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Twisted holomorphic symplectic forms

Twisted holomorphic symplectic forms Abstract We show that a compact Kähler manifold admitting a non-degenerate holomorphic 2-form valued in a line bundle is a finite cyclic cover of a hyperkähler manifold. With respect to the connection induced by the locally hyperkähler metric, the form is parallel. We then describe the structure of the fundamental group of such manifolds and derive some consequences. 1. Introduction In this note, we are concerned with compact complex manifolds that admit a particular kind of structure: holomorphic non-degenerate 2-forms valued in a line bundle. Manifolds admitting such a structure will be called twisted holomorphic symplectic (THS). The problem has different analogues that have been intensively studied. On the one hand, there is the non-twisted problem concerning holomorphic symplectic forms. On the other hand, its symmetric avatar consists in the study of holomorphic (conformal) metrics. In the compact setting, the class of complex manifolds of Kähler type admitting holomorphic symplectic forms coincides with the class of hyperkähler manifolds, as shown in [2]. There is a rich literature concerning this subject, and its study is ongoing. Turning to the symmetric counterpart, the situation is somewhat different. Although the class of compact Kähler manifolds admitting a holomorphic metric is rather small (it consists in all the finite coverings of tori, as shown in [4]), as soon as one allows the structure to be twisted, thus studying holomorphic conformal structures, one enters a very rich class of manifolds. A complete classification of these has been reached only in dimension 2 and 3, in [5, 6]. Although one could expect that the class of THS manifolds is also wide, it turns out that the situation is not much different from the non-twisted case. More precisely, we show in Theorem 2.5 that compact THS manifolds of Kähler type are locally hyperkähler. In particular, the presence of such a structure ensures the existence of a Ricci-flat Kähler metric, and with respect to the connection induced by this metric the form is parallel. Roughly speaking, the proof goes as follows: we first note that the THS form induces local Lefschetz-type operators acting on the sheaves of holomorphic forms $$\Omega ^*$$, which then determine a local splitting of $$\Omega ^3$$ into $$\Omega ^1$$ and some other summand. This, in turn, allows us to find local holomorphic 1-forms that behave like connection forms on the line bundle where the twisted form takes its values. Finally, this means that the bundle admits a holomorphic connection, thus also a flat one, and that the manifold is Ricci-flat locally holomorphic symplectic, thus locally hyperkähler. In the next section, we give a more precise description of THS manifolds. In Theorem 3.1, we show that they are finite cyclic quotients of hyperkähler manifolds. Then we investigate under which conditions a locally hyperkähler manifold admits a THS form. The two classes do not coincide, and this is essentially because locally hyperkähler manifolds behave well to products, while THS manifolds never do, as shown in Corollary 3.2. Still, for locally irreducible manifolds, the two classes coincide by Corollary 3.3. Finally, for the intermediate case of irreducible, locally reducible manifolds, a discussion depending on the compactness of the universal cover is carried out in the remaining part of Section 3. As a consequence, we also obtain that strict THS manifolds with finite fundamental group are necessarily projective. 