Periods of integrals on algebraic manifolds, III (some global differential-geometric properties of the period mapping)

Phillip Griffiths

doi:10.1007/BF02684654

Periods of integrals on algebraic manifolds, III (some global differential-geometric properties of the period mapping)

Griffiths, Phillip 2007-08-06 00:00:00 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III (SOME GLOBAL DIFFERENTIAL-GEOMETRIC PROPERTIES OF THE PERIOD MAPPING) by PHILLIP A. GRIFFITHS (i) TABLE OF CONTENTS PAGF~ o. Introduction ................................................................ ~6 Part I. --- Differentlal-Geometric Properties of Variation of Hodge Structure .............. ~3 o I. Algebraic families of algebraic varieties ........................................ 13o 2. Variation of Hodge structure ................................................. z3t 3. Remarks on the homology of algebraic fibre spaces ............................. ~34 4. Remarks on Hermitian differential geometry .................................... x36 5. Statement of main differential-geometric properties of Hodge bundles ............. I39 6. Structure equations for variation of Hodge structure ............................ 14I 7. Applications ................................................................ x45 a) Invariant cycle and rigidity theorems ....................................... x45 b) Negative bundles and variation of Hodge structure .......................... I46 c) A Mordell-Weil theorem for families of intermediate Jacobians ................ x47 Part II. --- Differential-Geometric Properties of the Period Mapping ....................... 8. Classifying spaces tbr Hodge structures ........................................ 9- Statement of results on variation of Hodge structure and period mappings ........ ~55 to. The generalized Schwarz lemma .............................................. I58 II. Proof of Propositions (9. Io) and (9. xx) ........................................ x62 Appendix A. -- A result on algebraic cycles and intermediate Jacobians .......................... ~65 Appendix B. -- Two examples: ,~) A family of curves .................................................................. I7I b) I,efschetz pencils of surfaces .......................................................... x7 ~ Appendix C. -- Discussion of some open questions: a) Statement of conjectures ............................................................. b) Proof of the invariant cycle theorem (7. I) for n=t ................................... ~75 c) Proof of the usual Mordell-Weil over function fields .................................... x77 Appendix D. -- A result on the monodromy of K 3 surfaces ...................................... x78 (a) Supported in part by National Science Foundation grant GP7952XI. 125 126 PHILLIP A. GRIFFITHS o. Introduction. a) In this paper we shall study some global properties of the periods of integrals in an algebraic family of algebraic varieties. Although our results are mostly in (alge- braic) geometry, the proofs are purely transcendental. In fact, we may roughly describe our methods as giving various applications of the maximum principle to problems in algebraic geometry. For the most part these methods have only succeeded in treating the situation when the parameter space for the non-singular varieties is complete. While the results should be true in general, it appears that new methods will be required to handle the situation when singular varieties are permitted in our family. These questions are discussed from time to time as they occur in the text below. The paper divides naturally into two parts. The first treats linear problems and is a study of the differential-geometric properties of the Hodge bundles as defined in w 2. The use of the maximum principle here is similar to the classical Bochner method [3], and is based on the rather remarkable structure equations and curvature properties of the Hodge bundles. The second part deals with global properties of the period mapping [: :], and the methods are those of hyperbolic complex analysis which, to paraphrase Chern [7], is the philosophy that suitable curvature conditions on complex manifolds impose strong restrictions on holomorphic mappings between these manifolds. A more detailed introduction to the two parts of the paper will now be given. b) Wc consider an algebraic family of algebraic varieties {V,}se s as defined in w :. For the time being we may think of the parameter space S as being a (generally non- compact) algebraic curve. The algebraic varieties V: corresponding to the points Y at infinity in S may be thought of as the specializations of a generic Vs having acquired singularities. If we replace V, by the cohomology groups H"(V,, C) and the various subspaces H~.q(V,) cH~(V~, 13) (p-t-q=n), then we find that the algebraic family {V~}~a s gives rise to a whole colh:ction of holomorphic vector bundles over S. We can abstract the data of theses bundles and arrive at what we call a variation of Hodge structure (w 2). Now the bundles which turn up in a variation of Hodge structure have intrinsic Hermitian differential geometries (w 4), and we use more or less standard methods in differential geometry to deduce results about the variation of Hodge structure which, in case this variation of Hodge structure arises from a family of algebraic varieties, have interpretations as theorems on invariant cycles and on holomorphic cross-sections of families of intermediate Jacobians. The only real twist is that the Hermitian vector bundles which appear generally have indefinite Hermitian metrics. In such a situation, the maximum principle does not usually apply. We are only able to push things through by using the so-called infini- tesimal period relation [I I] satisfied by periods of integrals and which is incorporated into the definition of variation of Hodge structure. It is perhaps worth pointing out that 126 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III ~27 the maximum principle is used to show that certain differential equations are satisfied, rather than to show that a " harmonic tensor " is zero as was the classical case [3]. In w 4 we give a review of Hermitian differential geometry and, in particular, discuss the second fundamental form of a holomorphic vector bundle embedded in an Hermitian vector bundle. The main differential geometric results on variation of Hodge structure (Theorems (5.2) and (5.9)) are stated and discussed in w 5. The proofs of these theorems are given in w 6 where we derive the structure equations for a variation of Hodge structure. This section is the heart of Part I of the paper, and we have used the Cartan method of moving frames ([6], [8]) to expose the structure equations (6.4)'(6-8), (6.12), and (6.18) of a variation of Hodge structure. These equations are to me quite remarkable and are much richer than one might have thought from just the classical case when the V 8 are curves. For example, if {V,}se s forms an algebraic ti~mily of algebraic surfaces with complete parameter space S and if y, eH~(V,, Z) is an invariant 2-cycle, then there is a non-negative function qb(s) whose vanishing at seS is necessary and sufficient that y, be the homology class of an algebraic curve on V,. It was a pleasant surprise to me that ~ turns out to be pluri-subharmonic on S. Even in ease S is not complete, ~ should be bounded, but this depends on the local invariant cycle conjecture (3-3). In w 7 we give three applications of the results in section 5. The first of these are some rigidity properties of variation of Hodge structures with complete base space (Corollary (7-3) and (7.4))- These particular results were motivated by a question of Grothendieck [17] and have appeared previously in the preprint [I2] with the same proof as given here. In this paper the rigidity theorems are given as consequences of Theorem (7.I), which was also in w 8 of [12] but was poorly stated there. The much better formulation given below is due to Deligne, whose paper [9] has several points of contact with this one, which are discussed in w 3 below. The second application is the positivity of certain bundles arising from a variation of Hodge structure (Propositions (7.7) and (7.15)). The third application is a Mordell-Weil type of theorem for cross-sections of families of intermediate Jacobians (Theorem (7.19)). Again this is a result purely about variation of Hodge structure but which is suggested by algebraic geometry. In this case the motivation comes from the study of intermediate cycles on algebraic varieties and the connection with Theorem (7-I9) is explained in Appendix A. c) Associated to any variation of Hodge structure, with parameter space S, there is a period matrix domain D [II], which is a homogeneous complex manifold D = G/H of a non-compact simple Lie group G divided by a compact subgroup H, together with a denumerable subgroup P of G and a holomorphic period mapping [r r] : (o.x) 9 : S-~r\D. In fact, the giving of a variation of Hodge structure over S is equivalent to giving a period mapping (o. I) satisfying an infinitesimal period relation which can be stated 127 x~8 PHILLIP A. GRIFFITHS purely in terms of D. These period matrix domains are discussed in w 8, and the correspondence between variations of Hodge structure and period mappings is given by Proposition (9.3). In many interesting cases, such as when the variation of Hodge structure arises from an algebraic family of algebraic varieties, the monodromy group P is a discrete subgroup of G and consequently r\D is a complex analytic variety. The point of view we have taken in Part II is to apply hyperbolic complex analysis to study the period mapping (o. t). We are especially interested in the asymptotic behavior of the period mapping q) as we go to infinity in S. In case dimcS=I , a neighborhood of S at infinity is a punctured disc A*, and the period mapping (o. I) may be localized at infinity and lifted to the universal covering of A* to yield a holomorphic mapping: : H-~D from the upper half-plane H={z=x+~y :y~'o} to thc period matrix domain D, and which satisfies the equivariance condition: ~(Z+I)=T.O(Z) where TaP is the Picard-Lefschetz transformation associated to the local monodromy around the origin in the punctured disc A*. In case dimeS> x, we can use Hironaka's resolution of singularities to have a similar localization at infinity given by a holomorphic mapping: (o."-) q) : HX... X H-+D which satisfies : *(zl, ..., + ..., = Tj. r ..., where the TgG are commuting automorphisms of D. To use metric methods for the study of the mapping (o.2), we introduce the standard Poincard metric ds~ on H � � H and the G-invariant metric ds~ on D deduced from the Cartan-Killing form on the Lie algebra of G. Now the metric ds~ does not have the (negative) curvature properties necessary to make hyperbolic complex analysis work on an arbitrary holomorphic mapping (o. 2). However, if we use the infinitesimal period relation, then the necessary curvature conditions will be satisfied relative to the mapping r Using this together with a formula of Chern [7], in w IO we prove a generalized Schwarz lemma (Theorem (IO, I)) which says that the period mapping is both distance and volume decreasing with respect to ds~ and ds~, cf. [~6]. The main geometric applications of the Schwarz lemma are Theorems (9-5) and (9.6), both of whose proofs are given in w I I. The first of these is a sort of Riemann extension theorem, and says that a period mapping r : A*~D from the punctured disc to a period matrix domain D extends holomorphically across the origin. Our proof makes essential use of an ingenious argument from [~5] (Proposition (iI. I)). The second result is that the period mapping (o.2) is (essentially) a proper mapping, and 128 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III x~9 consequently the closure of the image of 9 is an analytic set containing ~(S) as the complement of an analytic subvariety. A third geometric theorem is Theorem (9-7), which says that the image q~(S) is canonically a projective algebraic variety in case S is complete. Our final result (Theorem (9.8)) in this section is a theorem about the global monodromy group I' of a variation of Hodge structure with complete parameter space. The statements that I' is completely reducible, and that F is finite if it is solvable, are simply adaptations of similar results of Deligne [9] in the geometric case, which are discussed in w 3 below. The characterization of the case when F is a finite group was given in [I2]. d) As mentioned above, Appendix A contains a result about algebraic cycles and intermediate Jacobians varying in an algebraic family of algebraic varieties. In Appendix B we give some examples. In Appendix C we discuss some conjectures which should be true but which we are unable to prove. Finally, in Appendix D we give an application of the results in w 9 to the global monodromy group of certain (algebraic) K 3 surfaces. e) This paper is a successor to [I I]. However, our point of view has evolved somewhat and perhaps a more appropriate general reference is the survey article [I3], which in particular discusses most of the results in this paper and takes up many related problems and conjectures. Finally, this paper is essentially self-contained, except for w IO where we use a formula from [7] and a result from [I6] about the curvature of the metric ds~ discusscd above. It is my pleasure to thank the referee for many helpful suggestions and comments. 17 PART I DIFFERENTIAL-GEOMETRIC PROPERTIES OF VARIATION OF HODGE STRUCTURE x. Algebraic fan~ilies of algebraic varieties. By an algfbraicfamily of algebraic varieties we shall mean that we are given connected and smooth algebraic varieties X, S and a morphism f: X~S such that (i) fis smoolh, proper, and connected, and (ii) There is a distinguished projective embedding XcP s. Setting V~=-f-a(s) (seS) we may think of f: X-+S as the algebraic family {V~}se s of smooth, complete, connected, and projective algebraic manifolds paiametrized by S. The parameter space S is generally not complete, and we shall want to consider smooth compactificatzons of the situation f:X-->S. Such a smooth compactification is given by a diagram: X cX Cx.,) 4 Sr S c S where X, S are smooth, complete, and projective algebraic varieties which contain X, S respectively as Zariski open sets, and where X--X and S--S are each divisors with normal crossings. Thus, for example, g--S is It,early given by: (i.z) ~i--.sk:o where sl, ..., s~ are part of a local holomorphic coordinate system on S. The divisors Dj given locally by sj =~-o in (I. ~) will be called the irreducible branches of S--S. We then have S=--S-D where I)=D I+...+D m is the divisor with normal crossings, As another example, if dim S=-I and if -S--S is locally given by s=o, then f:X~S will be given locally by: where xt, ..., x~ is part of a local holomorphic coordinate system on X. Such smooth compactifications exist by the fundamental work of Hironaka [20]. We want now to say what it means to localize the situation (i. I) at infinity. Let 130 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, llI t3i S--S be given locally by (I.2) where sl, ..., s d is a holomorphic coordinate system on S. Denote by P the open polycylinder given by o<=[s~[<r (j-----i, ..., d) and let P*=PnS. Thus letting A be a disc in t3 and A* the corresponding punctured disc, we have P~(A) ~ and P*~(A*)~� d-k. Set Y=f-~(P) and Y=YnX. Then the localization of (I. i) at infinity is given by: Y c Y (x.4) ~ P* cP We will generally refer to P* as a punctured polycylinder. 2. Variation of Hodge structure. We shall linearize the situation (I. I). For this we now consider X, S as complex manifolds and f: X--->S as an analytic fibre space and topological fibre bundle. Fix a base point s0~S and consider the action of the fundamental group rh(S ) of S based at s o on the cohomology H"(Vs0 , 13). If L~He(Vs., Q) is the cohomology class of the hyperplane section relative to the given projective embedding X c PN, then L is invariant under rh(S ). Thus for n<_m = dimcV we may define the primitive cohomology pn(V~~ 13) to be the kernel of: Hm-'tV C)-~Hm+'+2tV C) (n=m--r). Lr+l : ~ 8~ ~ s0, Because of the Lefschetz decomposition [22]: [nt2] (2.x) H"(V,., C)= @ LkP"-~(V, ,0, C), k=0 which is a ~l(S)-invariant direct sum (over Q) decomposition of Hn(V~., C), it will suffice to consider the primitive cohomology. Let E=Pn(V~., C) and denote by E +S the complex vector bundle, with constant transition functions, associated to the action of 7h(S ) on E. There is the usual flat, holomorphic connection: D : Cs(E) which one has on any such vector bundle associated to a reprcsentation of the fundamental group. In fact we have a short exact sheaf scqucncc: D n (E) (2.*) o -+ ~(E) -+ (Vs(E) -~ where the sheaf ~(E) of locally constant sections of E has the following interpretation: Let R.~(C) be the usual Leray cohomology sheaf of f: X->S, which we recall is the sheaf arising from the presheaf: U-+ H"(f-t(U), C) 131 :3 2 PHILLIP A. GRIFFITHS where U runs through the family of M1 open sets in S, and define the Leray primitive cohomology sheaf P~.(C) to be the kernel of: L' : (n=m-r). Then %~ is just P~,(C). Now the fibre E s is tile vector space P"(V,, C) and as such has the structure of the primitive cohomology vector space of a K~hler manifold [3o]. Translating this structure into data on the flat bundle E-+S, what we find is the following [i3]: I) A flat conjugation e~'g (eeE). n) A flat, non-degenerate bilinear form (2.3) Q: E| Q(e, e')=(--I)"Q(e', e) called the Hodge bilinearform; and III) A fltration of E by holomorphic sub-bundles (2.4) F~ c:F'~-I cF "=E called the Hodge filtration. Remarks. -- (i) The conjugation on E is induced from the usual conjugation on H"(V~, C) = H"(V,, R) | C. (ii) The bilinear form (2.3) is given by: Q(e, e')=+fvL=-"ee' (m-- dime V,) (2-S)' where e, e'~P"(V~, C) cH"(V,, C). (iii) Letting pn- q, q(V~) = H"- q' q(Vs) n P'(Vs, C), we have for the fibre F~ that: (2.4)' F~ = P" ~ +... + P"- q' q(V.). We will denote the data of a flat bundle E with I)-III) above by 8---- (E, D, Q, { Fq}), and as conditions on this data we have: IV) The Hodge filtration (2.4) is isotropic, which means that: (2.5) "-q-1 where (Fq) : Q(e, Fq)=o}. V) The bilinear form (2.3) is real (i.e. Q=UD) and, if we let F,-q,q = Fqn~,-q =Fqn (~q-1) then we have the Hodge decomposition, which is a G ~~ direct sum decomposition: q=0 132 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III x33 VI) The Riemann-Hodge bilinear relations (q,r) ('~" 7) t (--i)" (--I)qQ(r"-q' q, F"-"q) >o are valid; and VII) The infinitesimal period relation [x I] ('~. 8) D : @s(Fq ) ~ Y~(F q+l) holds. Definition.- We will call the data d~=(E, D, Q, {Fq}) given by I)-III) and satisfying the conditions IV)-VII) a variation of Hodge structure. It is of course not necessary that a variation of Hodge structure come from an algebraic family of algebraic varieties f: X-+S. In case @=(E, D, Q, {Fq}) does arise from f: X-+S, we will say that the variation of Hodge structure arises from a geometric situation. Remarks (2.9). -- (i) Let d~=(E, D, Q, {Fq}) be a variation of Hodge structure. Referring to (2.5) , we have natural isomorphisms: (2. xo) (Fq/Fq- ~) v ~ Fn- q/F n-q-l, which are isomorphisms of holomorphic vector bundles. (ii) We may symbolically rewrite (2.8) as (2.xx) Q(D.F q, Fn-q-2)=o. (iii) Referring again to the infinitesimal period relation (2.8), we see that the connection D induces a linear bundle mapping of holomorphic bundles (2. x2) Grq : E q ~ Eq+l| " where Eq----Fq/'F q-1. The vector bundles E q will be called tile Hodge bundles, a terminology which we shall now try to justify. In case the variation of Hodge structure d" arises from a geometric situation f: X--~S, the fibre E, q is given by: (2'I3) E,q= H~(Vs, f~-q)0 (seS; q=o, I, ..., n), where Hq(V,, f~-q)0 is the kernel of the cup product: L '+1 : Hq(V,, ~z m-'+q) v, _~, Hq+ r +1(V8, f~+ l-q) (n=m--r) when we consider L as a class in Hi(V,, f~s)" In other words, in the geometric situation the fibre E q is just the primitive part of the Hodge cohomology space H n-q'q(V.). (iv) Referring to remark (iii) just above, we shall give a homological interpretation of the maps (~. I2) in case d" arises from a geometric situation f: X~S. To do this we recall the Kodaira-Spencer infinitesimal deformation class [~4] p~eH'(V~, Ov, ) | (seS). 133 i34 PHILLIP A, GRIFFITHS The pairing @v,| - q -+ ~,-q+l v, gives: (a. *4) 1t1(V,, |174176 f~-') -+ Hq+~(V,, f~S-e-'). Comparing (2.14) and (2.13) we see that cup product with the Kodaira-Spencer class gives : (a. *5) o~ : E~ -~ Eq+ t| From [II] it follows that Ps in (2.15) is the same as aq in (2. I2). Summarizing : Proposition (~'. i6). -- In case o ~ arises from a geometric situation, the linear mapping % in (~. I2) is the cup product with the Kodaira-Spencer class. 3. Remarks on the homology of algebraic fibre spaces. a) Consider the situation (I. I) and let: Y cY 4 Y P*c P be a localization at infinity as discussed just preceding (I -4). Since P" =(A')kx(A) d-k where A is a disc and N" is a punctured disc, the fundamental group rh(P ~ is free abelian and has as generators the paths around the deleted point in each of the factors A ~ The corresponding automorphisms of the cohomology H*(V,0, C) are called Picard-Lefschetz (P.-L.) transformations. In case k-----I we shall denote the P.-L. transformation on the primitive cohomology by TeAut(P"(V,~ C)). b) Let f: X § be an algebraic family of algebraic varieties as defined in w i. We consider the Leray cohomology sheaves R4,(C), and we recall the Leray spectral sequence {E~ 'q} which abuts to H'(X, C) and with E~'q=H'(S, R~,(C)). We will prove the following result of Blanchard and Deligne (cf. [2] and Deligne's paper in Publ. LH.E.S., vol. 35, PP" I~ Proposition (3- 9 -- The above spectral sequence degenerates at the E2-term. In particular the restriction mapping (3.2) H"(X, C) ~ H~ Rf",(C)) ~ o is surjective. Proof. -- The cohomology class L of the hyperplane section operates by cup product on the terms E, (r >2) of the spectral sequence and it commutes with the differentials d, : E r -+ Er+l. Using this let us show that d2----o , the argument for the other d~ (r_ >-3) being similar. Because of the Lefschetz decomposition (2. i), which in the present situation reads as: = LW2(c) uPS-, <el | 134 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III i35 it will suffice to show that d~: H~(S, P~.(C)) ~ HP+2(S, is zero for q~ m = dimcV ,. Writing q ~ m--t and using that L t + 1 : f, Rm-'-l(c) -, R U is an isomorphism [3o], we find a commutative diagram H~( S, Pr: '(t2)) d,> Hp+~(S, N~._, 1(12)) Ii 4, H'(S, R~.+'+~(C)) a,) Hp+2(S ' Rt~.+,+l(C)) Sincc the dotted vertical arrow is zero by the definition of the primitive cohomology sheaf, we see that our desired d~ is zero. Remark. -- Proposition (3-I) says that there is no transgression in the cohomology of algebraic fibre spaces. The result (3.2) was known classically in the following dual formulation [26] : Let yeI-~(Vs0 , C) bc a homology class on Vs, invariant under the action of the fundamental group rr ) on the homology of the fibres. (We will speak of "f as an invariant cycle.) Then there exists a cycle .s , 12) such that: .vs.=v. The cycle ,~qP(~,) is called the locus of y, and it is thought of intuitively as the locus of the cycle -(seH,(V,, 12) as s varies over S. Lefschetz's proof that s165 exists is really a homological version of the proof of (3. I) given above. We shall refer to (3. I) as the locus of an invariant cycle theorem. c) There are two variants of the locus of an invariant cycle theorem (3-2). The first is a somewhat interesting conjectural local result around an irreducible branch of S--S (cf. w167 8, t 5 in [I3] for further discussion). Conjecture (3.3) (local invariant cycle problem). -- Let: YcV r; P*c P be a localization of (i. I) around infinity as discussed in a) above, and let yeI-i"(V,., O) be a cohomology class invariant under nl(P*, So). Then there exists FeH"(Y, Q) with F[V,0 =y. Remarks. -- It is trivial that there exists reH"(Y, Q) with PlV..='r, so the conjecture has to do with the singular fibres of Y lying over P--P*. Thus far (3.3) 135 PHILLIP A. GRIFFITHS I36 has proved surprisingly difficult to handle and, in particular, it does not seem to be a topological result but will most likely require some sort of Hodge theory (w 15 in [I 3]) (1). The second variant is the following striking result of Deligne [9]: Theorem (3.4) (Deligne). -- Referring to (i. i), we have a commutative diagram (3.5) H"(X) )H"(V,.) H"(X) where the arrows are all restriction mappings of cohomology, and the image of r is equal to the image of? in (3.5). Remark. -- This result is a global version of (3.3). d) Let f: X---~S be an algebraic family of algebraic varieties and g=(E, D, Q, {Fq}) the resulting variation of Hodge structure (w 2). We shall use (3- 2) and (3-5) to deduce results about o ~ which will then later in w 7 be proved to hold for an arbitrary variation of Hodge structure which has a complete base space. It should be possible to prove the results of w 7 with no such assumptions, and this matter is taken up in Appendix C. The following are given in [9] by Deligne as consequences of (3.5) : (3.6) Let ?~H~ R~.(C)) be an invariant, locally constant cohomology class. Then the same is true of the Hodge (p, q) components of ~ (p-t-q = n). Proof. -- This is clear since we have: H"(X, C) -+ H~ RT.(C)) -+ o and X is a K~ihler manifold. (3.7) Let I~-----P"(Vs., Q),,,(s) be the invariant part of the primitive cohomology Pn(Vs. , Q) under the monodromy group F. Then there is an orthogonal direct sum decomposition (3.8) P"(V,., Q)-- I~| Proof. -- This follows from (3.6) and the properties of the Hodge inner product. From (3-7), Deligne has deduced: (3.9) The action of the monodromy group F on P"(V,0, Q) is completely reducible. Furthermore, if F is solvable, then it is a finite group. 4. Remarks on Hermltlan differential geometry. be a holomorphic vector a) Connections in Hermitian vector bundles. -- Let H-+S bundle and D~ : A~ -+ A'(H) (1) Added in proof -- This conjecture has now been proved for n = 2 by Katz, and then in the general case by Detigne, using his theory of mixed Hodge structure [9]. 136 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III t37 a C ~ connection. Then there is a decomposition D. = nk -[- D~ of D s into types (I, o) and (o, I), and D is said to be compatible with the complex structure D~ =0. Suppose that H has an Hermitian metric if " - (,) : H| (e,e')=(e',e). We do not require that ( , ) be positive definite, but it should of course be non-singular. Lemma (4. 9 -- There is a unique connection D r such that (i) the Hermitian metric (,) is flat, and (ii) D H is compatible with the complex structure. Proof. -- Let e~, ..., e, be a local holomorphic frame for H and let h~o = (e~, eo) }. denote the Hermitian metric. Then the required connection DH(ep)= ~ 0~eo is given a=l by O=h-lOh where 0=(0F) , h=(h~o) are the connection and metric matrices respectively. Now we consider a holomorphic vector bundle I-I-+S having a connection D r which is compatible with the complex structure. Let KcI-I be a holomorphic sub- bundle with quotient bundle L, so that we have an exact sequence (4.2) o~K-+H-+L-+o. The connection D~ induces, in the obvious way, a mapping (4.3) b : A~ -~ At(L) which is linear over the C * functions and is called the second fundamental form of K in H. Lemma {4-3)- -- The second fundamental form of K in H is of type (i, o), so that bzA~'~ L)). Proof. -- Given eeK,,, choose a C ~~ section f of K with f(so)=e. Then by definition b(e) is the projection on L of (Dsf) (So). Since we may choose f to be holo- morphic and since D~ =0, we have DHf=o so that b(e) is of type (i, o) as desired. Suppose now that H-+S is a holomorphic Hermitian vector bundle with holomorphic sub-bundle K as in (4.2)- Assume that the Hermitian metric (,) is non-singular when restricted to K. Then there is induced a C ~ splitting of (4.2) by considering L as being {eeH : (e, K)= o}, and so: (4.4) DK=DH--b induces a connection in H. Lemma (4-5)- -- DK=:Dn--b in (4.4) is the metric connection in K. Proof. -- By lemma (4.3), D~=0. For e, e'eA~ d(e, e')= (,Dne , e')+ (e, D.e')= (Dxe , e')+ (e, DKe' ) since L-: (K) as a C ~ sub-bundle of H. Q.E.D. 18 x38 PHILLIP A. GRIFFITtIS Similarly the connection D~, the holomorphic projection H-~ L--~ o, and the C a injection o-+ L ~ H induce a connection D L in L by (4.6) DI,(f) = :to Dnoi(f). Lemma (4.7).- The connection D~. in (4.6) is the metric connection in L. The proof is analogous to the proof of (4-5). b) Curvature in Hermitian vector bundles. -- Given a connection D x : A~ -+ AI(H), the curvature | is defined by (4.8) | e : (Dn) 2. e (e ~A~ In case D x is the metric connection for an Hermitian metric, | is of type (i, I) and satisfies the symmetry (One , e')+(e, | (e, e'~H). For us, the main use of the curvature is as it appears in the following: Lemma (4- 9)- - Let 9 and q~' be two local holomorphic sections of H-§ and d/--= (% 9') the inner product. Then (4. xo) 0~+ =(D~, D~?')--(| ~?'). proof. -- ~') (by Lemma (4.~) and since ?' ' '-1-( ' ' ' is holomorphic)=--(D~D~%?), ,D,%Dn.r ) (by Lemma (4.1) again). Now D~D~o:(D~Dn-{-D~D~)~0 (since qo is holomorphic)=@..q~ (by (4.8)). Q.E.D. If q~ is a holomorphic section of H-+S, then we may write locally: t (D'~(p, D'~)= ~g,,-jds' ^ dg j (4. IX ) -- (@a% q9) = .~.hi,?ds~^ d3 ~ where g--=- (gi, j) and h-- (h~, j) are Hermitian matric('s and s 1, . .., s d are local holomor- phic coordinates on S. From (4. IO) and the maximum principle for plurisubharmonicfunctions [I9] , we have Lemma (4. i2). -- Let qD be a holomorphic section of H---~S such that (i) the length +=(% q~) is bounded on S, and (ii) the Hermitian matrices g and h in (4. I I) are everywhere positive semi-definite. Then + is constant and we have (Dn% Dnq0)=o=(| q0). Now let H-+S be a holomorphic vector bundle with an Hcrmitian metric and metric connection D H. Suppose that KcH is a holomorphic sub-brindle such that the restriction of the metric on H to K is non-singular. Then wc are in the situation of Lcmmas (4.5) and (4.7)- Lemma (4.13). -- The curvature of the metric connection in K is given by (0 e, e') -(%e, -(be, be') (e, 138 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III t39 Proof. -- Choose holomorphic sections f, f' of K such that f(So)=e, f'(So)=e'. Then by (4. I O) applied to H we have: (| f')=(D~f ' ' DHf)_OO(f,f ' , - , ) =((D~ + b)f, (D'K+b)f')--O-O(f,f') = (D~f, D~f') +(bf, bf')--O-O(f,f') = (| f, f') + (bf, bf'), where we have used Lemma (4.5) in the second step, the equation: (Dif, bf') = o = ( bf , D~f) in the third step, and (4-Io) applied to the bundle K in the last step. Q.E.D. To give the curvature in L, we use thc conjugate linear isomorphisms: LzL induced by the Hermitian metrics to define tea ~ K)) as the image of the second fundamental form b under the isomorphism At, ~174176174 Lemma (4. x4)- -- The metric curvature in L is given by : (| f, f') == (| f, f') + (cf, cf') (f, f'eL ~ (K) 5. Statement of m~.in d;ft'eren_ti~!-geometric properties of the Hodge bundles. The results stated in this section will be proved in w 6 below. Let 8-=(E, D, Q, {1~}) be a variation of Hodge structure as defined in w 2. Using the Hodge bilinear form Q, we have an Hermitian metric ( , ) in E given by (5. x ) (e, e') = (-i)'Q(e, F) (e, deE). This Hermitian metric induces non-singular Hermitian metrics in the holomorphic sub-bundles F e of E, and (-- I)q-l(, ) subsequently induces a positive-definite tiermitian metric in the Hodge bundle Eq=Fq/F q-t. Referring to (2.I2), we have linear bundle maps eq:Eq-+Eq+~(~)~ " and teq_l :Eq~Eq-t| induced by the flat holomorphic connection D. Theorem (5.2) (Curvature of Hodge bundles). -- The curvature of the metric connection D~q is given by : (| e')= (Gqe, Gqe')--('Gq_te , 'aq_le') (e, deE') where we agree that a_ I = o =: %. 139 PHILLIP A. GRIFFITHS 14o Remark. -- If we choose local frames for all of the Hodge bundles E q, then Theorem (5-2) gives for the curvature matrix that (5" 3) OEq = Aq^ tAq--Bq ^ t]~q, where Aq, Bq are matrices of (I, o) forms with B0:o , A~:o. Remark. -- From (2. Io) we have isomorphisms (5.4) Eq~-E"-q. From (5.I) it follows that the isomorphisms in (5.4) are all isometries. Using the isomorphism ~..q| Eq + 1| ~= ~,,- q- l Q Eq| ~ ' we see that ~q corresponds to %-q-1, and so = - (%_1, 1) = - which is the correct relation between the curvature of an Hermitian vector bundle and the curvature of its dual. Our second main application of the structure equations of variation of Hodge structure is Proposition (5-5). -- Let ~ be a holomorphic section of F q over an open set U c S and assume that the projection of D~ in E /F q is zero. Then ap induces a section ? of E/F q-1 ~ F"-q+ 1, and the differential forms l "-q " ?) (--I) (D,.-q+~?, (5 "6) (_i),-q+1(| ?) are positive, in the sense that the Hermitian matrices defined as in (4. x i) are positive (semi- definite). Corollary (5.7)- -- Let 9 and ? be as in Proposition (5.5) above and assume that (i) U /s all of S, and (ii) the length of the section ~ of EqcE/F q-x is bounded. Then D~.~q-~? -= o -~ DEq 0. This Proposition and Corollary will bc proved togcther in w 6 below. Proposition (5" 8). --- Let r be a holomorphic section of E q cE/F q-1 over an open set U r S, and assume that DE/Fq-I ? = o. Then there exists a unique section tF of F q satisfying (i) o; (ii) ~F projects onto ?; and (iii) the inner product (~F, Fq-1)----o. Combining (5-7) and (5.8) we find: Theorem (5.9) (The0rem on global sections of Hodge bundles). -- Let ap be a holomorphic section of Fq~S such that a) the projection of D(b in I;./Fq is zero, and b) the length of the induced section ~ of E q -~Fq/F q-1 is bounded. 140 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III x4x Then there exists a section ~F of F q satisfying (i) DW---- o; (ii) W--(I) is a section of Fq-l; and (iii) the inner product (~F, F q - 1) = o. 6. Structure equations for variation of Hodge structure. We want to prove (5.2) and (5.5)-(5.8). Our method of proof is to use the calculus of frames [6], [8], where by definition a frame is a C ~ basis, over an open set, of the vector bundle in question. Given a variation of Hodge structure 8=(E, D, Q, {Fq}), we shall consider unitary frames adapted to the Hodge filtration (2.4). By definition this is a frame (6. x) el, ..., eat ; ehl+t , ..., eh~ ; ... ; ehn_t+l , ..., eh, where hq =dim F~ q and where the following conditions are satisfied: (i) referring to the Hermitian inner product (5.x) we have (6.2) (ei, ej) =(-- I)q-l~ (hq_l<i,j<=hq) and all other inner products are zero; (ii) the vectors el, . . . , ehq give a basis for F~ q for all points seS where thc frame is defined; and (iii) under the conjugate linear isomorphism Eq~_En-q given by (5.4) and the metric (5. i), we have (6.3) -inq_l-~j=eh,_q_t+j (I~j~hq--hq-a). Remarks. -- We first observe that (6.2) and (6.3) are compatible: (--I)q-l~j=(eaq_x+i, enq_l+~)=(--i)nQ(ehq_1+i, ehq_t+ j) =(--x)"(--i)"Q(eh._q_l+ p en._q_l+,) = (-- I)" (-- I)"- q-l~/. Secondly, I should like to comment that the alternation of signs in (6. ~) is of extreme importance -- it is this plus the infinitesimal bilinear relation (2.8) which makes everything go through. The flat holomorphic connection D, which by Lemma (4. x ) is the metric connection for the metric (5. i) in E, is given by hn (6.4) De,=j~=xO~ej, 141 I49 PHILLIP A. GRIFFITHS where the differential i-forms O~ satisfy the integrability condition hn i i k (6.s) doj +kY,.