a complete intersection. The local rings of a complete intersection are Cohen-Macaulay, hence satisfy Serre's condi- tion $2. The previous lemma shows that the space of representations is regular in codimension 1. By Serre's criterion, RB(X , x, n) is reduced and normal. [] Corollary 11.7. -- The representation spaces RDol(X, x, n), RD~(X, x, n), and R~(X, x, n) are normal varieties of dimension 2gn ~ -- n ~ + 1. The moduli spaces M~,, M,R , and M s are normal varieties of dimension 2gn ~ -- 2n ~ + 2. Proof. -- We have shown that RB(X , x, n) is normal of dimension 2gn 2 -- n ~ + 1. By the isosingularity principle (Theorem 10.6 and the following remark), the same is true for the de Rham and Dolbeault spaces. Good quotients of normal varieties are normal. (This can be seen by proving that if A is a ring which is integrally closed in its field of fractions K and a group acts, then A ~ is integrally closed in its field of frac- tions K~ Thus the moduli spaces are normal varieties. To calculate their dimensions, note that there is a Zariski dense open set of points of the representation space where Sl(n, 12) acts with finite stabilizer. The dimension of the quotient is the dimension of the representation space minus the dimension of Sl(n, 12). [] This corollary provides the first half of the proof of Theorem 11.1. To complete the proof, it suffices to prove that these varieties are connected. Connectedness of the representation spaces is equivalent to connectedness of their universal categorical quotients, the moduli spaces. By Proposition 7.8 and Theorem 7.18, the three moduli spaces are homeomorphic. Thus it suffices to prove that MDo,(X , n) is connected. The idea for the proof of connectedness comes from Hitchin's calculation of the cohomology of the moduli space of rank 2 projective bundles with odd degree. In that case, the moduli space is smooth, and Hitchin uses a Morse function, the moment map for the action of S x, to calculate the cohomology. The lowest stratum is the space of 75 MODULI OF REPRESENTATIONS. ]I unitary representations, known to be connected by the work of Narasimhan and Seshadri using a lemma of Atiyah. Hitchin deduces from Morse theory that the moduli space is connected and hence irredtlcible, and in fact he calculates the Betti numbers. It would be good to carry through this program for higher ranks and for the case when there are singularities. We will not attempt this here, but we will do enough to show that the moduli space is connected. Because of the presence of singularities, we will avoid Morse theory and instead proceed by algebraic geometry, using the C* action discussed in w fi (wbAch is the complexificat~on of H~tcbAn's S t action). The relationship betweex: these approaches is that the critical point set of the moment map is equal to the fixed point set of the C* action. In ([Si5] w 4), the fixed point set was identified with the set of complex variations of Hodge structure. We remark that in order to apply Hitchin's method to compute the Betti numbers, one would have to be able to compute the Betti numbers of the moduli spaces of variations of Hodge structure. Actions of C* Suppose that Y is a quasiprojective variety on which G* acts algebraically. Suppose that L is an ample line bundle w~th a tinear]zation of the action. Then C" acts locally finitely on H~ L ~ ~). Therefore we may choose n and a subspace V C H~ L~-) which is preserved by G" and which embedds Y into the projective space P(V*), so that C* acts on this projective space and acts compatibly on the very ample 0(1). Thin action is compatible with the embedding of Y and with the isomorphism L| @~(1). Write V = O V~ where the sum is over integers ~, and t e C* acts by t" on V,. Then the fixed point sets of C* on P(V*) are the subspaces P(V~). The fixed point sets in Y are Y~ ---= Y c~ P(V*), The action oft e C* on L IY~ is given by t ~. Let Z be the closure of Y in P(V*). Then Z is preserved by C*, and its fixed point sets are Z~ = Z tn P(V[). If z ~ Z then there are unique points lim,_,o tz and lim4_~ tz in Z. These are fixed points, kence are in some fixed p()ints set Z~ and Z~ respecfively. We will describe the weights ~~ and ~(z) explicitly (see the discussion near the end of w 1, Part I). Lift z to a point weV*, and write w=Zw, with w, aV*. Then ~~ (resp. ~(z)) is the smallest (resp. largest) integer ~ such that w, 4= 0. From this description, if z e Z then ~~ ,< ~(z) and equality holds if and only if z is a fixed point. Assume that Y has the property that lira t -~0 tx exists in Y for all x e Y. Let f~ ---- ~0(y) be the smallest integer such that Ya is nonempty. Then ~ is also the smallest integer such that Z~ + 0. If x E Y~ and there exists y e Y such that y # x and lim t _~ ~ ty = x, then a > ~. In particular, we obtain the following criterion. Lemma 11.8. -- Suppose that Y has the property that limt_~0 tx exists in Y for all x E Y. Suppose U C Y is a connected subset of the fixed point set of C', and suppose that for any fixed point x not in U, there exists y * x in Y such that lim t _~ ~o ty = x. Then Y is connected. 