2. THS manifolds We start by discussing the complex symplectic case. For this, let us first define the objects we will be interested in the following definition. Definition 1. A Riemannian manifold $$(M,g)$$ is called hyperkähler if it admits three complex structures $$I,J$$ and $$K$$ that are compatible with the metric, that is,   \[ g(\cdot,\cdot)=g(I\cdot,I\cdot)=g(J\cdot,J\cdot)=g(K\cdot,K\cdot); \] verify the quaternionic relations:   \[ IJ=-JI=K; \] are parallel with respect to the Levi-Civita connection given by $$g$$. In particular, a hyperkähler manifold is Kähler with respect to its fixed metric and any complex structure $$aI+bJ+cK$$, with $$a, b$$ and $$c$$ real constants verifying $$a^2+b^2+c^2=1$$. Equivalently, we could say that a $$4n$$-dimensional Riemannian manifold $$(M,g)$$ is hyperkähler if and only if its holonomy group is a subgroup of Sp$$(n)$$. Definition 2. A holomorphic 2-form on a complex manifold $$M,$$$$\omega \in H^0(M,\Omega _M^2),$$ is called a holomorphic symplectic form if it is non-degenerate in the following sense:   \[ i_v\omega_x=0 \Rightarrow v=0\quad \forall x\in M,\enspace \forall v\in T^{1,0}_xM,\quad \hbox{where}\ i_v\ \hbox{is the contraction with}\ v. \] We call a manifold admitting such a form a holomorphic symplectic manifold. In particular, a holomorphic symplectic manifold $$(M,\omega )$$ has even complex dimension $$2m$$ and $$\omega ^m$$ is a nowhere vanishing holomorphic section of the canonical bundle $$K_M={\rm det} \Omega _M^1$$. Thus, $$K_M$$ is holomorphically trivial and $$c_1(M)=0$$. It can be easily seen that, once we fix a complex structure on a hyperkähler manifold $$M$$, say $$I$$, there exists a holomorphic symplectic form $$\omega $$ on $$(M,I)$$ defined by   \[ \omega(\cdot,\cdot)=g(J\cdot,\cdot)+ig(K\cdot,\cdot) \] Thus, a hyperkähler manifold is a holomorphic symplectic manifold (but not in a canonical way). In the compact case, the converse is also true. Theorem 2.1 (Beauville [2]). Let$$(M,I)$$be a compact complex manifold of Kähler type admitting a holomorphic symplectic form. Then, for any Kähler class$$\alpha \in H^2(M,\mathbb {R}),$$there exists a unique metric$$g$$on M that is Kähler with respect to$$I,$$representing$$\alpha ,$$so that$$(M,g)$$is hyperkähler. Moreover, the manifold$$(M,I)$$admits a metric with holonomy exactly$${\rm Sp}(m)$$if and only if it is simply connected and admits a unique holomorphic symplectic form up to multiplication by a scalar. Remark 2.2. The existence and uniqueness of the Kähler metric representing the given Kähler class comes from Yau's theorem: it is exactly the unique representative in the class that has vanishing Ricci curvature. Consequently, the holomorphic symplectic form in the theorem is parallel with respect to the Levi-Civita connection given by this Ricci-flat metric. We will now concentrate on the twisted case, and see that the situation is similar to the non-twisted one. Specifically, we will show that a Kähler manifold admitting a non-degenerate twisted holomorphic form admits a locally hyperkähler metric that is moreover Kähler for the given complex structure. With respect to the connection induced by this metric, the form will be parallel. Definition 3. A Riemannian manifold $$(M^{4m},g)$$ is called locally hyperkähler if its universal cover with the pullback metric is hyperkähler