(o,,^ = o~) =o. From (6.2) and the flatness of the metric we find Ohp ,-,-i . q-h ,+~ (1<i<h~,--h~ ~" I<j<hq--hq_t). (6.6) h -x+j -~-(--I) ,+ 0 ~ - l = O _ hp_l+l -- __ _ ~ __ __ From (6.3) we have ~hp-l+i t ~n~hn p 1 +/ (6-7) hq_l+j=(-I) Uhn'-q-l+ j. As remarked below (6.3) , it is unfortunatcly the case that the signs are quite important and so must be kept track of carefully. The infinitesimal period relation (2.8) gives , AhP -l+k (6 8) ~^q_~+j =o for p>=q+i, i<_j<hq--hq_~, k>=o. At this point we have used all of the information in I)-VII) of w 2. From Lemmas (4.3) and (4.5) we have Lemma (6.9). -- The second fundamental form of F q in E is given by (6.,0) bq= X O~| 0('=o; I~i~hq hq-t" l ~j ~ hn and the metric connection in F ~ is given by hq (6. xx ) D~ei = Z Ojep i_~.._i~hq. j=t Using the Cartan structure equation [6], the curvature of the metric connection in F q is given by hq (i~i,j<=hq). From (6.5) and (6.8) we obtain hq. 1 --hq ,,,~ ,,hq + k, ( I < i, j < hq). (6.x2) (| ~ tVh,+~^V~ ) -- k:=l -- ---~ We shall now prove Proposition (5-5) by proving the following three lemmas. L~mma (6. "3). -- Let ~ be a vector in ~ such that ~ projects to zero in ~--,~-'--,o. Then the differential form (-x),(o~. ~, ~) is positive in the sense of Proposition (5.5). 142 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III ~43 hq --hq_ 1 -- ~ e%_l+z, we have from (6.I2) that Proof. Writing ~= ~ ~ 1=1 t~hq l+j ~hq+k t'r v ((~q~, ~)= , Z l(Jhq-{_k A I~hq l_{..~.~ehq 1.t.i, ehq_l_ ~ k)~J~ ' j,k, - - ~ q t ~.-~ ~hq-l- k :(--i) / z~ o h +~^ Ahq-~+j,~j72~ i,j,k q-1 Vhq -j-k Y '~ } ((fOhq l_~.,qO)A (Z0h~ l+j~)) ) (by (6.6)) - j - :(--I)q~k (+kA~)k) where +k=Z,%+k vhq_~+, -~ ? is of type (I, o) (by (6.Io)). This proves the Lcmma. Lemma (6. I4). -- Let ~ be as in Lemma (6. I3) and assume that D~q~ projects to zero in Fq-+Fq-t-+o. Then the differential form (--~)q-~(D~q~, D~) is positive. hq--hq-1 h --iv ~hq_l+i Proof. -- By assumption D-~q~ = w + q-~- e%_~ + ~ where is of type (i, o). i=1 Then (--~)q -~(D~q~, D~q~) = Y~ (+ ha-l+'^ ~hq_x +,) as required. Lemma (6.15). -- Let tF be a holomorphic section of F such that DW projects to zero in E/F. Denote by + the section of ElF -~ induced by qP, and let ~ be the holomorphic section of ~'~-~+~ which corresponds to + under the isomorphism E/Fv--~-~F "-v+~. Then D~,_~§ projects to zero in F'-P+I-~F"-V-+o. Proof. -- What we must prove is that D~.tF~_~ + projects to zero in E/FP-I~E/FV-+o. hp Now ~F= x~ +Se~ and by assumption we have J=l hp I j tQlP 4- k (6.I6) Z V vj =o (I<k<hp,~t--hp). j=l "-- ":: - hp" - ~lp_ 1 The section + ofE/F p-a is E ~) hp-l+J and the Lemma follows from (6 8), (4 7) j=l ehP -~ ~-j " " and (6. I6). It is clear that Proposition (5.5) follows from Lemmas (6. I3), (6. I4) and (6. I5). We now prove Corollary (5-7)- We use the notation of Proposition (5.5) and Corollary (5.7). From Lemma (4- I ~) it follows that the length (% q~) of q) is constant and (D~./~q_a% Dv.~q-~q))=o. By Lemma (6. I5) we have that the projection of Dv4rq-~? on E/F q is zero, from which it follows that D~/vq-~q)=o. It now follows from the exact sequence: 0 -+Eq---~E/F q - ~ -->E/Fq--> o and Lemma (4.5) that D~?=o. This proves the Corollary. 143 PHILLIP A. GRIFFITHS r44 We now prove Proposition (5-8)- We may assume that r is a unit vector, i.e. that the length (% r ~-1, and may then take: q) ~ ~hq_l+ 1 h~ in our frames (6. I). From (6.4) and Lemma (4.7), we havc Z 0t. l+, j=o j ~hq_l+ 1 which gives : 0 5 --o for j>hq_l~-I. (6. x7) hq_l+ 1 -- Differentiating (6. I7) and using (6.5) gives: hn ~Ah~ 1+1 x-~ ,~h _i+1 t,j o:-- "- : z~ (u~ q ^t~nq_l+l ) ~hq -1+1 j=l = (O q-t+ t^ 0 (by (6. i7) ) J<%-1 ^~ ~+t) hq_t --hq_2 ,~hq l+l ~hq_ t + 1. ~ : Y" (%,(~+j^~hq ~+i: (by (6.8) and (6.6)). j=l Ahq-1 +1 Since ~hq_s+J is of type (I, o), we must have: ~- , hq-l--hq-2. %-2+J=~ for j=I, ... This gives De%_1+, ---= o, from which Proposition (5.8) follows. We now prove Theorem (5-9). Taking the frame: ehq_l+l, 9 .., ehq in Eq, we see from Lemmas (4-5) and (4.7) that: hq--hq-1 h -t-i Z 0~ q-1 , .e i DEqehq-l+J~ i=1 "q-lt3 " From the Cartan structure equation: ~hq-l-b J ~ ~k k~hq-l-t- k A IJhq-l-~ j) and from (6.5), (6.6), (6.8) it follows that: (6.x8) f0 r247 %-~%~mb-:~ zb ~+J, b+~-b-~b+" ~h.+,, , k Eq]hq_l"4"j I,~hq_2-1-1AUhq-~+I)-~-m~'~l (Vhq 1 + ~-- lffil J A ~h;-l+ i)" Now Theorem (5.2) follows from (6. I8) and the equation hq+l--hq h +m ~q(e%_,+j)----- m~==l On~_i+~e%+,,, which says that aq given by (2.12) is just the second fundamental form orE q in E/F ~-1. Finally, we shall prove a Lemma for use in the proof of Theorem (7-I9) below. 144 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, IlI I45 Lemma (6. x9). -- Assume that S is complete and let ~ be a global holomorphic section 0fE/F q-1 such that (i) q) projects to zero in E/Fq-I~ E/F q, and (ii) the projection of Dq~ to ElF q is zero (this makes sense since D-d)s(Fq-~ ) cf2~(Fq)). Then there exists a constant section @ of E such that a) 9 projects to *O in E--*-E/F q-a and b) the inner product (r F q-t) =o. Proof. -- Referring to Theorem (5-2), we have: (62o) (o~, ~)=-('~_~. ~, '~, _~.~) when we consider ~ as a holomorphic section of E q (by (i)) and when we use crq. q) = o (by (ii)). From (6.2o) and Lemma (4. I~) we have DEq-~o. It follows that DE~rq-~ ~-~o and then our result follows from Proposition (5.8). 7" Applications. a) Invariant cycle and rigidity theorems. Theorem (7-i) ([nvariant cycle theorem). -- Let ~= (E, D, Q, {Fq}) be a variation of IIodge structure and assume that (i) S is complete, or (ii) the Picard-Lefschetz transformations around the irreducible branches of S--S are trivial. Let a) be a flat section of E~S. Then the Hodge (p, q) components of 9 are flat sections of E. Proof. -- We shall prove in w i I below that, with the assumption (ii) above, the variation of Hodge structure d ~ and section 9 of E both extend to S. Thus we may assume that S is complete. Referring to the theorem (5.9) on global sections of Hodge bundles, we may find a section ud, orE satisfying Dq~',=o, the projection of ~--~F, in E/F" -a is zero, and the inner product (~,, F"-l)=o. Writing q)= @,_l +~,, we may apply the same reasoning to find a flat section tF,_ 1 of F "-1 such that (q~,-1, F"-2) =~ and --=- q),-2 -r + q", where (I),_ 2 is a flat section of F"-2. Continuing in this way we find our theorem. Remarks.- (i) In [I2] we gave the above proof of Theorem (7-i) but formulated the result in a clumsy way. The above formulation was given by Deligne [9], who, as remarked in w 3, has proved Theorem (7-2) (Deligne). -- With no a.~sumptions on S but with the assumption that g arises from a geometric situation, the same conclusion as in Theorem (7. I) is valid. In fact Theorem (7-2) follows immediately from Deligne's result (3.5)- (ii) Of course, we would conjecture that (7. I)is true with no assumptions on S. 19 PHILLIP A. GRIFFITHS z46 Corollary (7.3). -- With the notations and assumptions of Theorem (7. I), we suppose further that n = 2m is even and that r is a flat section of E~S which is of type (m, m) at one point (i.e. the Hodge components q)r'q=o for (p, q)Oe(m,m)). Then d) is everywhere of type (m, m). Remark. -- In [18] Grothendieck, as a (non-trivial) consequence of the Tate conjectures, was led to suggest that, if f: X-+S is an algebraic family of algebraic varieties and ~ a section in H~ P~Q) which is an algebraic cycle at one point sots , then ~(s)~H~m(vs, O~) is everywhere an algebraic cycle. It was this problem which initially started me looking into sections of Hodge bundles. Corollary (7.4). -- Let g and if' be two variations of Hodge structure which satisfy the assumptions of Theorem (7. i). Suppose there is a linear isomorphism ~:E~-->E~, (sots) which is equivariant with respect to the action of ~1(S) on E,, and E~,, and which commutes with the tIodge decompositions of E,, and E',,. Then there is a global isomorphism g ~= 8' of the variations of Hodge structure which induces ~ at SotS. Proof. -- Because of rct(S)-equivariance, we may consider e as a global flat section of F.| Also ~ is of type (n, n) at s o since it commutes with ttodge decompositions. The result now follows from (7.3). Remark. -- This corollary, which should be thought of as a rigidity theorem, was prow~d for n=i by Grothendieck [17] in the geometric case and by Borel- R. Narasimhan [5] in the general case. Because of Deligne's theorem (7. ~), the corollary is true in general when go and d ~' both arise from geometric situations. We can formulate an analogous result about homomorphisms (and not just isomorphisms) between variations of t-Iodge structure. As we see no applications for such, we shall not discuss the matter further. b) Negative bundles and variation of Hodge structure. -- Let H-~S be a holomorphic vector bundle. We say that H is negative (semi-definite) if there exists a (positive-definite) Hermitian metric ( , ) in H whose metric curvature | (cf. Lemma (4. I)) has the property that the differential forms: (7.5) (O.e, e) = Z.h,, j#^ (h,j = hs, are negative in the scnsc that the Hcrmitian matrix (h~, ~) is negativc. Obscrve that H is negative if the matrix of the metric curvature has a local expression: (7.6) O. =--AA'X where A is a matrix of (i, o) forms. From theorem (5.2) we have Proposition (7.7). -- Let g=(E, D, Q, {Fq}) be a variation of Hodge structure. Then the llodge bundle E" is negative. We say that a locally free coherent analytic sheaf is negative if this is true of the corresponding holomorphic vector bundle. 146 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III I47 Corollary (7.8}. -- Let f: X~S be an algebraic family of algebraic varieties. Then the direct image sheaf t~.(d)x) is negative. More generally, if q is the least integer with H"-q(V,, f~s)+o, then the direct image sheaf R~--q(f2~:/s ) is negative. Recall that a cohomology class c0~Hz~(X, R) is negative if we have (7.9) f,.,~<= o for all compact k-dimensional algebraic subvarieties Z of S. Corollary (7. xo). -- The Chern monomials fi ---- ql" 9 9 c~t of the Hodge bundle E" ~ S are negative. In particular the 1 "t Chern class q(RT.(dPx) ) is negative (semi-definite), and we furthermore have <o (7.") f c~ (R~.(0x)) in case S is a complete curve, n = I or 2, and g # not trivial. Proof. -- It is well known that the Ghern classes of a holomorphic vector bundle H~S can be computed from the curvature | of a metric connexion. In particular, if (7-6) holds, then it follows that locally -( !]'" {7. ,,) c~_\~,~ 1 ( ~ (~,^,i,,)) where [ I I ---= i~ 3... + i~ is the degree of c x and ~', are (I I I, o) forms. The first two statements of (7. zo) follow from (7. I2), and (7-i I) follows the fact (cf. Theorem (5.2)) that, for n=I or 2: c~(r;.(ox))=o =- ~._~=o. TO give our final application of Theorem (5-~), we define the canonical bundle K(@) of the variation of Hodge structure 8=(E, D, Q,{I~}) by: (7. 9 3 ) K(8) = (det E~174 (act El) "- x| | (det E"- 1). The first Chern class of the line bundle K(8) is given by the differential form z---(~(8)) where 2~ (7. x4) co(8) =n(Trace | +(n-- t)(Trace | +Trace | Proposition (7-I5). -- For a tangent vector ~ to S, we have ( ~, ~^ ~) ~o with equality if, and only if, %(~) =- o for q = o, ..., n -- T. Proof. -- This follows by direct computation from the formula ~1-.1 <~, ~A~>= E I~(~)I ~ q=O which results from (7-I4) and Theorem (5.2). c) A Mordell-Weil theorem for intermediate Jacobians. -- Let 8=(E, D, Q, {1~}) be a variation of Hodge structure where we assume that n,=2m+I is odd. Then (7. I6) E = F~)~ -~ , 1~ n ~7.'__-- o. 147 x48 PHILLIP A. GRIFFITHS We let E s ={eeE : e=e} be the set of real points in E and assume given a flat lattice AcE s. Equivalently, we are given a ~l(S)-invariant lattice A,, in (E,),. Letting E+-----E/F ~ and J =E+/A, we obtain an analytic fibre space ~ :j-+s, ~ l(s)=J, of complex tori J,-=I~,\E~/A s which we shall call the family of intermediate Jacobians associated to ~ and A. We note that the tori J, are abelian varieties if m = o, but not (in general) otherwise. Let J-+S be a family of intermediate Jacobians as above and J the sheaf of holomorphic sections of this fibre space of complex tori. There is an obvious exact sequence (7. x7) o-+ cg(a) -+r -+J-+o where Cg(A) is the (locally-constant) sheaf of sections of the lattice A over S. From (7- I7) and the relations DV(A)=o D. ~)s(F") c ~(F"+ 1), we obtain a sheaf mapping (7-x8) Dj :J ---> a~(E/]b "~+ 1). The algebro-geometric significance of (7. I8) will be discussed in Appendix A below (of. Theorem (A.8)). We denote by ~om(S, J) the sub-sheaf of sections ,~c] which satisfy Dj,~=-o, and shall refer to sections in Yt~ J) as being integrable. In the abelian variety case (m=o), all holomorphic sections are integrable. Suppose now that S is complete. Referring to Theorem (7. I) we see that: H~ ~'(E))= H~ ~(F")) @H~ cg(F')) and it follows that: H~ ~(I~))\H~ ~'(E))/H~ (g(A))=J(~') is a complex torus which we call the trace or fixed part of J-+S (cf. Proposition (A.7) and the succeeding remark for an algebro-geometric interpretation of this fixed part). Theorem (7. I9) (Mordell-Weil for families of intermediate ]acobians). -- Let J-->S be a family of intermediate f acobians associated to a variation of Hodge structure d ~ and lattice A over a complete base space S. Then the group Horn(S, J)=H~ o~om(S, j)) of global, integrable cross-sections of J-~S is an extension of the fixed part J(g) by a finitely generated abelian group. Remark. -- The integrability condition Djv =o will be satisfied for any cross- section v of J-+S " which arises from algebraic cycles in case J-+S comes from a geometric situation " where we refer to Appendix A for an explanation of the phrase in quotation marks (of. Theorem (A.8)). Proof. -- From the exact cohomology sequence of (7. I7) we have: o ---~.~f" -+ Hom(S, J) H'(S, (r 148 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, llI where ~ is the vector space ofholomorphic sections q~ of E+ which satisfy D~e~(l~+l). Since Hi(S, Cg(A)) is finitely generated, our theorem follows from Lemma (6. I9) in the same way that Theorem (7-I) followed from (5.9). Remark. -- The extension of this theorem to arbitrary base S is discussed in Appendix C (eft (C.3)). In particular Theorem (C. I2) in this appendix gives such an extension to arbitrary base in case n-~I, which is just the usual Mordell-Weil theorem (over function fields). 149 PART II DIFFERENTIAL-GEOMETRIC PROPERTIES OF THE PERIOD MAPPING 8. Classifying spaces for Hodge structures. Let E be a complex vector space and o<h0< hi<=... _<=h,_l<h . = dim E an increasing sequence of integers which is self-dual in the sense that h,_q_l=h,--h q for o<q<n. We also assume given a non-singular bilinear form Q: E| C, Q(e, e') = (- x)"Q(e', e), and consider the set [) of all filtrations F~162 oF"-1 oF"= E, dim Fq = hq, which satisfy the first Riemann bilinear relation t (Fq) or equivalently t Q.(Fq, F"-'-l)=o. We will say that such filtrations are isotropic or self-dual. Proposition (8.2). -- 1~ is, in a natural way, a projective and smooth complete algebraic variety which is a homogeneous space =GrB of a complex simple Lie group G divided by a parabolic subgroup B. Proof. -- Let G(h, E) be the Grassman variety of h-planes through the origin in E. Observe that the filtration: determines F~ cF" by using the first bilinear relation (8. I). From this we have an obvious projective embedding: (8.3) 5 -~ G(h0, E) � X G(h~, E) which cxhibits f) as a complete and projective algebraic variety. 15o PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III x5z We shall prove that l) is smooth by exhibiting the tangent space TF(f) ) to I) at a given point F-=(F ~ ..., Fn). First we recall the natural identification Ts(G(h , E))~ Som(S, E/S) (SeG(h, E)). The tangent space to FeG(h0, E)�215 E) is (8.4) @ Hom(Fq, E/Fq), m , = 0 -- [-Y-I ' from which we see that TF(I) ) is given by all f=q__@0f q (fqEHom(F q, E/F')) in (8.4) which satisfy the conditions that the diagrams F q -/q-~ E/F (8.5) ~ i F q+l > E/F q+l (q=o, I, ...,m--I) [q+l are commutative, and that we have (8.6) Q(fm(e), e')-~Q(e, fm(e'))=o for e, e'sF m. Now let GcGL(E) be the complex orthogonal group of the bilinear form Q; thus 13 is the complex simple Lie group of all linear transformations T : E---~E which satisfy: Q(Te, Te')=Q(e, e') (e, e'eE). Each TsG induces an automorphism T : f)--~f) by T. (F ~ c... c F n) = (TF ~ c... c (TFn). This action of t3 on f) is transitive and the isotropy group B of a given point F0ef) is a parabolic subgroup of 13. This gives the desired representation I)= 13/B. Remark (8.7). -- We define an important holomorphic sub-bundle IF(I") ) of the complex tangent bundle I(I)) as follows : IF(D ) consists of all f-=q@__0f q in (8.4) which satisfy the infinitesimal bilinear relation (cf. (2.1 I)) (8.8) Q(f:,e')=o (e~Fq, e'~Fn-q-2). We now assume given a conjugation e ~ 7 of E such that Q(~, 7')=O(e, e'). In other words we are given that E----ER| where Qis real on the real subspace E R of vectors eeE which satisfy e =7. Define the Hermitian inner product (,) in E by (8.9) (e, e')=(-i)~Q(e,-~ ') (e, e'~E). 151 152 PHILLIP A. GRIFFITHS We define the period matrix domain D c ID to consist of all isotropic filtrations F ~ c... c F ~ which satisfy the second Riemann bilinear relation: ( ) : Fq| is non-singular and (8. IO) i (- I)q( ' ) : Eq| is 1positive definite ~where Eq={esFq: (e,F q- )=o}. We may combine (8. x) and (8. IO) by saying that D is the set of filtrations F~ cF", with dim F q = hq, which satisfy: l' Q(Fq, F"-q-1) = ~ (8. xx) I (-- i)"Q(Fq' ffq) is non-singular (-- I) q (-- i)nQ(E q, gq) ~>0. Proposition (8. I2). -- D /s an open complex submanifold of 19 which is a homogeneous complex manifold D = G/H of a real, simple, non-compact Lie group G divided by a compact subgroup H. Proof. -- Let GcG be the real form of all real linear transformations T: E~E which preserve Q. Then, under the natural action of 13 on I), G leaves invariant, and acts transitively on, D. It is clear that the isotropy group H = GcaB of a point F0eD is a compact subgroup of G. Definition. -- As mentioned D is called a period matrix domain and 1 ~) will be termed the compact dual of D. It is clear that D parametrizes the universal family of Hodge structures determined by E, Q, the conjugation e~7, and the numbers hq. However, this universal family of Hodge structures over D is generally not a variation of Hodge structure in the sense ofw 2 because of the infinitesimal bilinear relation (2.8). We refer to [16] for a discussion of the group-theoretic properties of D and D, especially as regards the equivariant embedding: D ,I3 ,! G/H > G/B In the following examples we let G and H be as above, K will denote the maximal compact subgroup of G and R-----G/K the corresponding Riemannian symmetric space, and h q = hq-- hq_ t the tIodge numbers. Example (8. x3) ,. -- When n = 2m is even, G=SO(a, b; R) (a=h~ ~', b=ht+h3'--...=-h 2=-t) 152 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III I53 a b is the orthogonal group of the quadratic form i--~l(xi)2--j--~l (y~)z' the compact isotropy group is H = U(h ~ � � U(h re-l) � SO(h"*), and the maximal compact subgroup of G is K :-SO(a; R)� R). We may identify the ILiemannian symmetric space R.= G/K with the set of real a-planes ScE~ such that Q(S, S)>o. The equivariant fibering g:D > R G/H > G/K is given by g(F~162 cF 2") = E~174 E~| | 2"*. (8. x4) . Example. -- When n=2m-l-x is odd, G=Sp(2a; R) (a=h~ +h m) is the group leaving tile skew-form ~ (XjAXa+j) invariant, the compact isotropy group is j=l H =U(h~215 � U(h"), and the maximal compact subgroup of G is K =U(a). We may identify R with the set of complex a-planes ScE which satisfy i o(s, _s)= o !iQ(s, S)>o, and the equivariant fibering g : D-+R is given by (8. x4)o ~(F~162 9 9 9 cF 2"+1) = E~174174 9 9 9 | ~". Now according to (8.4), (8.5), (8.6), we may identify the tangent bundle to D as . m, ( (8.xS) TF(D )=@ ~Homq(E q,E "+q) FeD, m= . q=op~l The identification (8.I5) is G-invariant, and the positive definite metrics (--I)q(,) on E q induce a G-invariant Hermitian metric ds~ on D. Group theoretically, ds~ is the metric induced by the Caftan-Killing form on the Lie algebra of G Ix6]. 20 PHILLIP A, GRIFFITHS '54 Proposition (8. x6). -- In the equivariant.fibering : D ) R !! !! !t G/H > G/K of the period matrix domain D over the Riemannian symmetric space R, the fibre ZFo through each point F0eD is a compact complex submanifold, and we have (8. I7) IF.(D ) c (TF.(ZF.)) where Ir~ cTF0(D ) is given by (8.8). Proof. -- We treat first the case when n = 2m is even. The point ~ (F0) is the same as giving an orthogonal direct sum decomposition E s = S~S~ if E R such that Q is positive on S R and negative on SIR. We shall discuss the case when m = 2l is even -- the other case is similar. The fibre g-l(N(F0) ) is the homogeneous space ( SO(a;R) ) { SO(b;R) ~ ZF~ U(h0)x.. .xv(h~,_~)� ) x \V(h~)x: :_~U(h~,_~) ] and has the following geometric description: ZF. consists of all pairs of filtrations iT~ dimoT 2p =h~ 2r iTIcT3r cT~-IcS dimGT~v+l=hl+...+h2P +1 which satisfy Q(T 2'-2, T2r-2)=o or Q(T ~-1, T~-l)=o as the case may be. These filtrations define a point F~ cF 2" in ~-l(g(F0) ) cD by letting F~ ~ Fa=--T~ F2=T~+T ~, ..., on up through F"-I=T~-2+T ~-1. Then we let F"=-(F~-I) F ~;1 =(F"--2) etc. It is clear that ZFo is a compact, complex analytic submanifold passing through F o and that ~?;--1 (8. x8), TF.(ZF. ) = @ Hom(E q, E q+~) q=0 under the identification (8. I5). From this, Proposition (8. i6) and (8. x7) are clear. In case n='2m+t is odd, the point g(F0) is given by a subspace ScE, I . 9 dimeS = a =--dlmeE, which satisfies t Q(s, S)=o I iQ(S, S)>o. 154 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III z55 The fibre ~-I(N(F0) ) is the homogeneous space u(a) z,, = U(h0 ) x... x U(hm) ' and may be described as all filtrations: T ~ 2c. .. cT 2mcS, with dimeT ~ ~ .. +h 2q. These filtrations define a point F~ 2m+t in Zr~ by letting F~ ~ FI=TO+(T~177 etc., on up to Fm. Then Fm+I=(Fm)L,...,F ~m=(F~ z. Clearly (8. X8)o TF~ ) = (~) Hom(E q, Eq:2), q::0 and Propositions (8. I6) and (8. I7) follow. Remark. -- It may bc noted that, except for the cases n----1 or n=2 and h~ the fibres of g are non-trivial, so that D is not a bounded domain in C s. Also, except for the case n =2, the inclusion (8. I7) is strict, so that there are additional conditions on a variation of Hodge structure other than transversality to the fibres of ~. 9. Statement of results on variation of Hodge structure and period mappings. a) Let g=(E, D, Q,{Fq}) be a variation of Hodge structure, with base space S, as defined in w 2. Letting E be the complex vector space E,. and taking the conjugation, bilinear form, and Hodge numbers hq==dimcFq0/Fq: 1 induced on E by g, we may define a period matrix domain D as in w 8. We now recall that the holonomy group of the flat connection D induces, by parallel displacement of a flat frame, the monodromy representation : l(S) of the fundamental group of S (based at So) in the automorphism group G of D as defined in w 8. The image P of rq(S) in G will be called the monodromy group of d ~. A continuous mapping (I) : S-~F\D will bc said to be locally liftable if, given seS, there exists a neighborhood U of s and a continuous mapping ~:U-~D such that the diagram is commutative. A locally liftable mapping (I) is holomorphic if the local liftings arc holomorphic, and a locally liftable holomorphic mapping (I) is said to satisfy the infinitesimal period relation if the local liftings ~ satisfy (9.x) ~. (v) eI~,(u ) (ueU, veT~(U)), 155 I56 PHILLIP A. GRIFFITHS where ~. is the differential of ~ and I cT(D) is defined by (8.8). We may symboIically rewrite (9.1) as: (9.2) ep.: T(S) -+I(D). Proposition (9-3). -- The giving of a variation of Hodge structure ~ with monodromy group F is equivalent to giving a locally liftable holomorphic mapping (9.,t) o: S-+F\D which satisfies the infinitesimal period relation (9. ~). Definition. -- We call ~ in (9.4) the period mapping associated to g. This terminology is explained in [II]. b) Our first result on variation of Hodge structure as interpreted by the period mapping is Theorem (9.5) (Extension of period mapping around branches of finite order). - Let 8 be a variation of Hodge structure over S and let D be an irreducible branch of S--S such that the associated Picard-Lefschetz transformation T is of finite order (el. w 3). Localize the period mapping (9.4) around a simple point ~eD to obtain ~:P*~F\D (cf. w ~). Then P" ~ A* � A a-1 is the product of a punctured disc with a polycylinder, and there exists a finite covering and a lift~ng ~ : P" ~ D of the period mapping 9 such that ~ extends holomorphically to the closed polycylinder P ~ A � A ~- a. To state our second main result, we assume that the monodromy group F is a discrete subgroup of G. This is the case if g arises from a geometric situation f: X-+S. With this assumption, the quotient space F\D is a complex space or analytic space in the sense of [I9]. In fact, the projection P\G~F\G/H is a proper mapping and so F acts properly discontinuously on D. Thus P\D is a separated topological space which is locally the quotient of a polycylinder by a finite group. Let O:S~P\D be the period mapping (9-4). By Theorem (9-5) we may extend this period mapping to a holomorphic mapping (I) : S'~F\D, where S' is the union of S with those points at infinity around which the Picard-Lefschetz transformations are of finite order. Theorem (9.6) (Analyticity of the image of the period mapping). -- The image ~(S') is a closed analytic subvariety of P\D which contains O(S) as the complement of an analytic subvariety. Furthermore, the volume ~r\D(~(S')) of O(S'), computed with respect to the invariant metric on D, is finite. 156 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III I57 Our third result is Theorem (9.7). -- Let ~ be a variation of Hodge structure whose monodromy group I' is discrete and whose base S is complete. Then the image O(S) c FXD is a complete projective algebraic variety. In fact the canon#al bundle K(d ~) (of. (7. I3)) of g is ample over O(S). Remark. -- Using Theorem (9.6), we see that (9.7) remains true if we only assume that all the P.-L. transformations around the branches of S at infinity are of finite order. This theorem follows from (7. I5) and the results of Grauert [IO]. We refer also to [x3], w IO for a discussion of this theorem together with some related open questions. As our final result we give a theorem about the monodromy group P of a variation of Hodge structure d~ D, Q, {F~}). Theorem (9.8) (theorem about the monodromy group of a variation of Hodge structure). -- Assume that either the Picard-Lefschetz transformations are all of finite order or that ~ arises from a geometric situation. Then (i) the global monodromy group P is completely reducible; (ii) P is finite if, a,ut only if, the variation of Hodge structure is trivial; and (iii) /f F is solvable, then it is finite. Proof. -- The first statement follows from Theorem (7. I) in the same way as (3.7) followed from (3.6). The third statement follows from the first by Grothendieck's argument given in Deligne [9]. Finally, the second statement follows from (9.5) and the fact that a horizontal, holomorphic mapping 9 : Z--+D from a compact, complex manifold Z to a period matrix domain D is constant [I i]. c) We will give some local statements which will imply (9.5) and (9.6). For the first we let H(D) cT(D) be the horizontal sub-bundle defined by (9.8) HF~ J- (cf. (8. I8)). The word " horizontal " follows from the fact that H(D) is the complement to the bundle along the fibres in g : D~R.. A locally liftable holomorphic mapping (I) : S-+P\D is horizontal if we have {9.9) O. : T(S)-+H(D) in the same sense as (9.2). Theorem (9.5) follows from Proposition (9.3) and Proposition (9.xo). -- Let P'=A'� d-1 be the product of a punctured disc with a polycylinder, and let 9 : P'-+D be a horizontal, holomorphic mapping. Then 9 extends to a holomorphic mapping 9 : Ad~D. We now claim that Theorem (9.6) follows from the proper mapping theorem [I9] together with the following 157 PHII, I, IP A. GRIFFITHS I 5 8 Proposition (9. II ). -- Let A(O) be the disc o<=lz[<?, A'(p) the corresponding punctured disc, and P(k,l; p)~(A*(p))~� ~ the product. We shall refer to P(k,l; p) as a punctured polycylinder. Let * : P(k, l; p) ~ F\D be a locally liftable holomorphic mapping. (i) Let "h,..., Vk be the canonical generators of ~t(P(k, l; p)) and Tj=*.(yj)eP the corresponding Picard-Lefschetz transformations. Assume that T~,..., Tj are of infinite order and Tj+D..., T k are of finite order. Let {z,}-----{(z~,..., ~+~)}eP(k, l; ?) be a sequence of points with inf [z][--,o as n~oo. Then the sequence {(I)(z,)}eP\D does not converge. IA~K~ (ii) The volume ~i~v(q)(P(k, l; p/2))) isfinite. Proof of (9.6) from (9. I i). -- We claim that ~ : S'-+P\D is a proper mapping. If not, there is a divergent sequence {s,}~S' such that qS(s,) converges in F\D. We may assume that {s,,} converges to some point ~-eg--S'. By localizing around ~ and using (i) in Proposition (9. I I), we arrive at a contradiction. The proof that the volume ~rXD(q)(S')) is finite follows from (ii) in (9. I I) by localizing around g--S' and an obvious compactness argument. xo. The generalized Schwarz lemma. Let PcC e be the polycylinder {(zl, ...,za) :o<~[z~[<I} of unit radius, and denote by ds~ the standard Poincarg metric given by: Denote by w v the associated g-form ~ t,=t(i---~.~-)2 J so that (tot) a is the non- Euclidean volume of P. Let D be a period matrix domain with (suitably normalized) invariant metric ds~ and associated 2-form m~. We want to prove Theorem (xo., ) (generalized Schwarz lemma). -- Let ~:P--->D be a horizontal, holomorphic mapping. Then we have (,o.2) I r <ds Proof. -- We first show Lemma (xo.3). -- If the volume estimate ~*(mD)a~(mp) a holds for d=x, then we have the distance estimate (I)*(dsg)<_~ds e. Proof. -- Let +:A~P be the embedding of the unit z-disc into P given by dg(Z)=(~lZ,., o~aZ), with Y, 10q[~=I. "' i~l ( I ar so Then (x_f- lzl y I z , that at the origin z=o we find (zo. 4) ~" I -,,asv,03, _-- dzd-z = ',ds~)0- 168 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, Ill I59 If -: is a tangent vector to P at (o, ..., o), then we can find ~j as above such that kb. ~-~ =-r for a suitable X:~o. Then the length IX --I1-11 by the isometry l] IIA property (lO.4). Using the volume decreasing assumption applied to the holomorphic curve Oo+:A-+D, wehave r :t =ll,ll , or Ile.,[l <ll,ll, 1 0z , liD II ~"" II A which is what we want to prove. We will base our proof of the volume estimate on a formula of Chern [7]. To explain his formula we let 1Vi and N be d-dimensional complex Hermitian manifolds and f: M-+N a holomorphic mapping. Using unitary frames as in [7], we write Then f*(Oj)= ~,a~r and we havc {IO. 6) f. (ON)d = [ det (a~)12 (o~) d The where r M and 0 s are the respective 2-forms associated to the metrics (io.5). non-negative function u--ldet(a~)l 2 is the ratio of the volume elements, and we are looking for a formula for the Laplacian A log u near a point m0elV[ where u(mo)>o. (Recall that the Laplacian Af of a function f is defined by i~f=(Af). ~.) The desired formula involves the Ricciform Ric N of N and scalar curvature R~t of M. To explain these terms, we recall that the metrics (lO.5) induce intrinsic Hermitian geometries (cf. Lemma (4. I)) on N and M and we let (xo.7) i I 1 - = gj.k) O 0 = _ 52 Sign O k ^ O~ , ~2k, l ,a be the curvature forms of the metrics in M and N respectively (cf. (4.8) and [7]). The Ricci form is defined by: (xo 8) Ric~ . ~a = I( ~ Rii~COkA~ll, =~=1 ~ ~\~.k., ] and the scalar curvature is given by: (IO. 9) R~ = ~ P~'0J = Trace(PdcM) Theorem (xo. io) (Chern [7]). -- In a neighborhood of a point m0eM where U(mo)>O, we have -IA log u = R.~--Trace(f* Rics). 159 I60 PHILLIP A. GRIFFITHS Return now to our period mapping 4) : P-+D. Given z0eP, either = o and (IO.2) is trivially true, or else ~*(COD)(Z0)4:0 in which case the differential ~. of ~P is injective at z0 and the image W= ~(U) of a small neighborhood U of z0 is a d-dimensional complex manifold with Hermitian metric ds~ induced from dsg. Denote by o~ w the associated 2-form. Lemma (IO,II). -- Let Pdcit and o H be the restrictions of the Ried form RicD and 2-form co D to the horizontal sub-bundle tI(D)cT(D). Suppose that we have Ric <--c(coH) (c>o). Then, keeping the situation and notation from just above this lemma, we have the estimate : Ricw=<_-- c(o~w). Proof. -- By the definition (io.8), Ricn=Trace(| where | is the metric connexion of the given Hermitian metric in the tangent bundle of D. We have then an inclusion of bundles: T(W) oH(D) cT(D) (over W), and we need to compare Trace(| ) and Trace(| The comparison of the curva- tures | OH, 01) is given by Lemma (4. I3), from which it follows that: Trace(Ow) _< Trace (@~) <__ Trace(| (in T(W)). f X Since cow=coD restricted to T~Wj, our lemma is proved. We now use the computation given in [i6], w 7 to prove: Lemma (IO.,2). -- In the notation of Lemma (IO. i I), we have RicR=_<--o~. This gives that: Ricw~--cow for the image manifold W=~(U) as described just above Lemma (io. Ii). We are now ready to prove the generalized Schwarz lemma (IO. I). Let P(p) be the polycylinder of radius p given as usual by {z=(zl, ..., zd): o<__]z,l<p} and ds~(p)= 4 ,~x (pL~]~.~)2] the PoincarE metric on P(p). We have made a slight change of scale from our original definition. With this change of scale, the scalar curvature Rp of the metric ds~Ip) is the constant --d. When p =i we write ds~ for ds~(I) and let c% be the associated 2-form. Define the non-negative function u(z) on P by ~*(coD)a=u.(op) a. We want to show that u < i. The idea, which is originally due to Ahlfors, is to use the maximum principle. We first show that it suffices to consider the case when u attains its maximum at some point in the interior of P. Let z0~P. Then z0eP(p) for some p<x and we may define u~(z)in P(p) by q~'(coD)~=u~.(co~(p)) a. Then limu~(zo)=u(zo) because of lim ds~(p) Thus it suffices to prove that up(zc)<I_ for p<i. Now, -4-1 t~zoj~--=~,P~,.oj't,2I'~. -m't,-,~,~j~af-~.0j PERIODS OF INTEGRALS ON MANIFOLDS, III t6t for p<I, q~*(dsg.) is bounded on the closed polycylinder P(p), while clearly (o~p(p)) a goes to infinity at the boundary P--P. Thus up (Z) goes to zero as z goes to the boundary, and so uo has its maximum at an interior point. We now assume that u has a maximum at zo~P. Then by Chern's formula (io. Io) (io. I3) o> IA log u=--d--Trace(f'PdCw). Now using orthonormal co-frames ( I o. 5) in the situation when M = P and N = W = q> (U) with U a neighborhood of zo in P, we have from Lemma (IO. 12) that - =- >, = x and so C,o. x4) -f'(mCw)> Z :i, j, Letting A be the matrix (a~), from (I o. 14) we find that (x o. x 5 ) -- Trace (f* R.iCw)__> Trace (A. 'A). Now use the Hadaraard inequality Trace(A.tA)~dldetAI2/d=du t/d together with (IO. 15) and (Io. 13) to find I>U a/d, which is what we wanted to prove. Remark (to.x6). -- As in Proposition (9.II) we let P(k, l; p) be the product (A;)kx(A0)' where A; is the punctured disc o<]z]< p and A 0 is the usual disc o<[zl<p. We set P'----P(k, l; I) and P=P(k+l, o; I). Then P--+P* is, in the usuaI way, the universal covering and so the Poincard metric ds~ induces a metric ds~,. on P*. Letting z4=r~ exp 0~ be polar coordinates, we have explicitly that (tO.IT) ds~= \,=, ~(log r~)2 ] + \j=k+, (i~[~)2], and for the volume element (IO, In) @0P*)k+ 1--'=1 r,(1og r,)2J-k+ x (i--4)~" From (IO. I8) we have Lemma CIO.X9). -- For p<I, the volume tx~(P(k, l; p))< oo of the sub-polycylinder P(k, 1; p)cP* is finite. The use of the following lemma was first demonstrated by Mrs. Kwack [25]: Lemma (IO.2O). --For o<p<I, let % be the circle [ZI=P in the punctured disc A" given by o<[z[<I. Then the length IA.