76 CARLOS T. SIMPSON Proof. -- Suppose Y'is a connected component of Y not containing U. Let ~ = [3~ Choose x e Y~. By hypothesis there existsy # x in Y' such that lim t ~.~o ty = x. On the other hand, z ---- limt_,0 ty is also in Y', say in Y',. Buty is not a fixed point, so o~< ~, contradicting minimality of [3. [] Connectedness Lemma 11.9. -- Suppose E is a stable Iliggs bundle of degree zero, fixed up to isomorphism by the action of C'. Suppose that q~ + O. Then there is a I[iggs bundle F not isomorphic to E, such that lim, _~ ~o tF = E in the moduli space. Proof. -- Since E is a fixed point, it has the structure of system of Hodge bundles, in other words E O E ~ with 9:E~--+E ~-1 ' = | Assume that the indexing is normalized so that 0~<p~< r, and E ~ and E'# 0. Note that r>/ 1 since 9~:0. Furthermore, note that deg(E ~ < 0 and deg(E') > 0, since E is stable of degree zero. Hence Hom(E', E ~ has degree < 0. In particular the Riemann-Roch theorem implies that there exists a nonzero extension class "~ in Extl(E ", E ~ = H'(Hom(E', E~ For each t e C, let M~ be the extension 0 -+E ~ ~ M t -+E" -+0 given by the class t" ~. Let F t be the Higgs bundle M s | Oo< ~<, E ~. The Higgs field 9 is given by the usual maps for 1 < p < r and by the compositions E I -~ E~ | t2~ -+ M~ | and M~ ~ E' ~ E'- 1 | t2~. Note that F 0 ---- E. We have isomorphisms ~t : Ft --- t-' F 1 given as follows. On E ~, 0 < p < r, 9~ is given by multiplication by t ~. On M, qh is the isomorphism fitting into the middle of the diagram E ~ > M t > E' 1" E ~ > M 1 > E'. Note that 90 = t-1 0<p. Thus we have a family of Higgs bundles F, with F 0 = E and F t = t- ' F t for t # 0. Since E is stable, so are Ft, by the openness of the condition of stability. Hence lira t -~o tFt = E. To complete the verification, we will show that the vector bundles underlying E and Fx are not isomorphic. First we show that M 1 + M 0. Let A C E' be the [5-subsheaf, in other words the subsheaf of highest slopc, and highest rank among subsheaves of that slope. We may choose aq to be a nontrivial extension of A by E ~ Note that the degree of any subsheaf of E ~ is < 0, whereas the slope of A is > 0. Thus if f: A r M is an inclusion, we must MODULI OF REPRESENTATIONS. II have p of: A -~ A where p is the projection from M to E'. Thus (p of)'(~) 4: 0. But the mapfis a splitting of (p of)'(~). This contradiction shows that there is no inclusion of A into Mx, and hence there can be no isomorphism M 0 -~ M~. The bundle M 1 is a deformation of M 0. By semicontinuity, h~ M0) ) >/h~ M~)). Furthermore, the inequality is strict: for if not then t ~ H~ Mr) ) would form a vector bundle over the t-line, and 1 ~ H~ M0) ) could be lifted to ft ~H~ withft ~1 as t ~0; then ft:M0~ Mt = Mx for t near 0, which would contradict the conclusion of the previous paragraph. Now in our situation there exists a vector bundle B (the direct sum of the other Hodge components) such that E = M 0@B and F = M IQB. We get h~ E)) = h~ Mo) ) + h~ B)), while h~ F~)) ----- h~ M~)) + h~ B)). Thus h~ E)) > h~ Ft)), so the vector bundles E and F x are not isomorphic. [] Corollary 11.10. -- The moduli space M~I(X , n) is connected. Proof. -- The ample line bundle L on MDo,(X, n) has a linearization of the action of C ~ This is because we constructed MDol(X, n) as the moduli space of some sheaves on the cotangent bundle to X, and the action ofC ~ came from the action of multiplication on T" X, so C" acts functorially on the Hilbert schemes, the Grassmanians, and the line bundles over the Grassmanians. We apply the criterion of Lemma 11.8. Let U be the subspace corresponding to Higgs bundles with 0----0 (these are the ones corresponding to unitary representations). It is isomorphic to the moduli space of vector bundles of rank n on X. The moduli space of vector bundles is projective, so U is a closed subset of MDoI(X, n). The subset U is connected--this fact was used by Narasimhan and Seshadri [NS] and comes from a lemma of Atiyah. Suppose that E is a direct sum of stable components, representing a point in M~t(X , n) -- U fixed by C'. All of the stable components of E are then fixed by C ~ We can write E----E I| z with E1 stable and not unitary. Apply Lemma 11.9 to obtain F 1 with lim~ ~o FI = Ex, and F 1 4: E~. Set F = F 1| E 2. Then lim~..co F = E, but gr(F)4= gr(E). The criterion of Lemma 11.8 now implies that M~I(X ,n) is connected. [] Proof of Theorem 11.1. -- From the homeomorphism given by Proposition 7.8 and Theorem 7.18, the varieties M~(X, n) and MoR(X, n) are also connected. By Corollary 11.7, all the moduli varieties are normal, and a normal connected variety is irreducible. [] 78 CARLOS T. SIMPSON REFERENCES [Ar] M. ARTIN, Algebraic approximation of structures over complete local rings, Publ. Math. LH.E.S., 86 (1969), 23-58. [Be] J. BERSSTEm, Course on ~-modules, Harvard, 1983-1984. 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