or, equivalently, if the restricted holonomy group $$\hbox {Hol}^0(g)$$ is a subgroup of Sp$$(m)$$. If, moreover, the manifold admits a global complex structure $$I$$ that is parallel with respect to the Levi-Civita connection induced by $$g,$$ then we will call it Kähler locally hyperkähler, or KLH for short. Hence, a locally hyperkähler manifold is the one that admits locally three orthogonal complex structures parallel for the Levi-Civita connection and verifies the quaternionic relations. It can be shown that in the case of a KLH manifold $$(M,g,I)$$, one of these complex structures can be taken to be $$I$$, so that an equivalent definition for KLH is a Kähler manifold that admits two local parallel complex structures preserved by $$g$$ that verify the quaternionic relations together with $$I$$. Definition 4. A compact manifold $$(M,I,L,\omega )$$ is called twisted holomorphic symplectic, or THS, if $$I$$ is a complex structure on $$M$$ and there exists a holomorphic line bundle $$L$$ over $$M$$ and a non-degenerate $$L$$-valued holomorphic form:   \[ \omega\in H^0(M,\Omega^2_M\otimes L). \] Remark 2.3. The existence of a THS form implies that $$M$$ is of even complex dimension $$2m$$. Moreover, $$\omega ^m$$ is a nowhere vanishing holomorphic section of the line bundle $$K_M\otimes L^m$$. Thus, we have that $$L^m\cong K_M^*$$. In particular, any metric on $$M$$ naturally induces one on $$L$$, and we also have   \[ c_1(M)=mc_1(L). \] Remark 2.4. Any complex surface $$M$$, of Kähler type or not, is THS in a tautological way. Simply take $$L$$ to be $$K_M^*$$, so that $$\Omega ^{2}_M\otimes L=K_M\otimes K_M^*$$ is holomorphically trivial. Thus, any non-zero section of this bundle is a twisted-symplectic form. Therefore, the class of THS manifolds is interesting only starting from complex dimension 4. Our main result in this section is the following theorem. Theorem 2.5. Let$$(M^{2m},I,L,\omega ),$$$$m>1,$$be a compact THS manifold of Kähler type, and let$$\alpha \in H^2(M,\mathbb {R})$$be a Kähler class. Then there exists a unique Kähler metric$$g$$with respect to$$I$$representing$$\alpha $$so that$$(M,g,I)$$is KLH. Moreover,$$L$$is unitary flat and$$\omega $$is parallel with respect to the natural connection induced by$$g$$on$$L$$. Proof. Let $$\{U_i\}$$ be a trivializing open cover for the line bundle $$L$$ and $$\sigma _i \in H^0(U_i,L)$$ be holomorphic frames, so that the holomorphic transition functions $$\{g_{ij}\}$$ are given by $$\sigma _i = g_{ij}\sigma _j$$. Then, if we write over $$U_i$$  \[ \omega=\omega_i\otimes\sigma_i \] we get local holomorphic symplectic forms $$\omega _i$$ that verify, on $$U_i \cap U_j$$, $$\omega _i = g_{ji}\omega _j$$. The forms $$\omega _i$$, being holomorphic, induce the morphisms of sheaves of $$\mathcal {O}_{U_i}$$-modules over $$U_i$$:   \begin{align*} &L_k:\Omega^k_{U_i}\longrightarrow\Omega^{k+2}_{U_i},\\ &L_k=\omega_i\wedge\cdot \end{align*} Lemma 2.6. For$$m>1,$$we have an isomorphism of sheaves of$$\mathcal {O}_{U_i}$$-modules:  \[ \Omega^3_{U_i}\cong\Omega^1_{U_i}\oplus\Omega^3_{0,U_i}, \]where$$\Omega ^3_{0,U_i}$$is the sheaf$${\rm Ker} (L_3^{m-2}:\Omega ^3_{U_i}\rightarrow \Omega ^{n-1}_{U_i})$$and$$n=2m$$. Proof. We claim that $$L_1^{m-1}:\Omega ^1_{U_i}\rightarrow \Omega ^{n-1}_{U_i}$$ is an isomorphism of sheaves over $$U_i$$. We inspect this at the germ level, so we fix $$z\in U_i$$. Since the corresponding free $$\mathcal {O}_z$$-modules have the same rank, it suffices to prove the injectivity