(% ) of %, computed using the Poincar/ metric ds~. 2~ on A*, is given by lA,(%)----. In partieular, l(%)--+o as 0--+o. Proof. -- This follows immediately from (I o. 17), which gives that ds~. = dr~ + r2dO~ in the situation at hand. r2(l~ r)2 ALGEBRAIC x62 PHILLIP A. GRIFFITHS As a Corollary of Theorem (IO.I) and Lemmas (io. I9) and (IO.2O), we have: Corollary (IO.2I). -- Let q) : P*--->F\D be a horizontal, holomorphic mapping. Then the volume ~xr\D(r , l; p) ) ) of the image of the concentric punctured polycylinder P(k, l; p)cP* is finite. In particular, (ii) of Proposition (9. ii) is valid. Corollary (xo. ~'2). -- Let P* = A* � A e-1 and let (P : P*--->D be a horizontal holomorphic mapping. Suppose that Yo is a curve in P* given parametrically by O~(pe i~ z2(e~~ ..., z~(e~~ where the zj(e ~~ are smooth curves in the unit disc o_<lzj]<l. Then the length ID(~(yo) ) of the image curve tends to zero as p-->o. H. Proof of Propositions (9.to) and (9.IX.) a) We first prove (i) in Proposition (9. I I). For simplicity we will consider the case k=l, l=o. The general situation will be done by the exact same argument. Thus we have a locally liftable, horizontal, holomorphic mapping q) : A'~r\D such that the Picard-Lefschetz transformation TeP is of infinite order. We assume given a sequence {z,}eA* with Iz, l~o and such that {q)(z,)} converges in F\D. We want to show that this leads to a contradiction. Let % be the circle Izl =lznl and set wn=q)(z~)eF\D. We may assume that w. tends to a point weF\D. Choose a point FeD lying over w in the projection 7~ : D--->F\D. The stabilizer P~={geP:g.w=w} off is a finite group, and we may choose neighborhoods U of w in P\D and U of N in D such that F~.U =U and r~-l(U) is the disjoint union gear g.U. We may assume also that the distance dD(U , gU)>:r from U to its translates is bounded below for geP--F~. Finally, since TeP is of infinite order, we may assume that the intersection TUnU is empty. Choose n so large that the non-Euclidean length lA.(%) is less than r This is possible by Lemma (lO.2O). By Corollary (lO.22) the length ID(r of the image curve may also be assumed to be less than ~. Finally we may assume that the image w, of z,, under q) lies in U. Choose ~,eU which projects onto w.. Now take a local lifting ~ of q) in a neighborhood of z~ such that ~(z,)=w,. Analytic continuation of ~ around the circle % passing thru z, leads to the new local lifting T. ~ around z~. This is a contradiction since the length of the image curve ~(%) is less than ~, which implies that dD(~(Z~) , T~(Z,))<r while we have dD(U , T.U)>~. Because of this contradiction we have proved (i) in Proposition (9. ~ i), and the other part of this proposition has been given in Corollary (IO.21). b) We want to prove Proposition (9. IO). For simplicity we assume that d= i; the general case is done by essentially the same argument. Thus we assume given a horizontal, holomorphic mapping 9 : A*~D. We want 162 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III I63 to show that 9 extends to a continuous mapping of A into D. Our proof is based on the following result of .Mrs. Kwack [25]: Proposition (ix. x) (Mrs. Kwack). -- Let IV[ be a compact complex manifold with Hermitian metric ds~. Let f: A*-+ M be a holomorphic mapping with the property that if (~,} c A* is any sequence of circles ]z] =P, whose radii p, tend to zero, then the lengths lM(f(%) ) of the image circles tend to zero. Then f extends to a continuous mapping f: A-+M. Remark. -- The interesting thing about this result is that it is not at all a topological statement. The fact that M is a complex manifold seems to be quite essential. For completeness we shall give a proof of (I i. I) below. We now use (ii. i) to prove our extension theorem for r :A*-~D. Recalling that D=G/H is a homogeneous complex manifold of a real simple Lie group G by a compact subgroup H, we select a discrete subgroup A of G such that the quotient A\G is compact and such that A acts without fixed points on G/H. The existence of such a uniform subgroup A follows from a general result of Borel and Harish-Chandra [4]. Or in our case we could use the theorem in [28] to write down such a A. The quotient M=A\D is now a compact, complex manifold M with an Hermitian metric ds~ induced from the G-invariant ds~ on D. From Corollary (IO.22) it follows that the conditions of (II. I) are satisfied by the mapping f: A*-+A\D obtained by composing ~ with the projection D-+A\D. Thus f extends to give a continuous mapping f: A-+A\D From this it follows that ~ extends to give our desired continuous mapping 9 : A >D. Remark. -- The use of the uniform subgroup A in the abovc proof is not as absurd as it might at first appear. To explain what I mean, we recall thc embedding Dc]) of D as an open domain in its compact dual. It is not too hard to show that our mapping r extends to a continuous mapping ~:A-+D. The trouble is that the image ~(o) of the origin might lie in the boundary 0D=D--D of D in ]). So our extension theorem is really a question of the pseudo-convexity of D. Now for a bounded domain B in C', it is a theorem of Siegel [29] that the existence of a properly discontinuous group ~" of automorphisms of B such that ~\B is compact already implies that B is a domain of holomorphy. Thus, if B CC" is a boundcd domain such that we have a holomorphic mapping ap : A*-->B, and if there exists a uniform subgroup 9 cAut(B), then * cxtends to r because of the usual Riemann extension theorem plus Siegel's theorem. Our proof of Proposition (9. IO) is essentially a similar argument. c) We now give a proof, which is essentially that of [25], of Proposition (I i. i). We use the notation a(Zo) for the circle [Z[---Iz01 passing thru Z0EA*. Let {z,}cA* be a sequence of points with Iz,[-+o. If we set w,=f(z,), then by the compactness of M we may assume that w,-->weM. Let xl,..., x,, be local holomorphic coordinates centered at weM and denote by U(0 ) the polycylinder [xjl<0 163 x6 4 PHILLIP A. GRIFFITHS around w. We have to show that, given e>o, there exists ~ such that f(z)eU(s) if o<lz]<~. Let e>o be given. Since the lengths/e(f(~(Z,))) of the images of the circles ~(Z,) tend to zero, and since w,,Ef(~,~) tends to w, we may assume that f(~(z,))cU(~/2) for all n. If we cannot find the required S, then, by renumbering if necessary, we may find a sequence {y,,} ~A* with [Z.+ll<ly.l<l z.I such thatf(y.) does not lie in U(e). Let An be a maximal annulus ~.<lzl<~. around .(z.) such that f(A~)cU(e/2). Then the An are all disjoint, and we may choose a.e.(%) and b.ea(~.) with f(a.) and f(b.) lying in the boundary 0U(e/2) of the polycylinder U(e/~). Passing to subsequences, we may assume that f(a.) -+ aeOU(./~) and f(b.) ~ beOU(r Then f(a(a.)) ~ a and f(a(b.)) ~ b by the argument using lengths of circles. Write f(z)=(xl(z),...,Xm(Z)) and let e=xl(a), ~=Xl(b ). We may assume that ~4o, ~eo. Using the x~-coordinates, we have a picture ~(bn) xl(z) For n sufficiently large, we find from the argument principle that: fl xl(z)dz f, x~(z)dz I~ ol~.l x~(z)-xl(z.) - o = ~ I e oc~.l x~ (z) - xl (z.) This is a contradiction, since the difference of these two integrals is the integral: fo xl(z)dz A.,q(z~--~(z.) * o. 164 APPENDIX A A result on algebraic cycles and inte,-..,edlate Jacoblans a) Let V be a smooth, complete, and projective algebraic variety, and consider the odd degree cohomology Wm+l(V, C). For simplicity we will discuss the case when I-Pm+I(V, C) is all primitive -- the general situation is essentially a " direct sum " of such cases. We set H~m+ l(V C) = H2m+ 1, q_ Hm+l, + ,_, ~ ~(V) and define the (mth) intermediate facobian J(V) by (A. I) J(V)= ..+T42"~ I(V, C)\H2m+ I(V, C)/I-Pro+ t(V, Z). As referenccs on the theory of intermediate Jacobians we mention [14] , [27] and [23]. Denote by O(V) the group of algebraic cycles (modulo rational equivalence) on V which are of pure codimension m + i and which are homologous to zero. We will define an Abel-Jacobi homomorphism (A.2) ~b: 0(V) -+J(V), which generalizes the usual mapping for divisors on curves (m=o and dimcV----~ ). Before defining d~ we need a result of Dolbeault about Hodge filtrations (cf. the appendix to [I4] and the references given there). Let A ~'q be the C ~ forms of type (n, o)-[-... +(n--q, q) on V and Z n'q the d-closed forms in A ",q. Observe that d(A m, p) r A m + 1, p + 1 and set (A. 3) F" q = Z n' q/dA n-l' q-1 Proposition (A. 4). -- The natural mapping F" q --+ H"(V, C) is injeaive with image H"~ +Hn-q'q(v). Suppose nowthat dim0V----d (then m~d--I)2 and let r ..., r be a basis for (A.5) F2e-2,,-1,~-m~ Hd.e-2m-l(V)+....3rua-m,a-m-l(V). Observe that 1 ~d-2"-~'d-m is the dual space to the tangent space H '~''*+ ~(V) +... + H ~ ~m+ ~(V) 165 x66 PHII, LIP A. GRIFFITHS of J(V), and so we may think of ]72,Z-2m-1. d-m as the space of holomorphic differentials Letting Ze| we define the Abel-Jacobi map (A.2) by on J(V). (A.6) ~b(Z) = (fci=) (modulo periods) where C is a chain on V with 0C=Z. We recall from [14], [27], [23] that ~ is holomorphic (in a suitable sense), and + has nice functorial properties. In particular, suppose that X is a smooth, complete and projective algebraic variety which contains V as a smoothly embedded subvariety. of cohomology induces a homo- The restriction map H2"+I(X, C) ~ H2"+'(V, C) morphism of intermediate Jacobians r : J(X) -+J(V) with the following interpretation: Proposition (A. 7). -- Intersecting cycles on X with V induces a homomorphism ~: |174 such that the diagram: *, j(x) o(x) ,r o(v) , J(V) is commutative. algebraic family of algebraic varieties and Remark. -- Let f: X-->S be an assume that S is complete. Suppose that V is a fixed fibre of f:X~S, and let o~----(E, D, Q, {Fq}) be the variation of Hodge structure whose fibre corresponding to V is H2"+I(V, C). Then the image in the mapping J(X) -+j(v) is precisely the fixed part J ( @) as defined in w 7, c). Proposition (A.7) gives an algebro- geometric interpretation of this fixed part. b) Let f: X+S be an "algebraic family of algebraic varieties with fibres V, =f-l(s) (seS). We shall think of V, as just discussed in a) above, as being a typical fibre. For simplicity we shall continue to assume that all of H2"+t(V, C) is primitive. Let N=(E,D, Q, {1~}) be the variation of Hodge structure associated to f: X~S and H am f l(V, C), and let AcE R be the flat lattice given by the images of t-la"+'(V,, Z) -+ H2"+ a(V,, R). Referring to w 7, c), the corresponding family of intermediate Jacobians =: J+S, J,=J(V,) will be said to arise from a geometric situation. 166 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III Let now 0(X/S) be the sheaf which associates to each open set U cS the group of analytic cycles Z (modulo rational equivalence) of codimension m+I in f-l(U) such that the cohomology class of Z is zero in H~m+l(f-l(U), Z) (cf. [23] ). Theorem (A.8) (Integrability theorem for Abel-Jacobi maps). -- The Abel-Jacobi maps (A. 6) induce a sheaf mapping : o(x/s)-+j which satisfies the integrability condition DjW = o. Proof. -- Our proof of the existence of tF is based on [23]. We may assume that dimcS=i and we let f: Y-+A be the situation f: X-+S localized over a small disc A on S which has holomorphic coordinate s. We let ZcY be an algebraic cycle of codimension m + I and in general position with respect to the fibres V s. Then the intersections Z~ = Z. V s are algebraic cycles of codimension m + I on V~ which are homologous to zero there. In fact, we have: Z- 0C (modulo 0Y) for a suitable chain C on Y, and we may put things in general position so that Z s ~ 0C s where C 8 = C. V s (seA) ; (el. [23] for a complete discussion of the foundational points involved here). Let +(Z,) eJ(V,) be the point defined by the Abel-Jacobi map (A. 2). We want to prove that +(Z,) depends holomorphically on s. For this we choose C ~ differential forms col, ..., r z on Y such that (i) each o)~ is of type (d,d--~m--I)+...+(d--m,d--m--I); (ii) do)~^ds~-o; and (iii) the restrictions o)~]V, =o),(s) give a basis of F2a-2"-l'a-"-~(V,) (cf. Propo- sition (A. 4)). The existence of such forms is proved in [23], where it is also proved that the integrals fce%(s) may be assumed to depend continuously on s. Let co be any linear combination of o)x,..., co s . We want to show that the integral fc depends holomorphically on s. Let ~, be a simple, positively oriented, s ~ closed curve in the disc A. It will suffice to show that Let C v =Cnf-l(y) and Z v be the intersection of Z with the part of Y lying over the region inside y. Then by Stokes' theorem: f (foo))ds--fc since do) ^ds=o. But (~ o)^ds=o since o)^ds is of type (d+I, d--2m--I)+... +(d--m +i, d--m--i) 167 x68 PHILLIP A. GRIFFITHS whereas Z. e is an analytic set of complex dimension d--re. This proves the existence of the sheaf homomorphism q~: | obtained by fitting together the Abel- Jacobi maps along the fibres of f: X---~S. We want now to prove the integrability condition Dj~F=o. For this we first localize to have f: Y~A as before, and then choose C OO differential forms ~0,, ..., co~a on Y such that (i) deojAds--- o (j=I, ..., 2/); (ii) the restrictions coj]Vo give a basis of H2a-2m-t(Vs, C); (iii) ~t, "'-, ~ are of type (d, d--2m--I)+...+(d--m, d--m--l) and restrict to a basis of F2a-2~ -l'a-r"-t(V,); and (iv) o h .... , ,% are of type (d, d--2m--I)+... +(d--re+I, d--m--2) and give a basis of F~-2m-l'a-m-2(V,). We may think of oh, ..., ~0n as a holomorphic frame for the flat bundle E with fibres Eo = ~-2"- t(V ' C) and which is adapted to the filtration F a-,.-2 c F a-"-I cE. We let ca 9) i i Then 0~----~(s)ds where the functions a~(s) on A have the be the connection in E. following interpretation : Write dc~ = ~i ^ ds where the ~i are C ~ forms on Y. This is possible by the first property of the c0~s. Then d~i^ds=o so that ~i[V 8 is closed and gives a cohomology class in HZa-z''(V,, C), and we have in H2~-m-t(V0, C). (A 9 We can even assume that in l~a-~- t, a-'~-l(V,) (I_<_i< k). (A. 9 9 7, = X .{oj j=t seA. F, eH2a_~_t(V,, Z) be a cycle varying smoothly with Lemraa (A. x2). -- Let Then 9J z If i=l\JFs ] ~" Proof. -- We have: ,-~o t\Jr,§ Jr, ! 168 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, lII x% where we may restrict t to be real and positive. Let F(s,s~, t) be the union of the cycles F, for s <__ z < s + t. By Stokes' theorem, ~t) d%) tO 6) t~o t ,+ t)~q~ ^ds ) t ~ 0 "(s, 2/ (by (A. IO)). = (j,, i=1 This completes the proof of Lemma (A. 12). Lemma (A. x3). -- Using the notation established above, we have (jL ,)oj So, Proof. -- As in the proof of i.emma (A. 12) we h't O(s, s-r- t) the union of the C, for s_<z<s+t. Then eC(s, s + t)=c~+,-c,-z(~, s + t) where Z(s,s+t) is union of the cycles Z~ for s._<v<:s+t. Now fz(,,,+,)coj=o since c0j is of type (d,d--2m--I)+...-r-(d--m+I,d--m--2). Using Stokes' theorem we then have co -+ I ds) --= j (: "~j ~=, c CO~ ~j (by (A.1I)). This completes the proof of Lemma (A. i3). We now choose a frame el, ..., ezs for the dual bundle E such that (o~j,%+1 i)--~ ~ (I<i,j~M). Observe that the fibre E8 = H2m + I(V~, C) and that we have t ((~1, "" ", %)1 =(el ' ..., e,) ~(ox, ..., ~ = (ex, "-', % -k) so that et, . .., el is a basis for F 2"+ I"(V,) and el, 9 9 e2t k is a basis for F 2''+ ~' '" + ~(Vs). Writing 2'2 PHII, LIP A. GRIFFITHS x7o we have the relation 0i . 2/+1--/ (A. ~4) ~--~2/+l--j" We may now prove that D a ~F = o. Thc vector ~)'= (fCj=l ~ .t 6)j ) e2I+ 1-J is a section of E which projccts onto ~F in the mapping E---~J. We want to compute D~2 and then show that (A. x5) D~l-o modulo et, . .., eu_~.. Using the notation " = " for " congruent modulo et, . .., e21-k ", we have k l j=l \Jt~s $" -- j=l xat~s "~. - J I l 1 k j-ti=t j= i=l c60j ~+t-/e2/+l-.i (by Lemma (A. t3) ) -o (by (A. I4) ). This completes our proof. 170 APPENDIX B Two Examples a) A family of curves. -- We shall construct and then discuss an example due to Atiyah [I] of an algebraic family of curves f: X-+S where the parameter space S is itself a complete curve. To construct the example, we take a smooth, complete curve C having a fixed- point-free involution j: C-+C. Such curves exist whenever the genus p(C)_~ 3. Now we take S to be the finite unramified abelian covering of C given by the composite homomorphism nl(C) -+ Hi(C, Z) -+ HI(C , Z2). Let n : S-+C be the covering map, Y = S � C the product variety, and D = r. + Fjo= the (non-singular) curve on Y which is the sum of the graph of n and the graph of jon. Atiyah shows that there is a non-singular algebraic surface X which is a 2-sheeted covering of Y with branch curve D. The projection f: X-+S then gives X as an algebraic family of algebraic curves {V,}se s where V8 is a 2-sheeted covering of C with branch points at n(s) and jo~(s). Now for us the main important thing is the existence of a non-trivial family of non-singular curves with a complete parameter space. Let f: X-+S be one such family where the corresponding variation of Hodge structure is non-trivial (i.e., the fibres V 8 are not all birationally equivalent). The sheaf R~,(0x) is the sheaf 0(E ~ and from (7. I I) we have (8.,) ~ IS] >o. We want to compute the signature sign(X), and to do this we use the Hirzebruch index formula for X and the Grothendieck-Riemann-Roch formula for f : X~S and 0x as in Atiyah [I] to obtain d ~ sign(X) = ~- [X] (B.2) d 2 I+d+-~), ch(I-- R~. (0x)) =f,( where d~H2(X, Z) is the first Chern class of the tangent bundle along the fibres of f:X-+S. From (B.I) and (B.2)we have sign(X) = q(E ~ [S] >o, CB.3) so that the signature is not muhiplicative in the fibration f: X-+S. 22* PHILLIP A. GRIFFITHS I72 The exact same proof will give the Generalprinciple (B. 4). -- Let f: X-+S be an algebraic family of algebraic varieties with complete parameter space S. Then the Hirzebruch ;(v-genus [2I] is generally not multiplicative for the fibration f: X~S. Another point we are trying to illustrate is that there are interesting examples of algebraic families of algebraic varieties with a complete parameter space, although the most interesting case is certainly when the fibres are allowed to have arbitrary singularities. b) Lefschetz pencils of algebraic surfaces. -- In order to illustrate the existence of algebraic families of algebraic varieties f: X-+S whose parameter space need not be complete but where the Picard-Lefschetz transformations are of finite order, we consider a smooth, complete, and projective threefold WcP N. A generic pencil IP~_ ~(X)ixc-p ' of linear hyperplanes in Px traces out on W a pencil IVx[xc-,, of surfaces with critical points X~, ..., )'N- Letting S=P~--{X~, ..., X~}, the V x (Xc-S) are non-singular surfaces while the Vx~ are surfaces having one isolated ordinary double point. In the obvious way we may construct an algebraic family of algebraic varieties f: X~S with f-t(X)=Vx for Xc-S. This family has the property that there is a smooth compactification XcX re r sos such that f has one of the local forms if(x~, ,=, x~)=x, ! f (x,, ~,., x~)- (~,)~ + (x~) 2 + (x.p where xl, x2, x 3 are suitably chosen local holomorphic coordinates on X. We want to use the theorems in Lefschetz [26] to amplify two of our results above. Before doing this, we fix a base point X0c-S and paths l~ from X 0 to each critical point X~ (e=I, ..., N). We let y~c-~(S) be the closed curve obtained by going out l~, turning around X~, and then returning to X o along l~. Associated to each path l~, there is a vanishing cycle 8~c-H2(Vx~ Z) such that (~, ~)~ (B.5) T~q~=q~ where T~ is the automorphism of I-I2(Vx,, Z) corresponding to ~,=E~I(S ) ([26], p. 93). We have, furthermore, that (loc. cit., p. 93) (a~, a=)=- ~, CB.6) so that (T~)2=I and the Picard-Lefschetz transformations in our family of surfaces f: X~S are all of finite order. 172 PERIODS OF INTEGRAI.S ON ALGEBRAIC MANIFOLDS, III I73 Proposition (B. 7). -- There is a ~l(S)-invariant orthogonal direct sum decomposition (B. 8) H2(Vx~ Q) = I| where I = H~(Vz,, O) "~(s) are the invariant cycles and where ~x(S) acts irreducibly on E =(I) 1. Proof. -- Let E'c E be a non-trivial ~l(S)-invariant subspace and q~# o a vector in E'. From (B.5) we have that ..., N) while some (~,., q~)#o since q~ is orthogonal to the invariant cycles. Thus 8~.cE' and it follows that all 3,eE' since =1(S) acts transitively on the set {~1, .--, 81~} of vanishing cycles (loc. cit., p. 1o7). Thus E=E' since E is the span of 31, ..., ~N (loc. cit., p. 93)" Our second observation is Proposition (B.9). -- Let f: X-+S be the family of surfaces constructed above and let rcAut(H2(V,,, C)) be the monodromy group. Then r is a finite group if, and only if, the subspace H2'~ of H2(Vz., C) is elementwise invariant. Proof. -- If H~'~ is elementwise invariant, then we have EcHI, I(Vx.)o in (B.8). Since the intersection form is negative definite on E, we see that P is a finite group. Conversely, if I' is a finite group, then the subspace H2'~ of H2(Vx, C) is locally constant. In particular t-P,~ ) is a Z~l(S)-invariant subspace of H2(Vx,, C). Let ~I~'~ Then from (B.5) we have that (% ~)~H2,~176 for ~=i, ..., N. If some (% ~,.):t:o, then 8~~176176 which is impossible by (B.6). Thus all (% ~)= o and so q~ is an invariant cycle. 173 APPENDIX 12 Discussion of some open questions a) Statement of conjectures. -- Many of our results about a variation of Hodge structure had restrictions imposed concerning the Picard-Lefschetz transformations around the branches of S--S. We should like to suggest that these theorems should be valid under much more general circumstances. To state things precisely, we first need a few comments about the monodromy group P of the variation of Hodge structure 8, especially with regard to the Picard- Lefschetz transformations (w 3) of F. Kecall the monodromy theorem (cf. w 3 in [13] for discussion and references), which says that in case d' arises from a geometric situation f: X-+S, the Picard-Lefschetz transformations T are essentially unipotent of index n (same n as in w 2), which means that viewed as automorphisms T:E-+E they satisfy the equation (C.x) (TS--I)"+l=o for some N>o. We also recall that in the geometric case the monodromy group is a discrete subgroup of the automorphism group G of the variation of Hodge structure. Finally we recall the theorem of Borel (w 3 in [13] ) which says that in case F is an arithmetic subgroup of G, then the monodromy theorem holds, but without the estimate on the index of unipotency being established as yet. Our precise conjectures are (cf. the remark added in proof at the end of Appendix C) : (C.2) The invariant cycle theorem (7-1) is true if we assume only that the mono- dromy theorem (C. i) is valid. Remark. -- We shall prove this conjecture for n=i in a little while. If this conjecture is true, then the theorem (9.8) about the monodromy group would also hold. (C.3) The Mordell-Weil type theorem (7-19) is true if we only assume the monodromy theorem (C. I), but where we need to say what it means for a holomorphic section of J--~S to remain holomorphic at infinity. Remark. -- We shall also prove this conjecture for n = I below. (C.4) Theorem (9.7), which says that the image qb(S)cP\D under the period mapping is canonically a projective algebraic variety in case S is complete, is true if we only assume that F is a discrete subgroup of G. 174 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III t75 Remark. -- This conjecture is discussed in w167 IO, x i of [3]- We also refer to w 6 of [13] where it is pointed out that this conjecture (C.4) is valid in case D is a bounded, symmetric domain and I' is an arithmetic subgroup of G (Borel). Now how is one supposed to prove the above conjectures ? My own (obvious) feeling is that it should be possible to prove (C. 2) and (C.3) by sufficiently ingenious use of hyperbolic complex analysis so as to be able to give a good asymptotic form of the period mapping as we go to infinity. We shall give an illustration of this below. To prove (C.4) , one will need the estimates from hyperbolic complex analysis as well as a reduction theory for discrete subgroups of G. It also seems to me that " sufficiently ingenious use of hyperbolic complex analysis " will involve a detailed study of the geodesics of the metric dsg on the period matrix domain D as well as a more refined Schwarz lemma (io. i) which will give suitable estimates both ways in (lO.2). b) Proof of the invariant cycle theorem (7-i)Jbr n = i. -- Thus let d ~ be a variation of Hodge structure, with base space S, and let @ be a flat section of E-+S. The Hodge 0 1 filtration in this case is F, cFs=Es, and we let 9 be the projection of 9 in E/F~ 1. Using theorem (5-9), we want to prove that the length ]912 of 9 is uniformly bounded on S. We may assume that dimcS=I. We then localize over a punctured disc A* at infinity given by A*={s :o<lsl<i}. Choose a base point s0eA* andlet c01, ..., m2,~ be a flat frame for E in a neighborhood of s 0. Parallel displacement of this frame around the origin induces an automorphism 2m where T-----(T~) is the Picard-Lefschetz transformation around s = o. The monodromy theorem (C. I) is (TN--Iy=o, and by replacing s with s ~, we may as well assume that (T--I) 2=o. Now we may choose over A* a holomorphic frame 91(s), ..., 9,~(s) for the sub- bundle F~ Then we define the period matrix t2(s)=(n~j(s)) by 2m ..., m). The bilinear relations (2.7) now become the usual Riemann relations i ao_'a=o (C'5) ? iaQ~fi =j>o where tQ-l=(Q(o~, @). The matrix f~(s) is a multi-valued holomorphic matrix on A* such that analytic continuation around s = o changes f~ into Y~T. Let y be the flat section of the dual bundle E* defined by < y, e} = Q(O, e) (e~E). 175 PHILLIP A. GRIFF1THS Let ~ be the column vector given by t~ =(~,, ..., ~,,~) where ~ =(y, q~(s)}. Then, if we set H=J -1 in (C.5) , t~H~ is a well-defined function on A* and gives the length [q~] z of the projection q~ of 9 onto E/F ~ Thus we want to show that (C.6) t~H~ is bounded as s-~o. We will now use the results of [I3] , w 1 3 to put f2 in canonical form. Accordingly we can choose the frames col, ..., c%~ and q01, ..., q% such that the matrices Q, T, and f~ are given by (o i:). (C.7) Q= _im ' oo!) I,~_ k o (C. 8) T = f2/I A =*A, A>o; o I k o o Im-k f2 = (Ira, Z) (c.9) where Z=tZ has the form where the submatrices Z~ are holomorphic in the whole disc ]s]<I. Write Z~ = X~ + iY~. Then (C. II) j=/~-~QI~={Yll Y12] log,s, (~ ~) \tY12 YJ zz: where the Y,~ are continuous and Y,>o throughout the disc 5 given by Is I< i. From (C. 9) and (C. I o) it follows that the vector ~ is continuous on Using (C. 6) we will be done if we prove that j-1 is continuous on A. Now j-l= J* (det J) where J* is the usual matrix of minors of J. From (C. 1 I) we see that each entry in J* has the form (--log Is[) m-k. (continuous function of s), while from (C.8) and (C.II) we have (--1 og ~r~ tsI) '~+* A) + (lower terms). det J = __ (det Yn" det order Since det Yn.det A>o throughout A, we are done. 176 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III I77 c) Proof of the usual Mordell-Weil over function fields. -- We will give a transcendental proof of the usual Mordell-Weil theorem for abelian varieties defined over function fields (char. o of course). Theorem (C. x2). -- Let do, A, and J-+S be as in the statement of theorem (7.19) with nI. Then there exists an extension of J-+S to an analytic fibre space J~S ofabelian complex Lie groups such that the group Horn(S, j) of holomorphic cross-sections of J-+S is an extension of the fixed part J( d ~ by a finitely generated abelian group. Remark. -- The integrability condition Ds~ = o is vacuous in this case since n = i. Proof. -- Let E+-+S be the holomorphic vector bundle over S whose fibre at each point seg is the complex Lie algebra of Js. Then F.+ IS is what was denoted by E+ in the proof of theorem (7- :9), and just as was the case in that proof, we want to show that any holomorphic section of ~..+ -+S comes from a constant section of E-+S. Of course this presumes that we have already defined J~S, which we now shall do. Let ~(A) be the sheaf over S of sections of the lattice AcE. We extend ~'(A) to a sheaf over S by saying that the sections of Cg(A) over an open set U c g are just the usual sections of <g(A) over UnS. To define ~.+, we will say what the sheaf g(E+) of holomorphic sections of the dual bundle is. Thus a section of Og(E+) over an open set U c-S is given by a holomorphic section r of E+ over U n S such that, for any section y of ~(A) over U, the contraction (?(s), Y} is a holomorphic function on all of U. There is an obvious injection Cg(A)-->~..+ and j is defined to be the quotient E+/Cg(A). .. We must prove that 0~(E+), as defined just above, is a locally free sheaf on all of S, that the image ~(A)-+E+ is discrete, and finally that the holomorphic sections of F.+-+g come from constant sections of E-+S. This is all done using the formulae (C.7)-(C.::) above together with the observations that (i) the flat frame ~i, .-., {~ of l~---~t* may be chosen to be commensurable with the lattice A, and (ii) the holomorphic sections of E+-+A are just the linear combinations of qh, ..., % with coefficients which are analytic functions in the whole disc A. Remark added in proof. -- Recent results of W. Schmid seem to show that the mono- dromy theorem (C. 1) is true for an arbitrary variation of Hodge structure. It may be hoped that his methods will also have a bearing on (C. 2) and (C. 3). 177 APPENDIX D A result on the monodromy of K3 surfaces In Pn with homogeneous coordinates ~---'--[~0, ~a, ~, ~3], we consider quartic surfaces defined by an equation Z.~,)s~,~,~,~, ~- ~,~. = o. (~~ .... The set of all such surfaces is parametrized by points s = [..., s~.~,~,~,, ... ] in a big PN. We denote by S' the Zariski open set in Ps of points such that the corresponding surface V, is non-singular. Such surfaces V, are among the K3 surfaces; i.e. they are simply- connected algebraic surfaces with trivial canonical bundle. We let sotS' be a fixed point and V=V,, the corresponding K 3 surface. Denote by E=P2(V, Q) the primitive part of the 2 ~a cohomology of V, and let F~cAut(E) be the arithmetic group induced from the automorphisms of H~(V, Z) which preserve the bilinear cup-product form and polarizing cohomology class. We denote by I" c P~ the global monodromy group; i.e. the image of ~1(S', So) acting on E. Theorem (D. x ). -- r is of finite index in F~. Proof. -- The period matrix domain D is, in this case, a bounded domain in C 19 and, by the local Torelli theorem [i i], the period mapping q~ : S'-+D/F contains an open set in its image. We now choose a I9-dimensional smooth algebraic subvariety Sr such that the restricted period mapping : S-+D/F contains also an open set in its image. By Theorem (9.6) above, q~(S) is the complement of an analytic subvariety in D/I'. Furthermore, because of the finite volume statement, it follows that D/U has finite volume with respect to the canonical invariant measure on D. Now it follows that the index of P in P~ is given by [r; i'.]= --~(D/F) <oo. ~(D/ro) Remarks. -- From Theorem (9.8) it follows that P is irreducible. Observe that from (B. 5) we may deduce that P is generated by elements of order 2. 178 PI.;RIODS OF INI'EGRALS ON .\I.(;EBRAIC MANIFOLDS, III x79 In general, for an algebraic family of algebraic varieties as in w i, the position of the monodromy group r in the arithmetic group 17 is extremely interesting. I know of no example where F is not of finite index in its Zariski closure. In relation to this, we close by observing that the proof of Theorem (D. ~) in general gives the following: Theorem (D.~). -- Let f: X~S be an algebraic family of algebraic varieties, and designate by P the glol)al monodromy group. Denote by S the universal covering of S and let " )" O S > D/l" be the period mapping. Let P' be any discrete subgroup of G = Aut(D) such that I" c F' and such that P' leaves the closure 0fq~(S) invariant. Then r is of finite index in P'. R EFERENCES [l] M. F. ATtYAH, The signature of fibre-bundles, Global Analysis (papers in honor of K. Kodaira), Princeton Univ. Press (x969), 73-89. [2] A. BLAN~ARD, Sur les vari6t6s analytiques complexes, Ann. Sci. [5cole Norm. Sup., 73 (1956), 157-2o~. [3] S- BOC:q~ER and K. YANO, Curvature and Betti Numbers, Princeton Univ. Press, J953- [4] A. BOR~L and Arithmetic subgroups of algebraic groups, Ann. of Aiath., 75 (I962), 485-535- [5] A. BoR~.~. and R. NARASlMHAN, Uniqueness conditions for certain holomorphic mappings, Invent. Math., 2 (1966), z47-~55. [6] E. CARTA~, Lefons sur la g~oraltrie des espaces de Rievnann, Paris, Gauthier-Villars, r95~. [7] S. S. Ctlnm~, On holomorphic mappings of Hermitian manifolds of the same dimension, Proc. Syrup. in Purr Math., 11, American Mathematical Society, x968. [8l S. S. C~tErtN, Characteristic classes of Hermitian manifolds, Ann. of Math., 47 (t946), 85-~2I. [9] P" D~LIOr, tI~, Theorie de Hodge, to appear in Publ. LH.E.S. [IO] I-t. GRAIJIgRT, I)ber Modifikationen und exzeptionelle analyfische Mengen, Math. Annalen, 14.6 096~), 331-368. [11] P. A. GRr~r~nnas, Periods of integrals on algebraic manifolds, I and II, Amer. Jour. Math., 90 (I968), 568-6'-'6 and 805-865 . [i2] P. A. GnsFFrrHs, Monodromy of homology and periods of integrals on algebraic manifolds, lecture notes available from Princeton University, I968. [13] P. A. GRI~'Fn~S, Periods of integrals on algebraic manifolds, Bull. Amer. j14ath. Soc., 75 097o), 228-o.96. [14] P. A. GRIFFrrHs, Some results on algebraic cycles on algebraic manifolds, Algebraic Geometry (papers presented at Bombay Colloquium), Oxford University Press, I969, 93-19 z. [x5] P. A. GRI~ITHS, Periods of certain rational integrals, Ann. of Math., 90 (t969), 460-54 I. [t6] P. A. GmFFITHS and W. SCHMIn, Locally homogeneous complex manifolds, Aeta Math., 19.8 (197o), '~53-'3o'- '. [t7] A. GaO'rrmNDmCK, Un th~or~me sur les homomorphismes de schemas ab61iens, lnoent. Math., 9. (196b), 59-78. [18] A. Gao'neer~t~cK, On the de Kham cohomology of algebraic varieties, Publ. Math. I.H.E.S., 9.9 (~966), 95-m3. ['9] R. Gum, nN~; and H. Ross*, Analytic Functions of Several Variables, Prentice-Hall, ~965. [zo] H. HmO~A~ZA, Resolution of singularities of an algebraic variety over a field of characteristic zero, 1 and 11, Ann. of Math., 79 (~964), ~o9-326. [2~] F. HXRZg~RtrCa, .Neue Topologische Methoden in der Algebraisehen Geometric, Springer-Verlag, x956. [2'2-] W. V. D. HOD~, The Theory and Applications of Harmonic Integrals, Cambridge University Press, I959. [~3] J- K~N~, Families of intermediate Jacobians, thesis at University of California, Berkeley, x969 . HAmStI-CHAN1)RA, ~8o PHILLIP A. GRIFFITHS [24] K. KODAIRA and D. C. SPENCER, On deformations of complex analytic structures, I and II, Ann. of Math., 67 (I958), 328-466. [25] M. H. KWACK, Generalization of the big Picard theorem, Ann. of Math., 90 (I969), I3-22. [26] S. LEFSCHETZ, L'Analysis Situs et la Gdom~trie Algdbrique, Paris, Gauthier-Villars, 1924. [27] D. Higher Picard varieties, Amer. flour. Math., 90 (1968), Ix65-I I99. [28] G. MosTow and T. TAMAGAWA, On the compactness of arithmetically defined homogeneous spaces, Ann. of Math., 76 (I962), 446-463 . [29] C. L. SIEGEL, Analytic functions of several complex variables, lecture notes from Institute for Advanced Study, Princeton, I962. [3 O] A. WEIL, Varidt~s Kdhleriennes, Paris, Hermann, i958. Manuscrit refu le 12 novembre 1969. Revisg le 6 juillet 1970. 1970. -- Imprimerie des Presses Universitaives de France. -- Vend6me (France) ~DIT. N ~ 31 283 IMeRIM~; EN FRANCE IMP. N ~ 22 222 LIEBERMAN, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals http://www.deepdyve.com/lp/springer-journals/periods-of-integrals-on-algebraic-manifolds-iii-some-global-XJFdS8zyRK

Loading next page...

References (43)

(1968)
On holomorphic mappings of Hermitian manifolds of the same dimension
P. Griffiths (1968)
On the Periods of Integrals on Algebraic Manifolds
Rice Institute Pamphlet - Rice University Studies, 54
S. Lefschetz (1925)
L'analysis situs et la géométrie algébrique
The Mathematical Gazette, 18
P. A. Griffiths (1969)
Periods of certain rational integrals
Ann. of Math., 90
F. Hirzebruch (1956)
Neue topologische Methoden in der algebraischen Geometrie
The Mathematical Gazette, 41
A. Blanchard (1956)
Sur les variétés analytiques complexes
Annales Scientifiques De L Ecole Normale Superieure, 73
P. Griffiths (1969)
On the Periods of Certain Rational Integrals: II
Annals of Mathematics, 90
(1969)
Some results on algebraic cycles on algebraic manifolds, Algebraic Geometry (papers presented at Bombay Colloquium)
N. Bogolubov, A. Logunov, A. Oksak, I. Todorov, G. Gould (1965)
Analytic functions of several complex variables
P. Griffiths, W. Schmid (1969)
Locally homogeneous complex manifolds
Acta Mathematica, 123
P. Val (1941)
The Theory and Applications of Harmonic Integrals
Nature, 148
G. Mostow, Tsuneo Tamagawa (1962)
On the compactness of arith-metically defined homogeneous spaces
(1958)
Mannscrit recu Ie 12 novembre 1969, Revise Ie 6 juillet
S. Chern (1946)
Characteristic Classes of Hermitian Manifolds
Annals of Mathematics, 47
P. Griffiths (1970)
Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems
Bulletin of the American Mathematical Society, 76
A. Weil (1958)
Variétés Kähleriennes
(1968)
Monodromy of homology and periods of integrals on algebraic manifolds, lecture notes available from Princeton University
A. Grothendieck (1966)
On the de Rham cohomology of albebraic varieties
Publ. Math. I.H.E.S., 29
M. H. Kwack (1969)
Generalization of the big Picard theorem
Ann. of Math., 90
小平邦彦, D. Spencer, 弥永昌吉 (2015)
Global analysis : papers in honor of K. Kodaira
H. Grauert (1962)
Über Modifikationen und exzeptionelle analytische Mengen
Mathematische Annalen, 146
A. Borel, Harish-Chandra (1961)
Arithmetic subgroups of algebraic groups
Bulletin of the American Mathematical Society, 67
(1962)
Math. Annalen
K. Kodaira, D. Spencer (1958)
On Deformations of Complex Analytic Structures, II
Annals of Mathematics, 67
一松信 (1965)
R.C. Gunning and H.Rossi: Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, N.J., 1965, 317頁, 15×23cm, $12.50.
, 20
矢野健太郎, S. Bochner (1948)
Curvature and Betti numbers
Annals of Mathematics, 49
H. Hironaka (1964)
Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: II
Annals of Mathematics, 79
P. Griffiths (1968)
Periods of Integrals on Algebraic Manifolds, I. (Construction and Properties of the Modular Varieties)
American Journal of Mathematics, 90
J. King (1969)
Families of intermediate Jacobians
A. Borel, R. Narasimhan (1967)
Uniqueness conditions for certain holomorphic mappings
Inventiones mathematicae, 2
A. Borel, Harish-Chandra (1962)
Arithmetic subgroups of algebraic groups
Ann. of Math., 75
A. Blanchard (1956)
Sur les variétés analytiques complexes
Ann. Sci. École Norm. Sup., 73
P. A. Griffiths (1970)
Periods of integrals on algebraic manifolds
Bull. Amer. Math. Soc., 75
D. Lieberman (1968)
On higher Picard varieties
M. Kwack (1968)
Generalization of the big Picard theorem
Bulletin of the American Mathematical Society, 74
Vendome (France) EDIT. N° 31 283 IMPRIME EN FRANCE IMP
M. Hasse (1963)
F. Hirzebruch, Neue Topologische Methoden in der Algebraischen Geometrie. Zweite ergänzte Aufl. VIII + 181 S. Berlin/Göttingen/Heidelberg 1962. Springer-Verlag. Preis brosch. DM 30,80
Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik, 43
(1968)
Amer. Jour. Math
A. Grothendieck (1966)
Un theoreme sur les homomorphismes de schemas abeliens
Inventiones mathematicae, 2
P. Griffiths (1968)
PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, II. (Local Study of the Period Mapping)
American Journal of Mathematics, 90
S. Bochner, K. Yano (1954)
Curvature and Betti Numbers. (AM-32)
E. Cartan (1928)
Leçons sur la géométrie des espaces de Riemann
P. A. Griffiths (1968)
Periods of integrals on algebraic manifolds, I and II
Amer. Jour. Math., 90

Publisher: Springer Journals
Copyright: Copyright © 1970 by Publications mathématiques de l’I.H.É.S
Subject: Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
ISSN: 0073-8301
eISSN: 1618-1913
DOI: 10.1007/BF02684654
Publisher site: See Article on Publisher Site

Abstract

PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III (SOME GLOBAL DIFFERENTIAL-GEOMETRIC PROPERTIES OF THE PERIOD MAPPING) by PHILLIP A. GRIFFITHS (i) TABLE OF CONTENTS PAGF~ o. Introduction ................................................................ ~6 Part I. --- Differentlal-Geometric Properties of Variation of Hodge Structure .............. ~3 o I. Algebraic families of algebraic varieties ........................................ 13o 2. Variation of Hodge structure ................................................. z3t 3. Remarks on the homology of algebraic fibre spaces ............................. ~34 4. Remarks on Hermitian differential geometry .................................... x36 5. Statement of main differential-geometric properties of Hodge bundles ............. I39 6. Structure equations for variation of Hodge structure ............................ 14I 7. Applications ................................................................ x45 a) Invariant cycle and rigidity theorems ....................................... x45 b) Negative bundles and variation of Hodge structure .......................... I46 c) A Mordell-Weil theorem for families of intermediate Jacobians ................ x47 Part II. --- Differential-Geometric Properties of the Period Mapping ....................... 8. Classifying spaces tbr Hodge structures ........................................ 9- Statement of results on variation of Hodge structure and period mappings ........ ~55 to. The generalized Schwarz lemma .............................................. I58 II. Proof of Propositions (9. Io) and (9. xx) ........................................ x62 Appendix A. -- A result on algebraic cycles and intermediate Jacobians .......................... ~65 Appendix B. -- Two examples: ,~) A family of curves .................................................................. I7I b) I,efschetz pencils of surfaces .......................................................... x7 ~ Appendix C. -- Discussion of some open questions: a) Statement of conjectures ............................................................. b) Proof of the invariant cycle theorem (7. I) for n=t ................................... ~75 c) Proof of the usual Mordell-Weil over function fields .................................... x77 Appendix D. -- A result on the monodromy of K 3 surfaces ...................................... x78 (a) Supported in part by National Science Foundation grant GP7952XI. 125 126 PHILLIP A. GRIFFITHS o. Introduction. a) In this paper we shall study some global properties of the periods of integrals in an algebraic family of algebraic varieties. Although our results are mostly in (alge- braic) geometry, the proofs are purely transcendental. In fact, we may roughly describe our methods as giving various applications of the maximum principle to problems in algebraic geometry. For the most part these methods have only succeeded in treating the situation when the parameter space for the non-singular varieties is complete. While the results should be true in general, it appears that new methods will be required to handle the situation when singular varieties are permitted in our family. These questions are discussed from time to time as they occur in the text below. The paper divides naturally into two parts. The first treats linear problems and is a study of the differential-geometric properties of the Hodge bundles as defined in w 2. The use of the maximum principle here is similar to the classical Bochner method [3], and is based on the rather remarkable structure equations and curvature properties of the Hodge bundles. The second part deals with global properties of the period mapping [: :], and the methods are those of hyperbolic complex analysis which, to paraphrase Chern [7], is the philosophy that suitable curvature conditions on complex manifolds impose strong restrictions on holomorphic mappings between these manifolds. A more detailed introduction to the two parts of the paper will now be given. b) Wc consider an algebraic family of algebraic varieties {V,}se s as defined in w :. For the time being we may think of the parameter space S as being a (generally non- compact) algebraic curve. The algebraic varieties V: corresponding to the points Y at infinity in S may be thought of as the specializations of a generic Vs having acquired singularities. If we replace V, by the cohomology groups H"(V,, C) and the various subspaces H~.q(V,) cH~(V~, 13) (p-t-q=n), then we find that the algebraic family {V~}~a s gives rise to a whole colh:ction of holomorphic vector bundles over S. We can abstract the data of theses bundles and arrive at what we call a variation of Hodge structure (w 2). Now the bundles which turn up in a variation of Hodge structure have intrinsic Hermitian differential geometries (w 4), and we use more or less standard methods in differential geometry to deduce results about the variation of Hodge structure which, in case this variation of Hodge structure arises from a family of algebraic varieties, have interpretations as theorems on invariant cycles and on holomorphic cross-sections of families of intermediate Jacobians. The only real twist is that the Hermitian vector bundles which appear generally have indefinite Hermitian metrics. In such a situation, the maximum principle does not usually apply. We are only able to push things through by using the so-called infini- tesimal period relation [I I] satisfied by periods of integrals and which is incorporated into the definition of variation of Hodge structure. It is perhaps worth pointing out that 126 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III ~27 the maximum principle is used to show that certain differential equations are satisfied, rather than to show that a " harmonic tensor " is zero as was the classical case [3]. In w 4 we give a review of Hermitian differential geometry and, in particular, discuss the second fundamental form of a holomorphic vector bundle embedded in an Hermitian vector bundle. The main differential geometric results on variation of Hodge structure (Theorems (5.2) and (5.9)) are stated and discussed in w 5. The proofs of these theorems are given in w 6 where we derive the structure equations for a variation of Hodge structure. This section is the heart of Part I of the paper, and we have used the Cartan method of moving frames ([6], [8]) to expose the structure equations (6.4)'(6-8), (6.12), and (6.18) of a variation of Hodge structure. These equations are to me quite remarkable and are much richer than one might have thought from just the classical case when the V 8 are curves. For example, if {V,}se s forms an algebraic ti~mily of algebraic surfaces with complete parameter space S and if y, eH~(V,, Z) is an invariant 2-cycle, then there is a non-negative function qb(s) whose vanishing at seS is necessary and sufficient that y, be the homology class of an algebraic curve on V,. It was a pleasant surprise to me that ~ turns out to be pluri-subharmonic on S. Even in ease S is not complete, ~ should be bounded, but this depends on the local invariant cycle conjecture (3-3). In w 7 we give three applications of the results in section 5. The first of these are some rigidity properties of variation of Hodge structures with complete base space (Corollary (7-3) and (7.4))- These particular results were motivated by a question of Grothendieck [17] and have appeared previously in the preprint [I2] with the same proof as given here. In this paper the rigidity theorems are given as consequences of Theorem (7.I), which was also in w 8 of [12] but was poorly stated there. The much better formulation given below is due to Deligne, whose paper [9] has several points of contact with this one, which are discussed in w 3 below. The second application is the positivity of certain bundles arising from a variation of Hodge structure (Propositions (7.7) and (7.15)). The third application is a Mordell-Weil type of theorem for cross-sections of families of intermediate Jacobians (Theorem (7.19)). Again this is a result purely about variation of Hodge structure but which is suggested by algebraic geometry. In this case the motivation comes from the study of intermediate cycles on algebraic varieties and the connection with Theorem (7-I9) is explained in Appendix A. c) Associated to any variation of Hodge structure, with parameter space S, there is a period matrix domain D [II], which is a homogeneous complex manifold D = G/H of a non-compact simple Lie group G divided by a compact subgroup H, together with a denumerable subgroup P of G and a holomorphic period mapping [r r] : (o.x) 9 : S-~r\D. In fact, the giving of a variation of Hodge structure over S is equivalent to giving a period mapping (o. I) satisfying an infinitesimal period relation which can be stated 127 x~8 PHILLIP A. GRIFFITHS purely in terms of D. These period matrix domains are discussed in w 8, and the correspondence between variations of Hodge structure and period mappings is given by Proposition (9.3). In many interesting cases, such as when the variation of Hodge structure arises from an algebraic family of algebraic varieties, the monodromy group P is a discrete subgroup of G and consequently r\D is a complex analytic variety. The point of view we have taken in Part II is to apply hyperbolic complex analysis to study the period mapping (o. t). We are especially interested in the asymptotic behavior of the period mapping q) as we go to infinity in S. In case dimcS=I , a neighborhood of S at infinity is a punctured disc A*, and the period mapping (o. I) may be localized at infinity and lifted to the universal covering of A* to yield a holomorphic mapping: : H-~D from the upper half-plane H={z=x+~y :y~'o} to thc period matrix domain D, and which satisfies the equivariance condition: ~(Z+I)=T.O(Z) where TaP is the Picard-Lefschetz transformation associated to the local monodromy around the origin in the punctured disc A*. In case dimeS> x, we can use Hironaka's resolution of singularities to have a similar localization at infinity given by a holomorphic mapping: (o."-) q) : HX... X H-+D which satisfies : *(zl, ..., + ..., = Tj. r ..., where the TgG are commuting automorphisms of D. To use metric methods for the study of the mapping (o.2), we introduce the standard Poincard metric ds~ on H � � H and the G-invariant metric ds~ on D deduced from the Cartan-Killing form on the Lie algebra of G. Now the metric ds~ does not have the (negative) curvature properties necessary to make hyperbolic complex analysis work on an arbitrary holomorphic mapping (o. 2). However, if we use the infinitesimal period relation, then the necessary curvature conditions will be satisfied relative to the mapping r Using this together with a formula of Chern [7], in w IO we prove a generalized Schwarz lemma (Theorem (IO, I)) which says that the period mapping is both distance and volume decreasing with respect to ds~ and ds~, cf. [~6]. The main geometric applications of the Schwarz lemma are Theorems (9-5) and (9.6), both of whose proofs are given in w I I. The first of these is a sort of Riemann extension theorem, and says that a period mapping r : A*~D from the punctured disc to a period matrix domain D extends holomorphically across the origin. Our proof makes essential use of an ingenious argument from [~5] (Proposition (iI. I)). The second result is that the period mapping (o.2) is (essentially) a proper mapping, and 128 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III x~9 consequently the closure of the image of 9 is an analytic set containing ~(S) as the complement of an analytic subvariety. A third geometric theorem is Theorem (9-7), which says that the image q~(S) is canonically a projective algebraic variety in case S is complete. Our final result (Theorem (9.8)) in this section is a theorem about the global monodromy group I' of a variation of Hodge structure with complete parameter space. The statements that I' is completely reducible, and that F is finite if it is solvable, are simply adaptations of similar results of Deligne [9] in the geometric case, which are discussed in w 3 below. The characterization of the case when F is a finite group was given in [I2]. d) As mentioned above, Appendix A contains a result about algebraic cycles and intermediate Jacobians varying in an algebraic family of algebraic varieties. In Appendix B we give some examples. In Appendix C we discuss some conjectures which should be true but which we are unable to prove. Finally, in Appendix D we give an application of the results in w 9 to the global monodromy group of certain (algebraic) K 3 surfaces. e) This paper is a successor to [I I]. However, our point of view has evolved somewhat and perhaps a more appropriate general reference is the survey article [I3], which in particular discusses most of the results in this paper and takes up many related problems and conjectures. Finally, this paper is essentially self-contained, except for w IO where we use a formula from [7] and a result from [I6] about the curvature of the metric ds~ discusscd above. It is my pleasure to thank the referee for many helpful suggestions and comments. 17 PART I DIFFERENTIAL-GEOMETRIC PROPERTIES OF VARIATION OF HODGE STRUCTURE x. Algebraic fan~ilies of algebraic varieties. By an algfbraicfamily of algebraic varieties we shall mean that we are given connected and smooth algebraic varieties X, S and a morphism f: X~S such that (i) fis smoolh, proper, and connected, and (ii) There is a distinguished projective embedding XcP s. Setting V~=-f-a(s) (seS) we may think of f: X-+S as the algebraic family {V~}se s of smooth, complete, connected, and projective algebraic manifolds paiametrized by S. The parameter space S is generally not complete, and we shall want to consider smooth compactificatzons of the situation f:X-->S. Such a smooth compactification is given by a diagram: X cX Cx.,) 4 Sr S c S where X, S are smooth, complete, and projective algebraic varieties which contain X, S respectively as Zariski open sets, and where X--X and S--S are each divisors with normal crossings. Thus, for example, g--S is It,early given by: (i.z) ~i--.sk:o where sl, ..., s~ are part of a local holomorphic coordinate system on S. The divisors Dj given locally by sj =~-o in (I. ~) will be called the irreducible branches of S--S. We then have S=--S-D where I)=D I+...+D m is the divisor with normal crossings, As another example, if dim S=-I and if -S--S is locally given by s=o, then f:X~S will be given locally by: where xt, ..., x~ is part of a local holomorphic coordinate system on X. Such smooth compactifications exist by the fundamental work of Hironaka [20]. We want now to say what it means to localize the situation (i. I) at infinity. Let 130 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, llI t3i S--S be given locally by (I.2) where sl, ..., s d is a holomorphic coordinate system on S. Denote by P the open polycylinder given by o<=[s~[<r (j-----i, ..., d) and let P*=PnS. Thus letting A be a disc in t3 and A* the corresponding punctured disc, we have P~(A) ~ and P*~(A*)~� d-k. Set Y=f-~(P) and Y=YnX. Then the localization of (I. i) at infinity is given by: Y c Y (x.4) ~ P* cP We will generally refer to P* as a punctured polycylinder. 2. Variation of Hodge structure. We shall linearize the situation (I. I). For this we now consider X, S as complex manifolds and f: X--->S as an analytic fibre space and topological fibre bundle. Fix a base point s0~S and consider the action of the fundamental group rh(S ) of S based at s o on the cohomology H"(Vs0 , 13). If L~He(Vs., Q) is the cohomology class of the hyperplane section relative to the given projective embedding X c PN, then L is invariant under rh(S ). Thus for n<_m = dimcV we may define the primitive cohomology pn(V~~ 13) to be the kernel of: Hm-'tV C)-~Hm+'+2tV C) (n=m--r). Lr+l : ~ 8~ ~ s0, Because of the Lefschetz decomposition [22]: [nt2] (2.x) H"(V,., C)= @ LkP"-~(V, ,0, C), k=0 which is a ~l(S)-invariant direct sum (over Q) decomposition of Hn(V~., C), it will suffice to consider the primitive cohomology. Let E=Pn(V~., C) and denote by E +S the complex vector bundle, with constant transition functions, associated to the action of 7h(S ) on E. There is the usual flat, holomorphic connection: D : Cs(E) which one has on any such vector bundle associated to a reprcsentation of the fundamental group. In fact we have a short exact sheaf scqucncc: D n (E) (2.*) o -+ ~(E) -+ (Vs(E) -~ where the sheaf ~(E) of locally constant sections of E has the following interpretation: Let R.~(C) be the usual Leray cohomology sheaf of f: X->S, which we recall is the sheaf arising from the presheaf: U-+ H"(f-t(U), C) 131 :3 2 PHILLIP A. GRIFFITHS where U runs through the family of M1 open sets in S, and define the Leray primitive cohomology sheaf P~.(C) to be the kernel of: L' : (n=m-r). Then %~ is just P~,(C). Now the fibre E s is tile vector space P"(V,, C) and as such has the structure of the primitive cohomology vector space of a K~hler manifold [3o]. Translating this structure into data on the flat bundle E-+S, what we find is the following [i3]: I) A flat conjugation e~'g (eeE). n) A flat, non-degenerate bilinear form (2.3) Q: E| Q(e, e')=(--I)"Q(e', e) called the Hodge bilinearform; and III) A fltration of E by holomorphic sub-bundles (2.4) F~ c:F'~-I cF "=E called the Hodge filtration. Remarks. -- (i) The conjugation on E is induced from the usual conjugation on H"(V~, C) = H"(V,, R) | C. (ii) The bilinear form (2.3) is given by: Q(e, e')=+fvL=-"ee' (m-- dime V,) (2-S)' where e, e'~P"(V~, C) cH"(V,, C). (iii) Letting pn- q, q(V~) = H"- q' q(Vs) n P'(Vs, C), we have for the fibre F~ that: (2.4)' F~ = P" ~ +... + P"- q' q(V.). We will denote the data of a flat bundle E with I)-III) above by 8---- (E, D, Q, { Fq}), and as conditions on this data we have: IV) The Hodge filtration (2.4) is isotropic, which means that: (2.5) "-q-1 where (Fq) : Q(e, Fq)=o}. V) The bilinear form (2.3) is real (i.e. Q=UD) and, if we let F,-q,q = Fqn~,-q =Fqn (~q-1) then we have the Hodge decomposition, which is a G ~~ direct sum decomposition: q=0 132 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III x33 VI) The Riemann-Hodge bilinear relations (q,r) ('~" 7) t (--i)" (--I)qQ(r"-q' q, F"-"q) >o are valid; and VII) The infinitesimal period relation [x I] ('~. 8) D : @s(Fq ) ~ Y~(F q+l) holds. Definition.- We will call the data d~=(E, D, Q, {Fq}) given by I)-III) and satisfying the conditions IV)-VII) a variation of Hodge structure. It is of course not necessary that a variation of Hodge structure come from an algebraic family of algebraic varieties f: X-+S. In case @=(E, D, Q, {Fq}) does arise from f: X-+S, we will say that the variation of Hodge structure arises from a geometric situation. Remarks (2.9). -- (i) Let d~=(E, D, Q, {Fq}) be a variation of Hodge structure. Referring to (2.5) , we have natural isomorphisms: (2. xo) (Fq/Fq- ~) v ~ Fn- q/F n-q-l, which are isomorphisms of holomorphic vector bundles. (ii) We may symbolically rewrite (2.8) as (2.xx) Q(D.F q, Fn-q-2)=o. (iii) Referring again to the infinitesimal period relation (2.8), we see that the connection D induces a linear bundle mapping of holomorphic bundles (2. x2) Grq : E q ~ Eq+l| " where Eq----Fq/'F q-1. The vector bundles E q will be called tile Hodge bundles, a terminology which we shall now try to justify. In case the variation of Hodge structure d" arises from a geometric situation f: X--~S, the fibre E, q is given by: (2'I3) E,q= H~(Vs, f~-q)0 (seS; q=o, I, ..., n), where Hq(V,, f~-q)0 is the kernel of the cup product: L '+1 : Hq(V,, ~z m-'+q) v, _~, Hq+ r +1(V8, f~+ l-q) (n=m--r) when we consider L as a class in Hi(V,, f~s)" In other words, in the geometric situation the fibre E q is just the primitive part of the Hodge cohomology space H n-q'q(V.). (iv) Referring to remark (iii) just above, we shall give a homological interpretation of the maps (~. I2) in case d" arises from a geometric situation f: X~S. To do this we recall the Kodaira-Spencer infinitesimal deformation class [~4] p~eH'(V~, Ov, ) | (seS). 133 i34 PHILLIP A, GRIFFITHS The pairing @v,| - q -+ ~,-q+l v, gives: (a. *4) 1t1(V,, |174176 f~-') -+ Hq+~(V,, f~S-e-'). Comparing (2.14) and (2.13) we see that cup product with the Kodaira-Spencer class gives : (a. *5) o~ : E~ -~ Eq+ t| From [II] it follows that Ps in (2.15) is the same as aq in (2. I2). Summarizing : Proposition (~'. i6). -- In case o ~ arises from a geometric situation, the linear mapping % in (~. I2) is the cup product with the Kodaira-Spencer class. 3. Remarks on the homology of algebraic fibre spaces. a) Consider the situation (I. I) and let: Y cY 4 Y P*c P be a localization at infinity as discussed just preceding (I -4). Since P" =(A')kx(A) d-k where A is a disc and N" is a punctured disc, the fundamental group rh(P ~ is free abelian and has as generators the paths around the deleted point in each of the factors A ~ The corresponding automorphisms of the cohomology H*(V,0, C) are called Picard-Lefschetz (P.-L.) transformations. In case k-----I we shall denote the P.-L. transformation on the primitive cohomology by TeAut(P"(V,~ C)). b) Let f: X § be an algebraic family of algebraic varieties as defined in w i. We consider the Leray cohomology sheaves R4,(C), and we recall the Leray spectral sequence {E~ 'q} which abuts to H'(X, C) and with E~'q=H'(S, R~,(C)). We will prove the following result of Blanchard and Deligne (cf. [2] and Deligne's paper in Publ. LH.E.S., vol. 35, PP" I~ Proposition (3- 9 -- The above spectral sequence degenerates at the E2-term. In particular the restriction mapping (3.2) H"(X, C) ~ H~ Rf",(C)) ~ o is surjective. Proof. -- The cohomology class L of the hyperplane section operates by cup product on the terms E, (r >2) of the spectral sequence and it commutes with the differentials d, : E r -+ Er+l. Using this let us show that d2----o , the argument for the other d~ (r_ >-3) being similar. Because of the Lefschetz decomposition (2. i), which in the present situation reads as: = LW2(c) uPS-, <el | 134 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III i35 it will suffice to show that d~: H~(S, P~.(C)) ~ HP+2(S, is zero for q~ m = dimcV ,. Writing q ~ m--t and using that L t + 1 : f, Rm-'-l(c) -, R U is an isomorphism [3o], we find a commutative diagram H~( S, Pr: '(t2)) d,> Hp+~(S, N~._, 1(12)) Ii 4, H'(S, R~.+'+~(C)) a,) Hp+2(S ' Rt~.+,+l(C)) Sincc the dotted vertical arrow is zero by the definition of the primitive cohomology sheaf, we see that our desired d~ is zero. Remark. -- Proposition (3-I) says that there is no transgression in the cohomology of algebraic fibre spaces. The result (3.2) was known classically in the following dual formulation [26] : Let yeI-~(Vs0 , C) bc a homology class on Vs, invariant under the action of the fundamental group rr ) on the homology of the fibres. (We will speak of "f as an invariant cycle.) Then there exists a cycle .s , 12) such that: .vs.=v. The cycle ,~qP(~,) is called the locus of y, and it is thought of intuitively as the locus of the cycle -(seH,(V,, 12) as s varies over S. Lefschetz's proof that s165 exists is really a homological version of the proof of (3. I) given above. We shall refer to (3. I) as the locus of an invariant cycle theorem. c) There are two variants of the locus of an invariant cycle theorem (3-2). The first is a somewhat interesting conjectural local result around an irreducible branch of S--S (cf. w167 8, t 5 in [I3] for further discussion). Conjecture (3.3) (local invariant cycle problem). -- Let: YcV r; P*c P be a localization of (i. I) around infinity as discussed in a) above, and let yeI-i"(V,., O) be a cohomology class invariant under nl(P*, So). Then there exists FeH"(Y, Q) with F[V,0 =y. Remarks. -- It is trivial that there exists reH"(Y, Q) with PlV..='r, so the conjecture has to do with the singular fibres of Y lying over P--P*. Thus far (3.3) 135 PHILLIP A. GRIFFITHS I36 has proved surprisingly difficult to handle and, in particular, it does not seem to be a topological result but will most likely require some sort of Hodge theory (w 15 in [I 3]) (1). The second variant is the following striking result of Deligne [9]: Theorem (3.4) (Deligne). -- Referring to (i. i), we have a commutative diagram (3.5) H"(X) )H"(V,.) H"(X) where the arrows are all restriction mappings of cohomology, and the image of r is equal to the image of? in (3.5). Remark. -- This result is a global version of (3.3). d) Let f: X---~S be an algebraic family of algebraic varieties and g=(E, D, Q, {Fq}) the resulting variation of Hodge structure (w 2). We shall use (3- 2) and (3-5) to deduce results about o ~ which will then later in w 7 be proved to hold for an arbitrary variation of Hodge structure which has a complete base space. It should be possible to prove the results of w 7 with no such assumptions, and this matter is taken up in Appendix C. The following are given in [9] by Deligne as consequences of (3.5) : (3.6) Let ?~H~ R~.(C)) be an invariant, locally constant cohomology class. Then the same is true of the Hodge (p, q) components of ~ (p-t-q = n). Proof. -- This is clear since we have: H"(X, C) -+ H~ RT.(C)) -+ o and X is a K~ihler manifold. (3.7) Let I~-----P"(Vs., Q),,,(s) be the invariant part of the primitive cohomology Pn(Vs. , Q) under the monodromy group F. Then there is an orthogonal direct sum decomposition (3.8) P"(V,., Q)-- I~| Proof. -- This follows from (3.6) and the properties of the Hodge inner product. From (3-7), Deligne has deduced: (3.9) The action of the monodromy group F on P"(V,0, Q) is completely reducible. Furthermore, if F is solvable, then it is a finite group. 4. Remarks on Hermltlan differential geometry. be a holomorphic vector a) Connections in Hermitian vector bundles. -- Let H-+S bundle and D~ : A~ -+ A'(H) (1) Added in proof -- This conjecture has now been proved for n = 2 by Katz, and then in the general case by Detigne, using his theory of mixed Hodge structure [9]. 136 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III t37 a C ~ connection. Then there is a decomposition D. = nk -[- D~ of D s into types (I, o) and (o, I), and D is said to be compatible with the complex structure D~ =0. Suppose that H has an Hermitian metric if " - (,) : H| (e,e')=(e',e). We do not require that ( , ) be positive definite, but it should of course be non-singular. Lemma (4. 9 -- There is a unique connection D r such that (i) the Hermitian metric (,) is flat, and (ii) D H is compatible with the complex structure. Proof. -- Let e~, ..., e, be a local holomorphic frame for H and let h~o = (e~, eo) }. denote the Hermitian metric. Then the required connection DH(ep)= ~ 0~eo is given a=l by O=h-lOh where 0=(0F) , h=(h~o) are the connection and metric matrices respectively. Now we consider a holomorphic vector bundle I-I-+S having a connection D r which is compatible with the complex structure. Let KcI-I be a holomorphic sub- bundle with quotient bundle L, so that we have an exact sequence (4.2) o~K-+H-+L-+o. The connection D~ induces, in the obvious way, a mapping (4.3) b : A~ -~ At(L) which is linear over the C * functions and is called the second fundamental form of K in H. Lemma {4-3)- -- The second fundamental form of K in H is of type (i, o), so that bzA~'~ L)). Proof. -- Given eeK,,, choose a C ~~ section f of K with f(so)=e. Then by definition b(e) is the projection on L of (Dsf) (So). Since we may choose f to be holo- morphic and since D~ =0, we have DHf=o so that b(e) is of type (i, o) as desired. Suppose now that H-+S is a holomorphic Hermitian vector bundle with holomorphic sub-bundle K as in (4.2)- Assume that the Hermitian metric (,) is non-singular when restricted to K. Then there is induced a C ~ splitting of (4.2) by considering L as being {eeH : (e, K)= o}, and so: (4.4) DK=DH--b induces a connection in H. Lemma (4-5)- -- DK=:Dn--b in (4.4) is the metric connection in K. Proof. -- By lemma (4.3), D~=0. For e, e'eA~ d(e, e')= (,Dne , e')+ (e, D.e')= (Dxe , e')+ (e, DKe' ) since L-: (K) as a C ~ sub-bundle of H. Q.E.D. 18 x38 PHILLIP A. GRIFFITtIS Similarly the connection D~, the holomorphic projection H-~ L--~ o, and the C a injection o-+ L ~ H induce a connection D L in L by (4.6) DI,(f) = :to Dnoi(f). Lemma (4.7).- The connection D~. in (4.6) is the metric connection in L. The proof is analogous to the proof of (4-5). b) Curvature in Hermitian vector bundles. -- Given a connection D x : A~ -+ AI(H), the curvature | is defined by (4.8) | e : (Dn) 2. e (e ~A~ In case D x is the metric connection for an Hermitian metric, | is of type (i, I) and satisfies the symmetry (One , e')+(e, | (e, e'~H). For us, the main use of the curvature is as it appears in the following: Lemma (4- 9)- - Let 9 and q~' be two local holomorphic sections of H-§ and d/--= (% 9') the inner product. Then (4. xo) 0~+ =(D~, D~?')--(| ~?'). proof. -- ~') (by Lemma (4.~) and since ?' ' '-1-( ' ' ' is holomorphic)=--(D~D~%?), ,D,%Dn.r ) (by Lemma (4.1) again). Now D~D~o:(D~Dn-{-D~D~)~0 (since qo is holomorphic)=@..q~ (by (4.8)). Q.E.D. If q~ is a holomorphic section of H-+S, then we may write locally: t (D'~(p, D'~)= ~g,,-jds' ^ dg j (4. IX ) -- (@a% q9) = .~.hi,?ds~^ d3 ~ where g--=- (gi, j) and h-- (h~, j) are Hermitian matric('s and s 1, . .., s d are local holomor- phic coordinates on S. From (4. IO) and the maximum principle for plurisubharmonicfunctions [I9] , we have Lemma (4. i2). -- Let qD be a holomorphic section of H---~S such that (i) the length +=(% q~) is bounded on S, and (ii) the Hermitian matrices g and h in (4. I I) are everywhere positive semi-definite. Then + is constant and we have (Dn% Dnq0)=o=(| q0). Now let H-+S be a holomorphic vector bundle with an Hcrmitian metric and metric connection D H. Suppose that KcH is a holomorphic sub-brindle such that the restriction of the metric on H to K is non-singular. Then wc are in the situation of Lcmmas (4.5) and (4.7)- Lemma (4.13). -- The curvature of the metric connection in K is given by (0 e, e') -(%e, -(be, be') (e, 138 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III t39 Proof. -- Choose holomorphic sections f, f' of K such that f(So)=e, f'(So)=e'. Then by (4. I O) applied to H we have: (| f')=(D~f ' ' DHf)_OO(f,f ' , - , ) =((D~ + b)f, (D'K+b)f')--O-O(f,f') = (D~f, D~f') +(bf, bf')--O-O(f,f') = (| f, f') + (bf, bf'), where we have used Lemma (4.5) in the second step, the equation: (Dif, bf') = o = ( bf , D~f) in the third step, and (4-Io) applied to the bundle K in the last step. Q.E.D. To give the curvature in L, we use thc conjugate linear isomorphisms: LzL induced by the Hermitian metrics to define tea ~ K)) as the image of the second fundamental form b under the isomorphism At, ~174176174 Lemma (4. x4)- -- The metric curvature in L is given by : (| f, f') == (| f, f') + (cf, cf') (f, f'eL ~ (K) 5. Statement of m~.in d;ft'eren_ti~!-geometric properties of the Hodge bundles. The results stated in this section will be proved in w 6 below. Let 8-=(E, D, Q, {1~}) be a variation of Hodge structure as defined in w 2. Using the Hodge bilinear form Q, we have an Hermitian metric ( , ) in E given by (5. x ) (e, e') = (-i)'Q(e, F) (e, deE). This Hermitian metric induces non-singular Hermitian metrics in the holomorphic sub-bundles F e of E, and (-- I)q-l(, ) subsequently induces a positive-definite tiermitian metric in the Hodge bundle Eq=Fq/F q-t. Referring to (2.I2), we have linear bundle maps eq:Eq-+Eq+~(~)~ " and teq_l :Eq~Eq-t| induced by the flat holomorphic connection D. Theorem (5.2) (Curvature of Hodge bundles). -- The curvature of the metric connection D~q is given by : (| e')= (Gqe, Gqe')--('Gq_te , 'aq_le') (e, deE') where we agree that a_ I = o =: %. 139 PHILLIP A. GRIFFITHS 14o Remark. -- If we choose local frames for all of the Hodge bundles E q, then Theorem (5-2) gives for the curvature matrix that (5" 3) OEq = Aq^ tAq--Bq ^ t]~q, where Aq, Bq are matrices of (I, o) forms with B0:o , A~:o. Remark. -- From (2. Io) we have isomorphisms (5.4) Eq~-E"-q. From (5.I) it follows that the isomorphisms in (5.4) are all isometries. Using the isomorphism ~..q| Eq + 1| ~= ~,,- q- l Q Eq| ~ ' we see that ~q corresponds to %-q-1, and so = - (%_1, 1) = - which is the correct relation between the curvature of an Hermitian vector bundle and the curvature of its dual. Our second main application of the structure equations of variation of Hodge structure is Proposition (5-5). -- Let ~ be a holomorphic section of F q over an open set U c S and assume that the projection of D~ in E /F q is zero. Then ap induces a section ? of E/F q-1 ~ F"-q+ 1, and the differential forms l "-q " ?) (--I) (D,.-q+~?, (5 "6) (_i),-q+1(| ?) are positive, in the sense that the Hermitian matrices defined as in (4. x i) are positive (semi- definite). Corollary (5.7)- -- Let 9 and ? be as in Proposition (5.5) above and assume that (i) U /s all of S, and (ii) the length of the section ~ of EqcE/F q-x is bounded. Then D~.~q-~? -= o -~ DEq 0. This Proposition and Corollary will bc proved togcther in w 6 below. Proposition (5" 8). --- Let r be a holomorphic section of E q cE/F q-1 over an open set U r S, and assume that DE/Fq-I ? = o. Then there exists a unique section tF of F q satisfying (i) o; (ii) ~F projects onto ?; and (iii) the inner product (~F, Fq-1)----o. Combining (5-7) and (5.8) we find: Theorem (5.9) (The0rem on global sections of Hodge bundles). -- Let ap be a holomorphic section of Fq~S such that a) the projection of D(b in I;./Fq is zero, and b) the length of the induced section ~ of E q -~Fq/F q-1 is bounded. 140 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III x4x Then there exists a section ~F of F q satisfying (i) DW---- o; (ii) W--(I) is a section of Fq-l; and (iii) the inner product (~F, F q - 1) = o. 6. Structure equations for variation of Hodge structure. We want to prove (5.2) and (5.5)-(5.8). Our method of proof is to use the calculus of frames [6], [8], where by definition a frame is a C ~ basis, over an open set, of the vector bundle in question. Given a variation of Hodge structure 8=(E, D, Q, {Fq}), we shall consider unitary frames adapted to the Hodge filtration (2.4). By definition this is a frame (6. x) el, ..., eat ; ehl+t , ..., eh~ ; ... ; ehn_t+l , ..., eh, where hq =dim F~ q and where the following conditions are satisfied: (i) referring to the Hermitian inner product (5.x) we have (6.2) (ei, ej) =(-- I)q-l~ (hq_l<i,j<=hq) and all other inner products are zero; (ii) the vectors el, . . . , ehq give a basis for F~ q for all points seS where thc frame is defined; and (iii) under the conjugate linear isomorphism Eq~_En-q given by (5.4) and the metric (5. i), we have (6.3) -inq_l-~j=eh,_q_t+j (I~j~hq--hq-a). Remarks. -- We first observe that (6.2) and (6.3) are compatible: (--I)q-l~j=(eaq_x+i, enq_l+~)=(--i)nQ(ehq_1+i, ehq_t+ j) =(--x)"(--i)"Q(eh._q_l+ p en._q_l+,) = (-- I)" (-- I)"- q-l~/. Secondly, I should like to comment that the alternation of signs in (6. ~) is of extreme importance -- it is this plus the infinitesimal bilinear relation (2.8) which makes everything go through. The flat holomorphic connection D, which by Lemma (4. x ) is the metric connection for the metric (5. i) in E, is given by hn (6.4) De,=j~=xO~ej, 141 I49 PHILLIP A. GRIFFITHS where the differential i-forms O~ satisfy the integrability condition hn i i k (6.s) doj +kY,.(o,,^ = o~) =o. From (6.2) and the flatness of the metric we find Ohp ,-,-i . q-h ,+~ (1<i<h~,--h~ ~" I<j<hq--hq_t). (6.6) h -x+j -~-(--I) ,+ 0 ~ - l = O _ hp_l+l -- __ _ ~ __ __ From (6.3) we have ~hp-l+i t ~n~hn p 1 +/ (6-7) hq_l+j=(-I) Uhn'-q-l+ j. As remarked below (6.3) , it is unfortunatcly the case that the signs are quite important and so must be kept track of carefully. The infinitesimal period relation (2.8) gives , AhP -l+k (6 8) ~^q_~+j =o for p>=q+i, i<_j<hq--hq_~, k>=o. At this point we have used all of the information in I)-VII) of w 2. From Lemmas (4.3) and (4.5) we have Lemma (6.9). -- The second fundamental form of F q in E is given by (6.,0) bq= X O~| 0('=o; I~i~hq hq-t" l ~j ~ hn and the metric connection in F ~ is given by hq (6. xx ) D~ei = Z Ojep i_~.._i~hq. j=t Using the Cartan structure equation [6], the curvature of the metric connection in F q is given by hq (i~i,j<=hq). From (6.5) and (6.8) we obtain hq. 1 --hq ,,,~ ,,hq + k, ( I < i, j < hq). (6.x2) (| ~ tVh,+~^V~ ) -- k:=l -- ---~ We shall now prove Proposition (5-5) by proving the following three lemmas. L~mma (6. "3). -- Let ~ be a vector in ~ such that ~ projects to zero in ~--,~-'--,o. Then the differential form (-x),(o~. ~, ~) is positive in the sense of Proposition (5.5). 142 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III ~43 hq --hq_ 1 -- ~ e%_l+z, we have from (6.I2) that Proof. Writing ~= ~ ~ 1=1 t~hq l+j ~hq+k t'r v ((~q~, ~)= , Z l(Jhq-{_k A I~hq l_{..~.~ehq 1.t.i, ehq_l_ ~ k)~J~ ' j,k, - - ~ q t ~.-~ ~hq-l- k :(--i) / z~ o h +~^ Ahq-~+j,~j72~ i,j,k q-1 Vhq -j-k Y '~ } ((fOhq l_~.,qO)A (Z0h~ l+j~)) ) (by (6.6)) - j - :(--I)q~k (+kA~)k) where +k=Z,%+k vhq_~+, -~ ? is of type (I, o) (by (6.Io)). This proves the Lcmma. Lemma (6. I4). -- Let ~ be as in Lemma (6. I3) and assume that D~q~ projects to zero in Fq-+Fq-t-+o. Then the differential form (--~)q-~(D~q~, D~) is positive. hq--hq-1 h --iv ~hq_l+i Proof. -- By assumption D-~q~ = w + q-~- e%_~ + ~ where is of type (i, o). i=1 Then (--~)q -~(D~q~, D~q~) = Y~ (+ ha-l+'^ ~hq_x +,) as required. Lemma (6.15). -- Let tF be a holomorphic section of F such that DW projects to zero in E/F. Denote by + the section of ElF -~ induced by qP, and let ~ be the holomorphic section of ~'~-~+~ which corresponds to + under the isomorphism E/Fv--~-~F "-v+~. Then D~,_~§ projects to zero in F'-P+I-~F"-V-+o. Proof. -- What we must prove is that D~.tF~_~ + projects to zero in E/FP-I~E/FV-+o. hp Now ~F= x~ +Se~ and by assumption we have J=l hp I j tQlP 4- k (6.I6) Z V vj =o (I<k<hp,~t--hp). j=l "-- ":: - hp" - ~lp_ 1 The section + ofE/F p-a is E ~) hp-l+J and the Lemma follows from (6 8), (4 7) j=l ehP -~ ~-j " " and (6. I6). It is clear that Proposition (5.5) follows from Lemmas (6. I3), (6. I4) and (6. I5). We now prove Corollary (5-7)- We use the notation of Proposition (5.5) and Corollary (5.7). From Lemma (4- I ~) it follows that the length (% q~) of q) is constant and (D~./~q_a% Dv.~q-~q))=o. By Lemma (6. I5) we have that the projection of Dv4rq-~? on E/F q is zero, from which it follows that D~/vq-~q)=o. It now follows from the exact sequence: 0 -+Eq---~E/F q - ~ -->E/Fq--> o and Lemma (4.5) that D~?=o. This proves the Corollary. 143 PHILLIP A. GRIFFITHS r44 We now prove Proposition (5-8)- We may assume that r is a unit vector, i.e. that the length (% r ~-1, and may then take: q) ~ ~hq_l+ 1 h~ in our frames (6. I). From (6.4) and Lemma (4.7), we havc Z 0t. l+, j=o j ~hq_l+ 1 which gives : 0 5 --o for j>hq_l~-I. (6. x7) hq_l+ 1 -- Differentiating (6. I7) and using (6.5) gives: hn ~Ah~ 1+1 x-~ ,~h _i+1 t,j o:-- "- : z~ (u~ q ^t~nq_l+l ) ~hq -1+1 j=l = (O q-t+ t^ 0 (by (6. i7) ) J<%-1 ^~ ~+t) hq_t --hq_2 ,~hq l+l ~hq_ t + 1. ~ : Y" (%,(~+j^~hq ~+i: (by (6.8) and (6.6)). j=l Ahq-1 +1 Since ~hq_s+J is of type (I, o), we must have: ~- , hq-l--hq-2. %-2+J=~ for j=I, ... This gives De%_1+, ---= o, from which Proposition (5.8) follows. We now prove Theorem (5-9). Taking the frame: ehq_l+l, 9 .., ehq in Eq, we see from Lemmas (4-5) and (4.7) that: hq--hq-1 h -t-i Z 0~ q-1 , .e i DEqehq-l+J~ i=1 "q-lt3 " From the Cartan structure equation: ~hq-l-b J ~ ~k k~hq-l-t- k A IJhq-l-~ j) and from (6.5), (6.6), (6.8) it follows that: (6.x8) f0 r247 %-~%~mb-:~ zb ~+J, b+~-b-~b+" ~h.+,, , k Eq]hq_l"4"j I,~hq_2-1-1AUhq-~+I)-~-m~'~l (Vhq 1 + ~-- lffil J A ~h;-l+ i)" Now Theorem (5.2) follows from (6. I8) and the equation hq+l--hq h +m ~q(e%_,+j)----- m~==l On~_i+~e%+,,, which says that aq given by (2.12) is just the second fundamental form orE q in E/F ~-1. Finally, we shall prove a Lemma for use in the proof of Theorem (7-I9) below. 144 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, IlI I45 Lemma (6. x9). -- Assume that S is complete and let ~ be a global holomorphic section 0fE/F q-1 such that (i) q) projects to zero in E/Fq-I~ E/F q, and (ii) the projection of Dq~ to ElF q is zero (this makes sense since D-d)s(Fq-~ ) cf2~(Fq)). Then there exists a constant section @ of E such that a) 9 projects to *O in E--*-E/F q-a and b) the inner product (r F q-t) =o. Proof. -- Referring to Theorem (5-2), we have: (62o) (o~, ~)=-('~_~. ~, '~, _~.~) when we consider ~ as a holomorphic section of E q (by (i)) and when we use crq. q) = o (by (ii)). From (6.2o) and Lemma (4. I~) we have DEq-~o. It follows that DE~rq-~ ~-~o and then our result follows from Proposition (5.8). 7" Applications. a) Invariant cycle and rigidity theorems. Theorem (7-i) ([nvariant cycle theorem). -- Let ~= (E, D, Q, {Fq}) be a variation of IIodge structure and assume that (i) S is complete, or (ii) the Picard-Lefschetz transformations around the irreducible branches of S--S are trivial. Let a) be a flat section of E~S. Then the Hodge (p, q) components of 9 are flat sections of E. Proof. -- We shall prove in w i I below that, with the assumption (ii) above, the variation of Hodge structure d ~ and section 9 of E both extend to S. Thus we may assume that S is complete. Referring to the theorem (5.9) on global sections of Hodge bundles, we may find a section ud, orE satisfying Dq~',=o, the projection of ~--~F, in E/F" -a is zero, and the inner product (~,, F"-l)=o. Writing q)= @,_l +~,, we may apply the same reasoning to find a flat section tF,_ 1 of F "-1 such that (q~,-1, F"-2) =~ and --=- q),-2 -r + q", where (I),_ 2 is a flat section of F"-2. Continuing in this way we find our theorem. Remarks.- (i) In [I2] we gave the above proof of Theorem (7-i) but formulated the result in a clumsy way. The above formulation was given by Deligne [9], who, as remarked in w 3, has proved Theorem (7-2) (Deligne). -- With no a.~sumptions on S but with the assumption that g arises from a geometric situation, the same conclusion as in Theorem (7. I) is valid. In fact Theorem (7-2) follows immediately from Deligne's result (3.5)- (ii) Of course, we would conjecture that (7. I)is true with no assumptions on S. 19 PHILLIP A. GRIFFITHS z46 Corollary (7.3). -- With the notations and assumptions of Theorem (7. I), we suppose further that n = 2m is even and that r is a flat section of E~S which is of type (m, m) at one point (i.e. the Hodge components q)r'q=o for (p, q)Oe(m,m)). Then d) is everywhere of type (m, m). Remark. -- In [18] Grothendieck, as a (non-trivial) consequence of the Tate conjectures, was led to suggest that, if f: X-+S is an algebraic family of algebraic varieties and ~ a section in H~ P~Q) which is an algebraic cycle at one point sots , then ~(s)~H~m(vs, O~) is everywhere an algebraic cycle. It was this problem which initially started me looking into sections of Hodge bundles. Corollary (7.4). -- Let g and if' be two variations of Hodge structure which satisfy the assumptions of Theorem (7. i). Suppose there is a linear isomorphism ~:E~-->E~, (sots) which is equivariant with respect to the action of ~1(S) on E,, and E~,, and which commutes with the tIodge decompositions of E,, and E',,. Then there is a global isomorphism g ~= 8' of the variations of Hodge structure which induces ~ at SotS. Proof. -- Because of rct(S)-equivariance, we may consider e as a global flat section of F.| Also ~ is of type (n, n) at s o since it commutes with ttodge decompositions. The result now follows from (7.3). Remark. -- This corollary, which should be thought of as a rigidity theorem, was prow~d for n=i by Grothendieck [17] in the geometric case and by Borel- R. Narasimhan [5] in the general case. Because of Deligne's theorem (7. ~), the corollary is true in general when go and d ~' both arise from geometric situations. We can formulate an analogous result about homomorphisms (and not just isomorphisms) between variations of t-Iodge structure. As we see no applications for such, we shall not discuss the matter further. b) Negative bundles and variation of Hodge structure. -- Let H-~S be a holomorphic vector bundle. We say that H is negative (semi-definite) if there exists a (positive-definite) Hermitian metric ( , ) in H whose metric curvature | (cf. Lemma (4. I)) has the property that the differential forms: (7.5) (O.e, e) = Z.h,, j#^ (h,j = hs, are negative in the scnsc that the Hcrmitian matrix (h~, ~) is negativc. Obscrve that H is negative if the matrix of the metric curvature has a local expression: (7.6) O. =--AA'X where A is a matrix of (i, o) forms. From theorem (5.2) we have Proposition (7.7). -- Let g=(E, D, Q, {Fq}) be a variation of Hodge structure. Then the llodge bundle E" is negative. We say that a locally free coherent analytic sheaf is negative if this is true of the corresponding holomorphic vector bundle. 146 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III I47 Corollary (7.8}. -- Let f: X~S be an algebraic family of algebraic varieties. Then the direct image sheaf t~.(d)x) is negative. More generally, if q is the least integer with H"-q(V,, f~s)+o, then the direct image sheaf R~--q(f2~:/s ) is negative. Recall that a cohomology class c0~Hz~(X, R) is negative if we have (7.9) f,.,~<= o for all compact k-dimensional algebraic subvarieties Z of S. Corollary (7. xo). -- The Chern monomials fi ---- ql" 9 9 c~t of the Hodge bundle E" ~ S are negative. In particular the 1 "t Chern class q(RT.(dPx) ) is negative (semi-definite), and we furthermore have <o (7.") f c~ (R~.(0x)) in case S is a complete curve, n = I or 2, and g # not trivial. Proof. -- It is well known that the Ghern classes of a holomorphic vector bundle H~S can be computed from the curvature | of a metric connexion. In particular, if (7-6) holds, then it follows that locally -( !]'" {7. ,,) c~_\~,~ 1 ( ~ (~,^,i,,)) where [ I I ---= i~ 3... + i~ is the degree of c x and ~', are (I I I, o) forms. The first two statements of (7. zo) follow from (7. I2), and (7-i I) follows the fact (cf. Theorem (5.2)) that, for n=I or 2: c~(r;.(ox))=o =- ~._~=o. TO give our final application of Theorem (5-~), we define the canonical bundle K(@) of the variation of Hodge structure 8=(E, D, Q,{I~}) by: (7. 9 3 ) K(8) = (det E~174 (act El) "- x| | (det E"- 1). The first Chern class of the line bundle K(8) is given by the differential form z---(~(8)) where 2~ (7. x4) co(8) =n(Trace | +(n-- t)(Trace | +Trace | Proposition (7-I5). -- For a tangent vector ~ to S, we have ( ~, ~^ ~) ~o with equality if, and only if, %(~) =- o for q = o, ..., n -- T. Proof. -- This follows by direct computation from the formula ~1-.1 <~, ~A~>= E I~(~)I ~ q=O which results from (7-I4) and Theorem (5.2). c) A Mordell-Weil theorem for intermediate Jacobians. -- Let 8=(E, D, Q, {1~}) be a variation of Hodge structure where we assume that n,=2m+I is odd. Then (7. I6) E = F~)~ -~ , 1~ n ~7.'__-- o. 147 x48 PHILLIP A. GRIFFITHS We let E s ={eeE : e=e} be the set of real points in E and assume given a flat lattice AcE s. Equivalently, we are given a ~l(S)-invariant lattice A,, in (E,),. Letting E+-----E/F ~ and J =E+/A, we obtain an analytic fibre space ~ :j-+s, ~ l(s)=J, of complex tori J,-=I~,\E~/A s which we shall call the family of intermediate Jacobians associated to ~ and A. We note that the tori J, are abelian varieties if m = o, but not (in general) otherwise. Let J-+S be a family of intermediate Jacobians as above and J the sheaf of holomorphic sections of this fibre space of complex tori. There is an obvious exact sequence (7. x7) o-+ cg(a) -+r -+J-+o where Cg(A) is the (locally-constant) sheaf of sections of the lattice A over S. From (7- I7) and the relations DV(A)=o D. ~)s(F") c ~(F"+ 1), we obtain a sheaf mapping (7-x8) Dj :J ---> a~(E/]b "~+ 1). The algebro-geometric significance of (7. I8) will be discussed in Appendix A below (of. Theorem (A.8)). We denote by ~om(S, J) the sub-sheaf of sections ,~c] which satisfy Dj,~=-o, and shall refer to sections in Yt~ J) as being integrable. In the abelian variety case (m=o), all holomorphic sections are integrable. Suppose now that S is complete. Referring to Theorem (7. I) we see that: H~ ~'(E))= H~ ~(F")) @H~ cg(F')) and it follows that: H~ ~(I~))\H~ ~'(E))/H~ (g(A))=J(~') is a complex torus which we call the trace or fixed part of J-+S (cf. Proposition (A.7) and the succeeding remark for an algebro-geometric interpretation of this fixed part). Theorem (7. I9) (Mordell-Weil for families of intermediate ]acobians). -- Let J-->S be a family of intermediate f acobians associated to a variation of Hodge structure d ~ and lattice A over a complete base space S. Then the group Horn(S, J)=H~ o~om(S, j)) of global, integrable cross-sections of J-~S is an extension of the fixed part J(g) by a finitely generated abelian group. Remark. -- The integrability condition Djv =o will be satisfied for any cross- section v of J-+S " which arises from algebraic cycles in case J-+S comes from a geometric situation " where we refer to Appendix A for an explanation of the phrase in quotation marks (of. Theorem (A.8)). Proof. -- From the exact cohomology sequence of (7. I7) we have: o ---~.~f" -+ Hom(S, J) H'(S, (r 148 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, llI where ~ is the vector space ofholomorphic sections q~ of E+ which satisfy D~e~(l~+l). Since Hi(S, Cg(A)) is finitely generated, our theorem follows from Lemma (6. I9) in the same way that Theorem (7-I) followed from (5.9). Remark. -- The extension of this theorem to arbitrary base S is discussed in Appendix C (eft (C.3)). In particular Theorem (C. I2) in this appendix gives such an extension to arbitrary base in case n-~I, which is just the usual Mordell-Weil theorem (over function fields). 149 PART II DIFFERENTIAL-GEOMETRIC PROPERTIES OF THE PERIOD MAPPING 8. Classifying spaces for Hodge structures. Let E be a complex vector space and o<h0< hi<=... _<=h,_l<h . = dim E an increasing sequence of integers which is self-dual in the sense that h,_q_l=h,--h q for o<q<n. We also assume given a non-singular bilinear form Q: E| C, Q(e, e') = (- x)"Q(e', e), and consider the set [) of all filtrations F~162 oF"-1 oF"= E, dim Fq = hq, which satisfy the first Riemann bilinear relation t (Fq) or equivalently t Q.(Fq, F"-'-l)=o. We will say that such filtrations are isotropic or self-dual. Proposition (8.2). -- 1~ is, in a natural way, a projective and smooth complete algebraic variety which is a homogeneous space =GrB of a complex simple Lie group G divided by a parabolic subgroup B. Proof. -- Let G(h, E) be the Grassman variety of h-planes through the origin in E. Observe that the filtration: determines F~ cF" by using the first bilinear relation (8. I). From this we have an obvious projective embedding: (8.3) 5 -~ G(h0, E) � X G(h~, E) which cxhibits f) as a complete and projective algebraic variety. 15o PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III x5z We shall prove that l) is smooth by exhibiting the tangent space TF(f) ) to I) at a given point F-=(F ~ ..., Fn). First we recall the natural identification Ts(G(h , E))~ Som(S, E/S) (SeG(h, E)). The tangent space to FeG(h0, E)�215 E) is (8.4) @ Hom(Fq, E/Fq), m , = 0 -- [-Y-I ' from which we see that TF(I) ) is given by all f=q__@0f q (fqEHom(F q, E/F')) in (8.4) which satisfy the conditions that the diagrams F q -/q-~ E/F (8.5) ~ i F q+l > E/F q+l (q=o, I, ...,m--I) [q+l are commutative, and that we have (8.6) Q(fm(e), e')-~Q(e, fm(e'))=o for e, e'sF m. Now let GcGL(E) be the complex orthogonal group of the bilinear form Q; thus 13 is the complex simple Lie group of all linear transformations T : E---~E which satisfy: Q(Te, Te')=Q(e, e') (e, e'eE). Each TsG induces an automorphism T : f)--~f) by T. (F ~ c... c F n) = (TF ~ c... c (TFn). This action of t3 on f) is transitive and the isotropy group B of a given point F0ef) is a parabolic subgroup of 13. This gives the desired representation I)= 13/B. Remark (8.7). -- We define an important holomorphic sub-bundle IF(I") ) of the complex tangent bundle I(I)) as follows : IF(D ) consists of all f-=q@__0f q in (8.4) which satisfy the infinitesimal bilinear relation (cf. (2.1 I)) (8.8) Q(f:,e')=o (e~Fq, e'~Fn-q-2). We now assume given a conjugation e ~ 7 of E such that Q(~, 7')=O(e, e'). In other words we are given that E----ER| where Qis real on the real subspace E R of vectors eeE which satisfy e =7. Define the Hermitian inner product (,) in E by (8.9) (e, e')=(-i)~Q(e,-~ ') (e, e'~E). 151 152 PHILLIP A. GRIFFITHS We define the period matrix domain D c ID to consist of all isotropic filtrations F ~ c... c F ~ which satisfy the second Riemann bilinear relation: ( ) : Fq| is non-singular and (8. IO) i (- I)q( ' ) : Eq| is 1positive definite ~where Eq={esFq: (e,F q- )=o}. We may combine (8. x) and (8. IO) by saying that D is the set of filtrations F~ cF", with dim F q = hq, which satisfy: l' Q(Fq, F"-q-1) = ~ (8. xx) I (-- i)"Q(Fq' ffq) is non-singular (-- I) q (-- i)nQ(E q, gq) ~>0. Proposition (8. I2). -- D /s an open complex submanifold of 19 which is a homogeneous complex manifold D = G/H of a real, simple, non-compact Lie group G divided by a compact subgroup H. Proof. -- Let GcG be the real form of all real linear transformations T: E~E which preserve Q. Then, under the natural action of 13 on I), G leaves invariant, and acts transitively on, D. It is clear that the isotropy group H = GcaB of a point F0eD is a compact subgroup of G. Definition. -- As mentioned D is called a period matrix domain and 1 ~) will be termed the compact dual of D. It is clear that D parametrizes the universal family of Hodge structures determined by E, Q, the conjugation e~7, and the numbers hq. However, this universal family of Hodge structures over D is generally not a variation of Hodge structure in the sense ofw 2 because of the infinitesimal bilinear relation (2.8). We refer to [16] for a discussion of the group-theoretic properties of D and D, especially as regards the equivariant embedding: D ,I3 ,! G/H > G/B In the following examples we let G and H be as above, K will denote the maximal compact subgroup of G and R-----G/K the corresponding Riemannian symmetric space, and h q = hq-- hq_ t the tIodge numbers. Example (8. x3) ,. -- When n = 2m is even, G=SO(a, b; R) (a=h~ ~', b=ht+h3'--...=-h 2=-t) 152 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III I53 a b is the orthogonal group of the quadratic form i--~l(xi)2--j--~l (y~)z' the compact isotropy group is H = U(h ~ � � U(h re-l) � SO(h"*), and the maximal compact subgroup of G is K :-SO(a; R)� R). We may identify the ILiemannian symmetric space R.= G/K with the set of real a-planes ScE~ such that Q(S, S)>o. The equivariant fibering g:D > R G/H > G/K is given by g(F~162 cF 2") = E~174 E~| | 2"*. (8. x4) . Example. -- When n=2m-l-x is odd, G=Sp(2a; R) (a=h~ +h m) is the group leaving tile skew-form ~ (XjAXa+j) invariant, the compact isotropy group is j=l H =U(h~215 � U(h"), and the maximal compact subgroup of G is K =U(a). We may identify R with the set of complex a-planes ScE which satisfy i o(s, _s)= o !iQ(s, S)>o, and the equivariant fibering g : D-+R is given by (8. x4)o ~(F~162 9 9 9 cF 2"+1) = E~174174 9 9 9 | ~". Now according to (8.4), (8.5), (8.6), we may identify the tangent bundle to D as . m, ( (8.xS) TF(D )=@ ~Homq(E q,E "+q) FeD, m= . q=op~l The identification (8.I5) is G-invariant, and the positive definite metrics (--I)q(,) on E q induce a G-invariant Hermitian metric ds~ on D. Group theoretically, ds~ is the metric induced by the Caftan-Killing form on the Lie algebra of G Ix6]. 20 PHILLIP A, GRIFFITHS '54 Proposition (8. x6). -- In the equivariant.fibering : D ) R !! !! !t G/H > G/K of the period matrix domain D over the Riemannian symmetric space R, the fibre ZFo through each point F0eD is a compact complex submanifold, and we have (8. I7) IF.(D ) c (TF.(ZF.)) where Ir~ cTF0(D ) is given by (8.8). Proof. -- We treat first the case when n = 2m is even. The point ~ (F0) is the same as giving an orthogonal direct sum decomposition E s = S~S~ if E R such that Q is positive on S R and negative on SIR. We shall discuss the case when m = 2l is even -- the other case is similar. The fibre g-l(N(F0) ) is the homogeneous space ( SO(a;R) ) { SO(b;R) ~ ZF~ U(h0)x.. .xv(h~,_~)� ) x \V(h~)x: :_~U(h~,_~) ] and has the following geometric description: ZF. consists of all pairs of filtrations iT~ dimoT 2p =h~ 2r iTIcT3r cT~-IcS dimGT~v+l=hl+...+h2P +1 which satisfy Q(T 2'-2, T2r-2)=o or Q(T ~-1, T~-l)=o as the case may be. These filtrations define a point F~ cF 2" in ~-l(g(F0) ) cD by letting F~ ~ Fa=--T~ F2=T~+T ~, ..., on up through F"-I=T~-2+T ~-1. Then we let F"=-(F~-I) F ~;1 =(F"--2) etc. It is clear that ZFo is a compact, complex analytic submanifold passing through F o and that ~?;--1 (8. x8), TF.(ZF. ) = @ Hom(E q, E q+~) q=0 under the identification (8. I5). From this, Proposition (8. i6) and (8. x7) are clear. In case n='2m+t is odd, the point g(F0) is given by a subspace ScE, I . 9 dimeS = a =--dlmeE, which satisfies t Q(s, S)=o I iQ(S, S)>o. 154 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III z55 The fibre ~-I(N(F0) ) is the homogeneous space u(a) z,, = U(h0 ) x... x U(hm) ' and may be described as all filtrations: T ~ 2c. .. cT 2mcS, with dimeT ~ ~ .. +h 2q. These filtrations define a point F~ 2m+t in Zr~ by letting F~ ~ FI=TO+(T~177 etc., on up to Fm. Then Fm+I=(Fm)L,...,F ~m=(F~ z. Clearly (8. X8)o TF~ ) = (~) Hom(E q, Eq:2), q::0 and Propositions (8. I6) and (8. I7) follow. Remark. -- It may bc noted that, except for the cases n----1 or n=2 and h~ the fibres of g are non-trivial, so that D is not a bounded domain in C s. Also, except for the case n =2, the inclusion (8. I7) is strict, so that there are additional conditions on a variation of Hodge structure other than transversality to the fibres of ~. 9. Statement of results on variation of Hodge structure and period mappings. a) Let g=(E, D, Q,{Fq}) be a variation of Hodge structure, with base space S, as defined in w 2. Letting E be the complex vector space E,. and taking the conjugation, bilinear form, and Hodge numbers hq==dimcFq0/Fq: 1 induced on E by g, we may define a period matrix domain D as in w 8. We now recall that the holonomy group of the flat connection D induces, by parallel displacement of a flat frame, the monodromy representation : l(S) of the fundamental group of S (based at So) in the automorphism group G of D as defined in w 8. The image P of rq(S) in G will be called the monodromy group of d ~. A continuous mapping (I) : S-~F\D will bc said to be locally liftable if, given seS, there exists a neighborhood U of s and a continuous mapping ~:U-~D such that the diagram is commutative. A locally liftable mapping (I) is holomorphic if the local liftings arc holomorphic, and a locally liftable holomorphic mapping (I) is said to satisfy the infinitesimal period relation if the local liftings ~ satisfy (9.x) ~. (v) eI~,(u ) (ueU, veT~(U)), 155 I56 PHILLIP A. GRIFFITHS where ~. is the differential of ~ and I cT(D) is defined by (8.8). We may symboIically rewrite (9.1) as: (9.2) ep.: T(S) -+I(D). Proposition (9-3). -- The giving of a variation of Hodge structure ~ with monodromy group F is equivalent to giving a locally liftable holomorphic mapping (9.,t) o: S-+F\D which satisfies the infinitesimal period relation (9. ~). Definition. -- We call ~ in (9.4) the period mapping associated to g. This terminology is explained in [II]. b) Our first result on variation of Hodge structure as interpreted by the period mapping is Theorem (9.5) (Extension of period mapping around branches of finite order). - Let 8 be a variation of Hodge structure over S and let D be an irreducible branch of S--S such that the associated Picard-Lefschetz transformation T is of finite order (el. w 3). Localize the period mapping (9.4) around a simple point ~eD to obtain ~:P*~F\D (cf. w ~). Then P" ~ A* � A a-1 is the product of a punctured disc with a polycylinder, and there exists a finite covering and a lift~ng ~ : P" ~ D of the period mapping 9 such that ~ extends holomorphically to the closed polycylinder P ~ A � A ~- a. To state our second main result, we assume that the monodromy group F is a discrete subgroup of G. This is the case if g arises from a geometric situation f: X-+S. With this assumption, the quotient space F\D is a complex space or analytic space in the sense of [I9]. In fact, the projection P\G~F\G/H is a proper mapping and so F acts properly discontinuously on D. Thus P\D is a separated topological space which is locally the quotient of a polycylinder by a finite group. Let O:S~P\D be the period mapping (9-4). By Theorem (9-5) we may extend this period mapping to a holomorphic mapping (I) : S'~F\D, where S' is the union of S with those points at infinity around which the Picard-Lefschetz transformations are of finite order. Theorem (9.6) (Analyticity of the image of the period mapping). -- The image ~(S') is a closed analytic subvariety of P\D which contains O(S) as the complement of an analytic subvariety. Furthermore, the volume ~r\D(~(S')) of O(S'), computed with respect to the invariant metric on D, is finite. 156 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III I57 Our third result is Theorem (9.7). -- Let ~ be a variation of Hodge structure whose monodromy group I' is discrete and whose base S is complete. Then the image O(S) c FXD is a complete projective algebraic variety. In fact the canon#al bundle K(d ~) (of. (7. I3)) of g is ample over O(S). Remark. -- Using Theorem (9.6), we see that (9.7) remains true if we only assume that all the P.-L. transformations around the branches of S at infinity are of finite order. This theorem follows from (7. I5) and the results of Grauert [IO]. We refer also to [x3], w IO for a discussion of this theorem together with some related open questions. As our final result we give a theorem about the monodromy group P of a variation of Hodge structure d~ D, Q, {F~}). Theorem (9.8) (theorem about the monodromy group of a variation of Hodge structure). -- Assume that either the Picard-Lefschetz transformations are all of finite order or that ~ arises from a geometric situation. Then (i) the global monodromy group P is completely reducible; (ii) P is finite if, a,ut only if, the variation of Hodge structure is trivial; and (iii) /f F is solvable, then it is finite. Proof. -- The first statement follows from Theorem (7. I) in the same way as (3.7) followed from (3.6). The third statement follows from the first by Grothendieck's argument given in Deligne [9]. Finally, the second statement follows from (9.5) and the fact that a horizontal, holomorphic mapping 9 : Z--+D from a compact, complex manifold Z to a period matrix domain D is constant [I i]. c) We will give some local statements which will imply (9.5) and (9.6). For the first we let H(D) cT(D) be the horizontal sub-bundle defined by (9.8) HF~ J- (cf. (8. I8)). The word " horizontal " follows from the fact that H(D) is the complement to the bundle along the fibres in g : D~R.. A locally liftable holomorphic mapping (I) : S-+P\D is horizontal if we have {9.9) O. : T(S)-+H(D) in the same sense as (9.2). Theorem (9.5) follows from Proposition (9.3) and Proposition (9.xo). -- Let P'=A'� d-1 be the product of a punctured disc with a polycylinder, and let 9 : P'-+D be a horizontal, holomorphic mapping. Then 9 extends to a holomorphic mapping 9 : Ad~D. We now claim that Theorem (9.6) follows from the proper mapping theorem [I9] together with the following 157 PHII, I, IP A. GRIFFITHS I 5 8 Proposition (9. II ). -- Let A(O) be the disc o<=lz[<?, A'(p) the corresponding punctured disc, and P(k,l; p)~(A*(p))~� ~ the product. We shall refer to P(k,l; p) as a punctured polycylinder. Let * : P(k, l; p) ~ F\D be a locally liftable holomorphic mapping. (i) Let "h,..., Vk be the canonical generators of ~t(P(k, l; p)) and Tj=*.(yj)eP the corresponding Picard-Lefschetz transformations. Assume that T~,..., Tj are of infinite order and Tj+D..., T k are of finite order. Let {z,}-----{(z~,..., ~+~)}eP(k, l; ?) be a sequence of points with inf [z][--,o as n~oo. Then the sequence {(I)(z,)}eP\D does not converge. IA~K~ (ii) The volume ~i~v(q)(P(k, l; p/2))) isfinite. Proof of (9.6) from (9. I i). -- We claim that ~ : S'-+P\D is a proper mapping. If not, there is a divergent sequence {s,}~S' such that qS(s,) converges in F\D. We may assume that {s,,} converges to some point ~-eg--S'. By localizing around ~ and using (i) in Proposition (9. I I), we arrive at a contradiction. The proof that the volume ~rXD(q)(S')) is finite follows from (ii) in (9. I I) by localizing around g--S' and an obvious compactness argument. xo. The generalized Schwarz lemma. Let PcC e be the polycylinder {(zl, ...,za) :o<~[z~[<I} of unit radius, and denote by ds~ the standard Poincarg metric given by: Denote by w v the associated g-form ~ t,=t(i---~.~-)2 J so that (tot) a is the non- Euclidean volume of P. Let D be a period matrix domain with (suitably normalized) invariant metric ds~ and associated 2-form m~. We want to prove Theorem (xo., ) (generalized Schwarz lemma). -- Let ~:P--->D be a horizontal, holomorphic mapping. Then we have (,o.2) I r <ds Proof. -- We first show Lemma (xo.3). -- If the volume estimate ~*(mD)a~(mp) a holds for d=x, then we have the distance estimate (I)*(dsg)<_~ds e. Proof. -- Let +:A~P be the embedding of the unit z-disc into P given by dg(Z)=(~lZ,., o~aZ), with Y, 10q[~=I. "' i~l ( I ar so Then (x_f- lzl y I z , that at the origin z=o we find (zo. 4) ~" I -,,asv,03, _-- dzd-z = ',ds~)0- 168 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, Ill I59 If -: is a tangent vector to P at (o, ..., o), then we can find ~j as above such that kb. ~-~ =-r for a suitable X:~o. Then the length IX --I1-11 by the isometry l] IIA property (lO.4). Using the volume decreasing assumption applied to the holomorphic curve Oo+:A-+D, wehave r :t =ll,ll , or Ile.,[l <ll,ll, 1 0z , liD II ~"" II A which is what we want to prove. We will base our proof of the volume estimate on a formula of Chern [7]. To explain his formula we let 1Vi and N be d-dimensional complex Hermitian manifolds and f: M-+N a holomorphic mapping. Using unitary frames as in [7], we write Then f*(Oj)= ~,a~r and we havc {IO. 6) f. (ON)d = [ det (a~)12 (o~) d The where r M and 0 s are the respective 2-forms associated to the metrics (io.5). non-negative function u--ldet(a~)l 2 is the ratio of the volume elements, and we are looking for a formula for the Laplacian A log u near a point m0elV[ where u(mo)>o. (Recall that the Laplacian Af of a function f is defined by i~f=(Af). ~.) The desired formula involves the Ricciform Ric N of N and scalar curvature R~t of M. To explain these terms, we recall that the metrics (lO.5) induce intrinsic Hermitian geometries (cf. Lemma (4. I)) on N and M and we let (xo.7) i I 1 - = gj.k) O 0 = _ 52 Sign O k ^ O~ , ~2k, l ,a be the curvature forms of the metrics in M and N respectively (cf. (4.8) and [7]). The Ricci form is defined by: (xo 8) Ric~ . ~a = I( ~ Rii~COkA~ll, =~=1 ~ ~\~.k., ] and the scalar curvature is given by: (IO. 9) R~ = ~ P~'0J = Trace(PdcM) Theorem (xo. io) (Chern [7]). -- In a neighborhood of a point m0eM where U(mo)>O, we have -IA log u = R.~--Trace(f* Rics). 159 I60 PHILLIP A. GRIFFITHS Return now to our period mapping 4) : P-+D. Given z0eP, either = o and (IO.2) is trivially true, or else ~*(COD)(Z0)4:0 in which case the differential ~. of ~P is injective at z0 and the image W= ~(U) of a small neighborhood U of z0 is a d-dimensional complex manifold with Hermitian metric ds~ induced from dsg. Denote by o~ w the associated 2-form. Lemma (IO,II). -- Let Pdcit and o H be the restrictions of the Ried form RicD and 2-form co D to the horizontal sub-bundle tI(D)cT(D). Suppose that we have Ric <--c(coH) (c>o). Then, keeping the situation and notation from just above this lemma, we have the estimate : Ricw=<_-- c(o~w). Proof. -- By the definition (io.8), Ricn=Trace(| where | is the metric connexion of the given Hermitian metric in the tangent bundle of D. We have then an inclusion of bundles: T(W) oH(D) cT(D) (over W), and we need to compare Trace(| ) and Trace(| The comparison of the curva- tures | OH, 01) is given by Lemma (4. I3), from which it follows that: Trace(Ow) _< Trace (@~) <__ Trace(| (in T(W)). f X Since cow=coD restricted to T~Wj, our lemma is proved. We now use the computation given in [i6], w 7 to prove: Lemma (IO.,2). -- In the notation of Lemma (IO. i I), we have RicR=_<--o~. This gives that: Ricw~--cow for the image manifold W=~(U) as described just above Lemma (io. Ii). We are now ready to prove the generalized Schwarz lemma (IO. I). Let P(p) be the polycylinder of radius p given as usual by {z=(zl, ..., zd): o<__]z,l<p} and ds~(p)= 4 ,~x (pL~]~.~)2] the PoincarE metric on P(p). We have made a slight change of scale from our original definition. With this change of scale, the scalar curvature Rp of the metric ds~Ip) is the constant --d. When p =i we write ds~ for ds~(I) and let c% be the associated 2-form. Define the non-negative function u(z) on P by ~*(coD)a=u.(op) a. We want to show that u < i. The idea, which is originally due to Ahlfors, is to use the maximum principle. We first show that it suffices to consider the case when u attains its maximum at some point in the interior of P. Let z0~P. Then z0eP(p) for some p<x and we may define u~(z)in P(p) by q~'(coD)~=u~.(co~(p)) a. Then limu~(zo)=u(zo) because of lim ds~(p) Thus it suffices to prove that up(zc)<I_ for p<i. Now, -4-1 t~zoj~--=~,P~,.oj't,2I'~. -m't,-,~,~j~af-~.0j PERIODS OF INTEGRALS ON MANIFOLDS, III t6t for p<I, q~*(dsg.) is bounded on the closed polycylinder P(p), while clearly (o~p(p)) a goes to infinity at the boundary P--P. Thus up (Z) goes to zero as z goes to the boundary, and so uo has its maximum at an interior point. We now assume that u has a maximum at zo~P. Then by Chern's formula (io. Io) (io. I3) o> IA log u=--d--Trace(f'PdCw). Now using orthonormal co-frames ( I o. 5) in the situation when M = P and N = W = q> (U) with U a neighborhood of zo in P, we have from Lemma (IO. 12) that - =- >, = x and so C,o. x4) -f'(mCw)> Z :i, j, Letting A be the matrix (a~), from (I o. 14) we find that (x o. x 5 ) -- Trace (f* R.iCw)__> Trace (A. 'A). Now use the Hadaraard inequality Trace(A.tA)~dldetAI2/d=du t/d together with (IO. 15) and (Io. 13) to find I>U a/d, which is what we wanted to prove. Remark (to.x6). -- As in Proposition (9.II) we let P(k, l; p) be the product (A;)kx(A0)' where A; is the punctured disc o<]z]< p and A 0 is the usual disc o<[zl<p. We set P'----P(k, l; I) and P=P(k+l, o; I). Then P--+P* is, in the usuaI way, the universal covering and so the Poincard metric ds~ induces a metric ds~,. on P*. Letting z4=r~ exp 0~ be polar coordinates, we have explicitly that (tO.IT) ds~= \,=, ~(log r~)2 ] + \j=k+, (i~[~)2], and for the volume element (IO, In) @0P*)k+ 1--'=1 r,(1og r,)2J-k+ x (i--4)~" From (IO. I8) we have Lemma CIO.X9). -- For p<I, the volume tx~(P(k, l; p))< oo of the sub-polycylinder P(k, 1; p)cP* is finite. The use of the following lemma was first demonstrated by Mrs. Kwack [25]: Lemma (IO.2O). --For o<p<I, let % be the circle [ZI=P in the punctured disc A" given by o<[z[<I. Then the length IA.(% ) of %, computed using the Poincar/ metric ds~. 2~ on A*, is given by lA,(%)----. In partieular, l(%)--+o as 0--+o. Proof. -- This follows immediately from (I o. 17), which gives that ds~. = dr~ + r2dO~ in the situation at hand. r2(l~ r)2 ALGEBRAIC x62 PHILLIP A. GRIFFITHS As a Corollary of Theorem (IO.I) and Lemmas (io. I9) and (IO.2O), we have: Corollary (IO.2I). -- Let q) : P*--->F\D be a horizontal, holomorphic mapping. Then the volume ~xr\D(r , l; p) ) ) of the image of the concentric punctured polycylinder P(k, l; p)cP* is finite. In particular, (ii) of Proposition (9. ii) is valid. Corollary (xo. ~'2). -- Let P* = A* � A e-1 and let (P : P*--->D be a horizontal holomorphic mapping. Suppose that Yo is a curve in P* given parametrically by O~(pe i~ z2(e~~ ..., z~(e~~ where the zj(e ~~ are smooth curves in the unit disc o_<lzj]<l. Then the length ID(~(yo) ) of the image curve tends to zero as p-->o. H. Proof of Propositions (9.to) and (9.IX.) a) We first prove (i) in Proposition (9. I I). For simplicity we will consider the case k=l, l=o. The general situation will be done by the exact same argument. Thus we have a locally liftable, horizontal, holomorphic mapping q) : A'~r\D such that the Picard-Lefschetz transformation TeP is of infinite order. We assume given a sequence {z,}eA* with Iz, l~o and such that {q)(z,)} converges in F\D. We want to show that this leads to a contradiction. Let % be the circle Izl =lznl and set wn=q)(z~)eF\D. We may assume that w. tends to a point weF\D. Choose a point FeD lying over w in the projection 7~ : D--->F\D. The stabilizer P~={geP:g.w=w} off is a finite group, and we may choose neighborhoods U of w in P\D and U of N in D such that F~.U =U and r~-l(U) is the disjoint union gear g.U. We may assume also that the distance dD(U , gU)>:r from U to its translates is bounded below for geP--F~. Finally, since TeP is of infinite order, we may assume that the intersection TUnU is empty. Choose n so large that the non-Euclidean length lA.(%) is less than r This is possible by Lemma (lO.2O). By Corollary (lO.22) the length ID(r of the image curve may also be assumed to be less than ~. Finally we may assume that the image w, of z,, under q) lies in U. Choose ~,eU which projects onto w.. Now take a local lifting ~ of q) in a neighborhood of z~ such that ~(z,)=w,. Analytic continuation of ~ around the circle % passing thru z, leads to the new local lifting T. ~ around z~. This is a contradiction since the length of the image curve ~(%) is less than ~, which implies that dD(~(Z~) , T~(Z,))<r while we have dD(U , T.U)>~. Because of this contradiction we have proved (i) in Proposition (9. ~ i), and the other part of this proposition has been given in Corollary (IO.21). b) We want to prove Proposition (9. IO). For simplicity we assume that d= i; the general case is done by essentially the same argument. Thus we assume given a horizontal, holomorphic mapping 9 : A*~D. We want 162 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III I63 to show that 9 extends to a continuous mapping of A into D. Our proof is based on the following result of .Mrs. Kwack [25]: Proposition (ix. x) (Mrs. Kwack). -- Let IV[ be a compact complex manifold with Hermitian metric ds~. Let f: A*-+ M be a holomorphic mapping with the property that if (~,} c A* is any sequence of circles ]z] =P, whose radii p, tend to zero, then the lengths lM(f(%) ) of the image circles tend to zero. Then f extends to a continuous mapping f: A-+M. Remark. -- The interesting thing about this result is that it is not at all a topological statement. The fact that M is a complex manifold seems to be quite essential. For completeness we shall give a proof of (I i. I) below. We now use (ii. i) to prove our extension theorem for r :A*-~D. Recalling that D=G/H is a homogeneous complex manifold of a real simple Lie group G by a compact subgroup H, we select a discrete subgroup A of G such that the quotient A\G is compact and such that A acts without fixed points on G/H. The existence of such a uniform subgroup A follows from a general result of Borel and Harish-Chandra [4]. Or in our case we could use the theorem in [28] to write down such a A. The quotient M=A\D is now a compact, complex manifold M with an Hermitian metric ds~ induced from the G-invariant ds~ on D. From Corollary (IO.22) it follows that the conditions of (II. I) are satisfied by the mapping f: A*-+A\D obtained by composing ~ with the projection D-+A\D. Thus f extends to give a continuous mapping f: A-+A\D From this it follows that ~ extends to give our desired continuous mapping 9 : A >D. Remark. -- The use of the uniform subgroup A in the abovc proof is not as absurd as it might at first appear. To explain what I mean, we recall thc embedding Dc]) of D as an open domain in its compact dual. It is not too hard to show that our mapping r extends to a continuous mapping ~:A-+D. The trouble is that the image ~(o) of the origin might lie in the boundary 0D=D--D of D in ]). So our extension theorem is really a question of the pseudo-convexity of D. Now for a bounded domain B in C', it is a theorem of Siegel [29] that the existence of a properly discontinuous group ~" of automorphisms of B such that ~\B is compact already implies that B is a domain of holomorphy. Thus, if B CC" is a boundcd domain such that we have a holomorphic mapping ap : A*-->B, and if there exists a uniform subgroup 9 cAut(B), then * cxtends to r because of the usual Riemann extension theorem plus Siegel's theorem. Our proof of Proposition (9. IO) is essentially a similar argument. c) We now give a proof, which is essentially that of [25], of Proposition (I i. i). We use the notation a(Zo) for the circle [Z[---Iz01 passing thru Z0EA*. Let {z,}cA* be a sequence of points with Iz,[-+o. If we set w,=f(z,), then by the compactness of M we may assume that w,-->weM. Let xl,..., x,, be local holomorphic coordinates centered at weM and denote by U(0 ) the polycylinder [xjl<0 163 x6 4 PHILLIP A. GRIFFITHS around w. We have to show that, given e>o, there exists ~ such that f(z)eU(s) if o<lz]<~. Let e>o be given. Since the lengths/e(f(~(Z,))) of the images of the circles ~(Z,) tend to zero, and since w,,Ef(~,~) tends to w, we may assume that f(~(z,))cU(~/2) for all n. If we cannot find the required S, then, by renumbering if necessary, we may find a sequence {y,,} ~A* with [Z.+ll<ly.l<l z.I such thatf(y.) does not lie in U(e). Let An be a maximal annulus ~.<lzl<~. around .(z.) such that f(A~)cU(e/2). Then the An are all disjoint, and we may choose a.e.(%) and b.ea(~.) with f(a.) and f(b.) lying in the boundary 0U(e/2) of the polycylinder U(e/~). Passing to subsequences, we may assume that f(a.) -+ aeOU(./~) and f(b.) ~ beOU(r Then f(a(a.)) ~ a and f(a(b.)) ~ b by the argument using lengths of circles. Write f(z)=(xl(z),...,Xm(Z)) and let e=xl(a), ~=Xl(b ). We may assume that ~4o, ~eo. Using the x~-coordinates, we have a picture ~(bn) xl(z) For n sufficiently large, we find from the argument principle that: fl xl(z)dz f, x~(z)dz I~ ol~.l x~(z)-xl(z.) - o = ~ I e oc~.l x~ (z) - xl (z.) This is a contradiction, since the difference of these two integrals is the integral: fo xl(z)dz A.,q(z~--~(z.) * o. 164 APPENDIX A A result on algebraic cycles and inte,-..,edlate Jacoblans a) Let V be a smooth, complete, and projective algebraic variety, and consider the odd degree cohomology Wm+l(V, C). For simplicity we will discuss the case when I-Pm+I(V, C) is all primitive -- the general situation is essentially a " direct sum " of such cases. We set H~m+ l(V C) = H2m+ 1, q_ Hm+l, + ,_, ~ ~(V) and define the (mth) intermediate facobian J(V) by (A. I) J(V)= ..+T42"~ I(V, C)\H2m+ I(V, C)/I-Pro+ t(V, Z). As referenccs on the theory of intermediate Jacobians we mention [14] , [27] and [23]. Denote by O(V) the group of algebraic cycles (modulo rational equivalence) on V which are of pure codimension m + i and which are homologous to zero. We will define an Abel-Jacobi homomorphism (A.2) ~b: 0(V) -+J(V), which generalizes the usual mapping for divisors on curves (m=o and dimcV----~ ). Before defining d~ we need a result of Dolbeault about Hodge filtrations (cf. the appendix to [I4] and the references given there). Let A ~'q be the C ~ forms of type (n, o)-[-... +(n--q, q) on V and Z n'q the d-closed forms in A ",q. Observe that d(A m, p) r A m + 1, p + 1 and set (A. 3) F" q = Z n' q/dA n-l' q-1 Proposition (A. 4). -- The natural mapping F" q --+ H"(V, C) is injeaive with image H"~ +Hn-q'q(v). Suppose nowthat dim0V----d (then m~d--I)2 and let r ..., r be a basis for (A.5) F2e-2,,-1,~-m~ Hd.e-2m-l(V)+....3rua-m,a-m-l(V). Observe that 1 ~d-2"-~'d-m is the dual space to the tangent space H '~''*+ ~(V) +... + H ~ ~m+ ~(V) 165 x66 PHII, LIP A. GRIFFITHS of J(V), and so we may think of ]72,Z-2m-1. d-m as the space of holomorphic differentials Letting Ze| we define the Abel-Jacobi map (A.2) by on J(V). (A.6) ~b(Z) = (fci=) (modulo periods) where C is a chain on V with 0C=Z. We recall from [14], [27], [23] that ~ is holomorphic (in a suitable sense), and + has nice functorial properties. In particular, suppose that X is a smooth, complete and projective algebraic variety which contains V as a smoothly embedded subvariety. of cohomology induces a homo- The restriction map H2"+I(X, C) ~ H2"+'(V, C) morphism of intermediate Jacobians r : J(X) -+J(V) with the following interpretation: Proposition (A. 7). -- Intersecting cycles on X with V induces a homomorphism ~: |174 such that the diagram: *, j(x) o(x) ,r o(v) , J(V) is commutative. algebraic family of algebraic varieties and Remark. -- Let f: X-->S be an assume that S is complete. Suppose that V is a fixed fibre of f:X~S, and let o~----(E, D, Q, {Fq}) be the variation of Hodge structure whose fibre corresponding to V is H2"+I(V, C). Then the image in the mapping J(X) -+j(v) is precisely the fixed part J ( @) as defined in w 7, c). Proposition (A.7) gives an algebro- geometric interpretation of this fixed part. b) Let f: X+S be an "algebraic family of algebraic varieties with fibres V, =f-l(s) (seS). We shall think of V, as just discussed in a) above, as being a typical fibre. For simplicity we shall continue to assume that all of H2"+t(V, C) is primitive. Let N=(E,D, Q, {1~}) be the variation of Hodge structure associated to f: X~S and H am f l(V, C), and let AcE R be the flat lattice given by the images of t-la"+'(V,, Z) -+ H2"+ a(V,, R). Referring to w 7, c), the corresponding family of intermediate Jacobians =: J+S, J,=J(V,) will be said to arise from a geometric situation. 166 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III Let now 0(X/S) be the sheaf which associates to each open set U cS the group of analytic cycles Z (modulo rational equivalence) of codimension m+I in f-l(U) such that the cohomology class of Z is zero in H~m+l(f-l(U), Z) (cf. [23] ). Theorem (A.8) (Integrability theorem for Abel-Jacobi maps). -- The Abel-Jacobi maps (A. 6) induce a sheaf mapping : o(x/s)-+j which satisfies the integrability condition DjW = o. Proof. -- Our proof of the existence of tF is based on [23]. We may assume that dimcS=i and we let f: Y-+A be the situation f: X-+S localized over a small disc A on S which has holomorphic coordinate s. We let ZcY be an algebraic cycle of codimension m + I and in general position with respect to the fibres V s. Then the intersections Z~ = Z. V s are algebraic cycles of codimension m + I on V~ which are homologous to zero there. In fact, we have: Z- 0C (modulo 0Y) for a suitable chain C on Y, and we may put things in general position so that Z s ~ 0C s where C 8 = C. V s (seA) ; (el. [23] for a complete discussion of the foundational points involved here). Let +(Z,) eJ(V,) be the point defined by the Abel-Jacobi map (A. 2). We want to prove that +(Z,) depends holomorphically on s. For this we choose C ~ differential forms col, ..., r z on Y such that (i) each o)~ is of type (d,d--~m--I)+...+(d--m,d--m--I); (ii) do)~^ds~-o; and (iii) the restrictions o)~]V, =o),(s) give a basis of F2a-2"-l'a-"-~(V,) (cf. Propo- sition (A. 4)). The existence of such forms is proved in [23], where it is also proved that the integrals fce%(s) may be assumed to depend continuously on s. Let co be any linear combination of o)x,..., co s . We want to show that the integral fc depends holomorphically on s. Let ~, be a simple, positively oriented, s ~ closed curve in the disc A. It will suffice to show that Let C v =Cnf-l(y) and Z v be the intersection of Z with the part of Y lying over the region inside y. Then by Stokes' theorem: f (foo))ds--fc since do) ^ds=o. But (~ o)^ds=o since o)^ds is of type (d+I, d--2m--I)+... +(d--m +i, d--m--i) 167 x68 PHILLIP A. GRIFFITHS whereas Z. e is an analytic set of complex dimension d--re. This proves the existence of the sheaf homomorphism q~: | obtained by fitting together the Abel- Jacobi maps along the fibres of f: X---~S. We want now to prove the integrability condition Dj~F=o. For this we first localize to have f: Y~A as before, and then choose C OO differential forms ~0,, ..., co~a on Y such that (i) deojAds--- o (j=I, ..., 2/); (ii) the restrictions coj]Vo give a basis of H2a-2m-t(Vs, C); (iii) ~t, "'-, ~ are of type (d, d--2m--I)+...+(d--m, d--m--l) and restrict to a basis of F2a-2~ -l'a-r"-t(V,); and (iv) o h .... , ,% are of type (d, d--2m--I)+... +(d--re+I, d--m--2) and give a basis of F~-2m-l'a-m-2(V,). We may think of oh, ..., ~0n as a holomorphic frame for the flat bundle E with fibres Eo = ~-2"- t(V ' C) and which is adapted to the filtration F a-,.-2 c F a-"-I cE. We let ca 9) i i Then 0~----~(s)ds where the functions a~(s) on A have the be the connection in E. following interpretation : Write dc~ = ~i ^ ds where the ~i are C ~ forms on Y. This is possible by the first property of the c0~s. Then d~i^ds=o so that ~i[V 8 is closed and gives a cohomology class in HZa-z''(V,, C), and we have in H2~-m-t(V0, C). (A 9 We can even assume that in l~a-~- t, a-'~-l(V,) (I_<_i< k). (A. 9 9 7, = X .{oj j=t seA. F, eH2a_~_t(V,, Z) be a cycle varying smoothly with Lemraa (A. x2). -- Let Then 9J z If i=l\JFs ] ~" Proof. -- We have: ,-~o t\Jr,§ Jr, ! 168 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, lII x% where we may restrict t to be real and positive. Let F(s,s~, t) be the union of the cycles F, for s <__ z < s + t. By Stokes' theorem, ~t) d%) tO 6) t~o t ,+ t)~q~ ^ds ) t ~ 0 "(s, 2/ (by (A. IO)). = (j,, i=1 This completes the proof of Lemma (A. 12). Lemma (A. x3). -- Using the notation established above, we have (jL ,)oj So, Proof. -- As in the proof of i.emma (A. 12) we h't O(s, s-r- t) the union of the C, for s_<z<s+t. Then eC(s, s + t)=c~+,-c,-z(~, s + t) where Z(s,s+t) is union of the cycles Z~ for s._<v<:s+t. Now fz(,,,+,)coj=o since c0j is of type (d,d--2m--I)+...-r-(d--m+I,d--m--2). Using Stokes' theorem we then have co -+ I ds) --= j (: "~j ~=, c CO~ ~j (by (A.1I)). This completes the proof of Lemma (A. i3). We now choose a frame el, ..., ezs for the dual bundle E such that (o~j,%+1 i)--~ ~ (I<i,j~M). Observe that the fibre E8 = H2m + I(V~, C) and that we have t ((~1, "" ", %)1 =(el ' ..., e,) ~(ox, ..., ~ = (ex, "-', % -k) so that et, . .., el is a basis for F 2"+ I"(V,) and el, 9 9 e2t k is a basis for F 2''+ ~' '" + ~(Vs). Writing 2'2 PHII, LIP A. GRIFFITHS x7o we have the relation 0i . 2/+1--/ (A. ~4) ~--~2/+l--j" We may now prove that D a ~F = o. Thc vector ~)'= (fCj=l ~ .t 6)j ) e2I+ 1-J is a section of E which projccts onto ~F in the mapping E---~J. We want to compute D~2 and then show that (A. x5) D~l-o modulo et, . .., eu_~.. Using the notation " = " for " congruent modulo et, . .., e21-k ", we have k l j=l \Jt~s $" -- j=l xat~s "~. - J I l 1 k j-ti=t j= i=l c60j ~+t-/e2/+l-.i (by Lemma (A. t3) ) -o (by (A. I4) ). This completes our proof. 170 APPENDIX B Two Examples a) A family of curves. -- We shall construct and then discuss an example due to Atiyah [I] of an algebraic family of curves f: X-+S where the parameter space S is itself a complete curve. To construct the example, we take a smooth, complete curve C having a fixed- point-free involution j: C-+C. Such curves exist whenever the genus p(C)_~ 3. Now we take S to be the finite unramified abelian covering of C given by the composite homomorphism nl(C) -+ Hi(C, Z) -+ HI(C , Z2). Let n : S-+C be the covering map, Y = S � C the product variety, and D = r. + Fjo= the (non-singular) curve on Y which is the sum of the graph of n and the graph of jon. Atiyah shows that there is a non-singular algebraic surface X which is a 2-sheeted covering of Y with branch curve D. The projection f: X-+S then gives X as an algebraic family of algebraic curves {V,}se s where V8 is a 2-sheeted covering of C with branch points at n(s) and jo~(s). Now for us the main important thing is the existence of a non-trivial family of non-singular curves with a complete parameter space. Let f: X-+S be one such family where the corresponding variation of Hodge structure is non-trivial (i.e., the fibres V 8 are not all birationally equivalent). The sheaf R~,(0x) is the sheaf 0(E ~ and from (7. I I) we have (8.,) ~ IS] >o. We want to compute the signature sign(X), and to do this we use the Hirzebruch index formula for X and the Grothendieck-Riemann-Roch formula for f : X~S and 0x as in Atiyah [I] to obtain d ~ sign(X) = ~- [X] (B.2) d 2 I+d+-~), ch(I-- R~. (0x)) =f,( where d~H2(X, Z) is the first Chern class of the tangent bundle along the fibres of f:X-+S. From (B.I) and (B.2)we have sign(X) = q(E ~ [S] >o, CB.3) so that the signature is not muhiplicative in the fibration f: X-+S. 22* PHILLIP A. GRIFFITHS I72 The exact same proof will give the Generalprinciple (B. 4). -- Let f: X-+S be an algebraic family of algebraic varieties with complete parameter space S. Then the Hirzebruch ;(v-genus [2I] is generally not multiplicative for the fibration f: X~S. Another point we are trying to illustrate is that there are interesting examples of algebraic families of algebraic varieties with a complete parameter space, although the most interesting case is certainly when the fibres are allowed to have arbitrary singularities. b) Lefschetz pencils of algebraic surfaces. -- In order to illustrate the existence of algebraic families of algebraic varieties f: X-+S whose parameter space need not be complete but where the Picard-Lefschetz transformations are of finite order, we consider a smooth, complete, and projective threefold WcP N. A generic pencil IP~_ ~(X)ixc-p ' of linear hyperplanes in Px traces out on W a pencil IVx[xc-,, of surfaces with critical points X~, ..., )'N- Letting S=P~--{X~, ..., X~}, the V x (Xc-S) are non-singular surfaces while the Vx~ are surfaces having one isolated ordinary double point. In the obvious way we may construct an algebraic family of algebraic varieties f: X~S with f-t(X)=Vx for Xc-S. This family has the property that there is a smooth compactification XcX re r sos such that f has one of the local forms if(x~, ,=, x~)=x, ! f (x,, ~,., x~)- (~,)~ + (x~) 2 + (x.p where xl, x2, x 3 are suitably chosen local holomorphic coordinates on X. We want to use the theorems in Lefschetz [26] to amplify two of our results above. Before doing this, we fix a base point X0c-S and paths l~ from X 0 to each critical point X~ (e=I, ..., N). We let y~c-~(S) be the closed curve obtained by going out l~, turning around X~, and then returning to X o along l~. Associated to each path l~, there is a vanishing cycle 8~c-H2(Vx~ Z) such that (~, ~)~ (B.5) T~q~=q~ where T~ is the automorphism of I-I2(Vx,, Z) corresponding to ~,=E~I(S ) ([26], p. 93). We have, furthermore, that (loc. cit., p. 93) (a~, a=)=- ~, CB.6) so that (T~)2=I and the Picard-Lefschetz transformations in our family of surfaces f: X~S are all of finite order. 172 PERIODS OF INTEGRAI.S ON ALGEBRAIC MANIFOLDS, III I73 Proposition (B. 7). -- There is a ~l(S)-invariant orthogonal direct sum decomposition (B. 8) H2(Vx~ Q) = I| where I = H~(Vz,, O) "~(s) are the invariant cycles and where ~x(S) acts irreducibly on E =(I) 1. Proof. -- Let E'c E be a non-trivial ~l(S)-invariant subspace and q~# o a vector in E'. From (B.5) we have that ..., N) while some (~,., q~)#o since q~ is orthogonal to the invariant cycles. Thus 8~.cE' and it follows that all 3,eE' since =1(S) acts transitively on the set {~1, .--, 81~} of vanishing cycles (loc. cit., p. 1o7). Thus E=E' since E is the span of 31, ..., ~N (loc. cit., p. 93)" Our second observation is Proposition (B.9). -- Let f: X-+S be the family of surfaces constructed above and let rcAut(H2(V,,, C)) be the monodromy group. Then r is a finite group if, and only if, the subspace H2'~ of H2(Vz., C) is elementwise invariant. Proof. -- If H~'~ is elementwise invariant, then we have EcHI, I(Vx.)o in (B.8). Since the intersection form is negative definite on E, we see that P is a finite group. Conversely, if I' is a finite group, then the subspace H2'~ of H2(Vx, C) is locally constant. In particular t-P,~ ) is a Z~l(S)-invariant subspace of H2(Vx,, C). Let ~I~'~ Then from (B.5) we have that (% ~)~H2,~176 for ~=i, ..., N. If some (% ~,.):t:o, then 8~~176176 which is impossible by (B.6). Thus all (% ~)= o and so q~ is an invariant cycle. 173 APPENDIX 12 Discussion of some open questions a) Statement of conjectures. -- Many of our results about a variation of Hodge structure had restrictions imposed concerning the Picard-Lefschetz transformations around the branches of S--S. We should like to suggest that these theorems should be valid under much more general circumstances. To state things precisely, we first need a few comments about the monodromy group P of the variation of Hodge structure 8, especially with regard to the Picard- Lefschetz transformations (w 3) of F. Kecall the monodromy theorem (cf. w 3 in [13] for discussion and references), which says that in case d' arises from a geometric situation f: X-+S, the Picard-Lefschetz transformations T are essentially unipotent of index n (same n as in w 2), which means that viewed as automorphisms T:E-+E they satisfy the equation (C.x) (TS--I)"+l=o for some N>o. We also recall that in the geometric case the monodromy group is a discrete subgroup of the automorphism group G of the variation of Hodge structure. Finally we recall the theorem of Borel (w 3 in [13] ) which says that in case F is an arithmetic subgroup of G, then the monodromy theorem holds, but without the estimate on the index of unipotency being established as yet. Our precise conjectures are (cf. the remark added in proof at the end of Appendix C) : (C.2) The invariant cycle theorem (7-1) is true if we assume only that the mono- dromy theorem (C. i) is valid. Remark. -- We shall prove this conjecture for n=i in a little while. If this conjecture is true, then the theorem (9.8) about the monodromy group would also hold. (C.3) The Mordell-Weil type theorem (7-19) is true if we only assume the monodromy theorem (C. I), but where we need to say what it means for a holomorphic section of J--~S to remain holomorphic at infinity. Remark. -- We shall also prove this conjecture for n = I below. (C.4) Theorem (9.7), which says that the image qb(S)cP\D under the period mapping is canonically a projective algebraic variety in case S is complete, is true if we only assume that F is a discrete subgroup of G. 174 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III t75 Remark. -- This conjecture is discussed in w167 IO, x i of [3]- We also refer to w 6 of [13] where it is pointed out that this conjecture (C.4) is valid in case D is a bounded, symmetric domain and I' is an arithmetic subgroup of G (Borel). Now how is one supposed to prove the above conjectures ? My own (obvious) feeling is that it should be possible to prove (C. 2) and (C.3) by sufficiently ingenious use of hyperbolic complex analysis so as to be able to give a good asymptotic form of the period mapping as we go to infinity. We shall give an illustration of this below. To prove (C.4) , one will need the estimates from hyperbolic complex analysis as well as a reduction theory for discrete subgroups of G. It also seems to me that " sufficiently ingenious use of hyperbolic complex analysis " will involve a detailed study of the geodesics of the metric dsg on the period matrix domain D as well as a more refined Schwarz lemma (io. i) which will give suitable estimates both ways in (lO.2). b) Proof of the invariant cycle theorem (7-i)Jbr n = i. -- Thus let d ~ be a variation of Hodge structure, with base space S, and let @ be a flat section of E-+S. The Hodge 0 1 filtration in this case is F, cFs=Es, and we let 9 be the projection of 9 in E/F~ 1. Using theorem (5-9), we want to prove that the length ]912 of 9 is uniformly bounded on S. We may assume that dimcS=I. We then localize over a punctured disc A* at infinity given by A*={s :o<lsl<i}. Choose a base point s0eA* andlet c01, ..., m2,~ be a flat frame for E in a neighborhood of s 0. Parallel displacement of this frame around the origin induces an automorphism 2m where T-----(T~) is the Picard-Lefschetz transformation around s = o. The monodromy theorem (C. I) is (TN--Iy=o, and by replacing s with s ~, we may as well assume that (T--I) 2=o. Now we may choose over A* a holomorphic frame 91(s), ..., 9,~(s) for the sub- bundle F~ Then we define the period matrix t2(s)=(n~j(s)) by 2m ..., m). The bilinear relations (2.7) now become the usual Riemann relations i ao_'a=o (C'5) ? iaQ~fi =j>o where tQ-l=(Q(o~, @). The matrix f~(s) is a multi-valued holomorphic matrix on A* such that analytic continuation around s = o changes f~ into Y~T. Let y be the flat section of the dual bundle E* defined by < y, e} = Q(O, e) (e~E). 175 PHILLIP A. GRIFF1THS Let ~ be the column vector given by t~ =(~,, ..., ~,,~) where ~ =(y, q~(s)}. Then, if we set H=J -1 in (C.5) , t~H~ is a well-defined function on A* and gives the length [q~] z of the projection q~ of 9 onto E/F ~ Thus we want to show that (C.6) t~H~ is bounded as s-~o. We will now use the results of [I3] , w 1 3 to put f2 in canonical form. Accordingly we can choose the frames col, ..., c%~ and q01, ..., q% such that the matrices Q, T, and f~ are given by (o i:). (C.7) Q= _im ' oo!) I,~_ k o (C. 8) T = f2/I A =*A, A>o; o I k o o Im-k f2 = (Ira, Z) (c.9) where Z=tZ has the form where the submatrices Z~ are holomorphic in the whole disc ]s]<I. Write Z~ = X~ + iY~. Then (C. II) j=/~-~QI~={Yll Y12] log,s, (~ ~) \tY12 YJ zz: where the Y,~ are continuous and Y,>o throughout the disc 5 given by Is I< i. From (C. 9) and (C. I o) it follows that the vector ~ is continuous on Using (C. 6) we will be done if we prove that j-1 is continuous on A. Now j-l= J* (det J) where J* is the usual matrix of minors of J. From (C. 1 I) we see that each entry in J* has the form (--log Is[) m-k. (continuous function of s), while from (C.8) and (C.II) we have (--1 og ~r~ tsI) '~+* A) + (lower terms). det J = __ (det Yn" det order Since det Yn.det A>o throughout A, we are done. 176 PERIODS OF INTEGRALS ON ALGEBRAIC MANIFOLDS, III I77 c) Proof of the usual Mordell-Weil over function fields. -- We will give a transcendental proof of the usual Mordell-Weil theorem for abelian varieties defined over function fields (char. o of course). Theorem (C. x2). -- Let do, A, and J-+S be as in the statement of theorem (7.19) with nI. Then there exists an extension of J-+S to an analytic fibre space J~S ofabelian complex Lie groups such that the group Horn(S, j) of holomorphic cross-sections of J-+S is an extension of the fixed part J( d ~ by a finitely generated abelian group. Remark. -- The integrability condition Ds~ = o is vacuous in this case since n = i. Proof. -- Let E+-+S be the holomorphic vector bundle over S whose fibre at each point seg is the complex Lie algebra of Js. Then F.+ IS is what was denoted by E+ in the proof of theorem (7- :9), and just as was the case in that proof, we want to show that any holomorphic section of ~..+ -+S comes from a constant section of E-+S. Of course this presumes that we have already defined J~S, which we now shall do. Let ~(A) be the sheaf over S of sections of the lattice AcE. We extend ~'(A) to a sheaf over S by saying that the sections of Cg(A) over an open set U c g are just the usual sections of <g(A) over UnS. To define ~.+, we will say what the sheaf g(E+) of holomorphic sections of the dual bundle is. Thus a section of Og(E+) over an open set U c-S is given by a holomorphic section r of E+ over U n S such that, for any section y of ~(A) over U, the contraction (?(s), Y} is a holomorphic function on all of U. There is an obvious injection Cg(A)-->~..+ and j is defined to be the quotient E+/Cg(A). .. We must prove that 0~(E+), as defined just above, is a locally free sheaf on all of S, that the image ~(A)-+E+ is discrete, and finally that the holomorphic sections of F.+-+g come from constant sections of E-+S. This is all done using the formulae (C.7)-(C.::) above together with the observations that (i) the flat frame ~i, .-., {~ of l~---~t* may be chosen to be commensurable with the lattice A, and (ii) the holomorphic sections of E+-+A are just the linear combinations of qh, ..., % with coefficients which are analytic functions in the whole disc A. Remark added in proof. -- Recent results of W. Schmid seem to show that the mono- dromy theorem (C. 1) is true for an arbitrary variation of Hodge structure. It may be hoped that his methods will also have a bearing on (C. 2) and (C. 3). 177 APPENDIX D A result on the monodromy of K3 surfaces In Pn with homogeneous coordinates ~---'--[~0, ~a, ~, ~3], we consider quartic surfaces defined by an equation Z.~,)s~,~,~,~, ~- ~,~. = o. (~~ .... The set of all such surfaces is parametrized by points s = [..., s~.~,~,~,, ... ] in a big PN. We denote by S' the Zariski open set in Ps of points such that the corresponding surface V, is non-singular. Such surfaces V, are among the K3 surfaces; i.e. they are simply- connected algebraic surfaces with trivial canonical bundle. We let sotS' be a fixed point and V=V,, the corresponding K 3 surface. Denote by E=P2(V, Q) the primitive part of the 2 ~a cohomology of V, and let F~cAut(E) be the arithmetic group induced from the automorphisms of H~(V, Z) which preserve the bilinear cup-product form and polarizing cohomology class. We denote by I" c P~ the global monodromy group; i.e. the image of ~1(S', So) acting on E. Theorem (D. x ). -- r is of finite index in F~. Proof. -- The period matrix domain D is, in this case, a bounded domain in C 19 and, by the local Torelli theorem [i i], the period mapping q~ : S'-+D/F contains an open set in its image. We now choose a I9-dimensional smooth algebraic subvariety Sr such that the restricted period mapping : S-+D/F contains also an open set in its image. By Theorem (9.6) above, q~(S) is the complement of an analytic subvariety in D/I'. Furthermore, because of the finite volume statement, it follows that D/U has finite volume with respect to the canonical invariant measure on D. Now it follows that the index of P in P~ is given by [r; i'.]= --~(D/F) <oo. ~(D/ro) Remarks. -- From Theorem (9.8) it follows that P is irreducible. Observe that from (B. 5) we may deduce that P is generated by elements of order 2. 178 PI.;RIODS OF INI'EGRALS ON .\I.(;EBRAIC MANIFOLDS, III x79 In general, for an algebraic family of algebraic varieties as in w i, the position of the monodromy group r in the arithmetic group 17 is extremely interesting. I know of no example where F is not of finite index in its Zariski closure. In relation to this, we close by observing that the proof of Theorem (D. ~) in general gives the following: Theorem (D.~). -- Let f: X~S be an algebraic family of algebraic varieties, and designate by P the glol)al monodromy group. Denote by S the universal covering of S and let " )" O S > D/l" be the period mapping. Let P' be any discrete subgroup of G = Aut(D) such that I" c F' and such that P' leaves the closure 0fq~(S) invariant. Then r is of finite index in P'. R EFERENCES [l] M. F. ATtYAH, The signature of fibre-bundles, Global Analysis (papers in honor of K. Kodaira), Princeton Univ. Press (x969), 73-89. [2] A. BLAN~ARD, Sur les vari6t6s analytiques complexes, Ann. Sci. [5cole Norm. Sup., 73 (1956), 157-2o~. [3] S- BOC:q~ER and K. YANO, Curvature and Betti Numbers, Princeton Univ. Press, J953- [4] A. BOR~L and Arithmetic subgroups of algebraic groups, Ann. of Aiath., 75 (I962), 485-535- [5] A. BoR~.~. and R. NARASlMHAN, Uniqueness conditions for certain holomorphic mappings, Invent. Math., 2 (1966), z47-~55. [6] E. CARTA~, Lefons sur la g~oraltrie des espaces de Rievnann, Paris, Gauthier-Villars, r95~. [7] S. S. Ctlnm~, On holomorphic mappings of Hermitian manifolds of the same dimension, Proc. Syrup. in Purr Math., 11, American Mathematical Society, x968. [8l S. S. C~tErtN, Characteristic classes of Hermitian manifolds, Ann. of Math., 47 (t946), 85-~2I. [9] P" D~LIOr, tI~, Theorie de Hodge, to appear in Publ. LH.E.S. [IO] I-t. GRAIJIgRT, I)ber Modifikationen und exzeptionelle analyfische Mengen, Math. Annalen, 14.6 096~), 331-368. [11] P. A. GRr~r~nnas, Periods of integrals on algebraic manifolds, I and II, Amer. Jour. Math., 90 (I968), 568-6'-'6 and 805-865 . [i2] P. A. GnsFFrrHs, Monodromy of homology and periods of integrals on algebraic manifolds, lecture notes available from Princeton University, I968. [13] P. A. GRI~'Fn~S, Periods of integrals on algebraic manifolds, Bull. Amer. j14ath. Soc., 75 097o), 228-o.96. [14] P. A. GRIFFrrHs, Some results on algebraic cycles on algebraic manifolds, Algebraic Geometry (papers presented at Bombay Colloquium), Oxford University Press, I969, 93-19 z. [x5] P. A. GRI~ITHS, Periods of certain rational integrals, Ann. of Math., 90 (t969), 460-54 I. [t6] P. A. GmFFITHS and W. SCHMIn, Locally homogeneous complex manifolds, Aeta Math., 19.8 (197o), '~53-'3o'- '. [t7] A. GaO'rrmNDmCK, Un th~or~me sur les homomorphismes de schemas ab61iens, lnoent. Math., 9. (196b), 59-78. [18] A. Gao'neer~t~cK, On the de Kham cohomology of algebraic varieties, Publ. Math. I.H.E.S., 9.9 (~966), 95-m3. ['9] R. Gum, nN~; and H. Ross*, Analytic Functions of Several Variables, Prentice-Hall, ~965. [zo] H. HmO~A~ZA, Resolution of singularities of an algebraic variety over a field of characteristic zero, 1 and 11, Ann. of Math., 79 (~964), ~o9-326. [2~] F. HXRZg~RtrCa, .Neue Topologische Methoden in der Algebraisehen Geometric, Springer-Verlag, x956. [2'2-] W. V. D. HOD~, The Theory and Applications of Harmonic Integrals, Cambridge University Press, I959. [~3] J- K~N~, Families of intermediate Jacobians, thesis at University of California, Berkeley, x969 . HAmStI-CHAN1)RA, ~8o PHILLIP A. GRIFFITHS [24] K. KODAIRA and D. C. SPENCER, On deformations of complex analytic structures, I and II, Ann. of Math., 67 (I958), 328-466. [25] M. H. KWACK, Generalization of the big Picard theorem, Ann. of Math., 90 (I969), I3-22. [26] S. LEFSCHETZ, L'Analysis Situs et la Gdom~trie Algdbrique, Paris, Gauthier-Villars, 1924. [27] D. Higher Picard varieties, Amer. flour. Math., 90 (1968), Ix65-I I99. [28] G. MosTow and T. TAMAGAWA, On the compactness of arithmetically defined homogeneous spaces, Ann. of Math., 76 (I962), 446-463 . [29] C. L. SIEGEL, Analytic functions of several complex variables, lecture notes from Institute for Advanced Study, Princeton, I962. [3 O] A. WEIL, Varidt~s Kdhleriennes, Paris, Hermann, i958. Manuscrit refu le 12 novembre 1969. Revisg le 6 juillet 1970. 1970. -- Imprimerie des Presses Universitaives de France. -- Vend6me (France) ~DIT. N ~ 31 283 IMeRIM~; EN FRANCE IMP. N ~ 22 222 LIEBERMAN,

Journal

Publications mathématiques de l'IHÉS – Springer Journals

Published: Aug 6, 2007

Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Periods of integrals on algebraic manifolds, III (some global differential-geometric properties of the period mapping)

Periods of integrals on algebraic manifolds, III (some global differential-geometric properties of the period mapping)

Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Periods of integrals on algebraic manifolds, III (some global differential-geometric properties of the period mapping)

Periods of integrals on algebraic manifolds, III (some global differential-geometric properties of the period mapping)

References (43)

Abstract

Journal

Recommended Articles

There are no references for this article.

Our policy towards the use of cookies