of $$L^{m-1}_{1,z}$$. But this becomes a trivial linear algebra problem, noting that we can always find a basis over $$\mathbb {C}$$ in $$T^{1,0}M_z^*$$$$\{e_1,\ldots ,e_m,f_1,\ldots , f_m\}$$ so that   \[ \omega_i(z)=\sum_{s=1}^me_s\wedge f_s. \] Next, since $$L_1^{m-1}=L_3^{m-2}\circ L_1$$ we get that $$L_1$$ is injective and $$L_3^{m-2}$$ is surjective. Hence, we have an exact sequence of sheaves:   \[ 0 \longrightarrow \Omega^3_{0,U_i} \longrightarrow \Omega^3_{U_i} \xrightarrow{T} \Omega^{1}_{U_i}\longrightarrow 0 \] where $$T:=(L_1^{m-1})^{-1}\circ L_3^{m-2}$$ admits as a section $$L_1:\Omega _{U_i}^1\rightarrow \Omega ^3_{U_i}$$. Thus, the sequence splits and we get the desired isomorphism. This ends the proof of the lemma. □ Now, we have $$d\omega _i\in \Omega ^3_M(U_i)$$, so we can write:   \begin{equation} d\omega_i=\omega_i\wedge\theta_i+\xi_i \end{equation} (2.1) with $$\theta _i\in \Omega ^1_M(U_i)$$ and $$\xi _i\in \Omega _{0,M}^3(U_i)$$ holomorphic sections uniquely determined by the previous lemma. Since $$\omega _i=g_{ji}\omega _j$$, we get the following:   \[ dg_{ji}\wedge\omega_j+g_{ji} d\omega_j=g_{ji} \omega_j \wedge \theta_i+\xi_i \] whence   \[ \omega_j \wedge \theta_j+ \xi_j=d\omega_j = \omega_j\wedge\theta_i+ \frac{1}{g_{ji}}\xi_i-\frac{dg_{ji}}{g_{ji}}\wedge\omega_j \] Thus the forms $$\theta _i$$ change by the rule:   \begin{equation} \theta_i=\theta_j+d\log g_{ji}. \end{equation} (2.2) Hence, the differential operator $$D:C^\infty (M,L) \rightarrow C^\infty (M, T^*M\otimes L)$$ given over $$U_i$$ by $$D(f\otimes \sigma _i)=(df-\theta _i)\otimes \sigma _i$$ is a well-defined connection on $$L$$. On the other hand, given some hermitian metric $$h$$ on $$L$$, its Chern connection $$D^h$$ must differ from $$D$$ by a linear operator:   \[ D^h=D+A, \quad A\in C^\infty(T^*M\otimes \hbox{End}\,L). \] Moreover, since $$D^{0,1}=(D^h)^{0,1}=\bar {\partial }_L$$, $$A$$ must be a global $$(1,0)$$ form on $$M$$. Now $$\Theta (D^h)=\Theta (D)+dA$$, and since $$\Theta (D)_{U_i}=-d\theta _i$$ is of type (2,0) and $$i\Theta (D^h)$$ is a real (1,1)-form, we have that $$i\Theta (D^h)=i\bar {\partial } A$$ is exact in $$H^{1,1}(M,\mathbb {R})$$. But on a compact Kähler manifold, $$H^{1,1}(M,\mathbb {R})\subset H^2_{dR}(M,\mathbb {R})$$, so $$c_1(L)=({i}/{2\pi })[\Theta (D^h)]=0\in H^2_{dR}(M,\mathbb {R})$$. Thus we also get $$c_1(M)=mc_1(L)=0$$. So, by Yau's theorem, there exists a unique Ricci-flat Kähler metric $$g$$ whose fundamental form $$\omega _g$$ represents the given class $$\alpha $$. Now, on $$\Omega ^{2,0}_M\otimes L$$ we have the Weitzenböck formula (see, for instance, [7]):   \[ 2\bar{\partial}^*\bar{\partial} =\nabla^*\nabla+\mathcal{R,} \] where $$\nabla $$ is the naturally induced connection by $$g$$ on $$\Omega ^{2,0}_M\otimes L$$ and $$\mathcal {R}$$ is a curvature operator that on decomposable sections is given by   \[ \mathcal{R}(\beta\otimes s)=i\rho_g\beta\otimes s + \beta\otimes \hbox{Tr}_{\omega_g}(i\Theta(L))s \] with $$\rho _g:\Omega ^{2,0}_M\rightarrow \Omega ^{2,0}_M$$ the induced action of the Ricci form on $$\Omega ^{2,0}_M$$. Now, since $$g$$ is Ricci-flat, $$\rho _g\equiv 0$$. Also, if we consider the curvatures induced by $$g$$, then we have   \[ 0=-i\rho=\Theta(K^*_M)=\Theta(L^m) \] so the induced connection on $$L$$ is flat and $$\mathcal {R}$$ vanishes. Hence, applying the Weitzenböck formula to $$\omega $$, we get $$0=\nabla ^*\nabla \omega $$ or also, after integrating over $$M$$, $$\|\nabla \omega \|^2_{L^2}=0$$. Thus $$\nabla \omega =0$$. Finally, if we let $$\pi :(\tilde {M},\tilde {g},\tilde {I})\rightarrow (M,g,I)$$ be the universal cover with the pullback metric and complex structure, then we have that $$\pi ^*L$$ is holomorphically trivial and $$\tilde \omega =\pi ^*\omega \in H^0(\tilde {M},\Omega _{\tilde M}^2)$$ is a holomorphic symplectic form. By the Cheeger–Gromoll theorem, $$\tilde M\cong \mathbb {C}^l\times M_0$$, where $$M_0$$ is compact, simply connected, Kähler, Ricci-flat and $$\mathbb {C}^l$$ has the standard Kähler metric. Moreover, by the theorems of de Rham and Berger, the holonomy of $$M_0$$ is a product of groups of type Sp$$(k)$$ and SU$$(k)$$. We have that $$\tilde \omega $$ is a parallel section of   \[ \bigwedge^2T^*\tilde M= \bigwedge^2 \hbox{pr}_1^*T^*\mathbb{C}^l\oplus (\hbox{pr}_1^*T^*\mathbb{C}^l\otimes \hbox{pr}_2^* T^*M_0)\oplus \bigwedge^2 \hbox{pr}_2^*T^*M_0. \] But $$\hbox {pr}_1^*T^*\mathbb {C}^l\otimes \hbox {pr}_2^* T^*M_0\cong (T^*M_0)^{\oplus l}$$ has no parallel sections by the holonomy principle, so $$\tilde \omega $$ is of the form $$\omega _c+\omega _0$$, with $$\omega _c$$, $$\omega _0$$ holomorphic symplectic forms on $$\mathbb {C}^l$$, $$M_0$$, respectively. Thus, $$l$$ is even, so $$\mathbb {C}^l$$ is hyperkähler, and also, by Theorem 2.1, $$M_0$$ is hyperkähler. It follows that $$(M,g,I)$$ is KLH. This concludes the proof of the theorem. □ Remark 2.7. Note that $$D^{1,0}$$ is actually a holomorphic connection on $$L$$, so this gives another reason of why $$L$$ must be unitary flat. Remark 2.8. The flat connection induced by $$g$$ on $$L$$ does not depend on the Kähler class $$\alpha $$. It is uniquely determined by $$\omega $$ and is equal to the connection $$D$$ given in the above proof. To see this, let $$D^g$$ be the Chern connection on $$L$$ induced by $$g$$ and write $$D^g\sigma _i=\tau _i\otimes \sigma _i$$. Then we have   \[ 0=\nabla\omega=\nabla\omega_i\otimes\sigma_i +\omega_i\otimes\tau_i \otimes\sigma_i. \] So, denoting by $$a: \Omega _M^2\otimes T^*M^c\otimes L \rightarrow (\Omega _M^{3,0}\oplus \Omega _M^{2,1})\otimes L$$ the antisymmetrization map, we get   \[ d\omega_i=a(\nabla\omega_i)= -\omega_i\wedge\tau_i. \] Thus, by (2.1) we deduce that $$\xi _i=0$$ and $$\tau _i=-\theta _i$$, that is, $$D^g=D$$. Remark 2.9. If we suppose only that $$\omega $$ is a non-degenerate $$(2,0)$$ twisted form, not necessarily holomorphic, then $$\omega $$ still induces a connection on $$L$$ in the same manner. This time, we have the morphisms of sheaves of $$\mathcal {E}_{U_i}$$-modules $$L_k:\Omega ^{k,0}_{U_i}\rightarrow \Omega ^{k+2,0}_{U_i}$$ that induce isomorphisms $$\Omega ^{3,0}_M(U_i)\cong \Omega ^{1,0}_M(U_i)\oplus \Omega ^{3,0}_{0,M}(U_i)$$. Writing   \[ \Omega^{3,0}_M(U_i)\ni\partial \omega_i=\omega_i\wedge\theta_i+\xi_i, \] we get the $$(1,0)$$ forms $$\theta _i$$ that define a connection $$D$$ just as before. It is only at this point that the holomorphicity of $$\omega $$ becomes essential to have that $$D$$ defines a holomorphic connection on $$L$$. Actually, the complex manifolds that admit a non-degenerate (2,0)-form valued in a complex line bundle are exactly those that have a topological Sp($$m$$)U(1) structure. As expected, these are not necessarily locally hyperkähler : a counterexample is given by the quadric $$\mathbb {Q}_6={\rm SO}7)/{\rm U}(3)\subset \mathbb {P}^7\mathbb {C}$$, which is a Kähler manifold with topological Sp(3)U(1) structure, see [8], but is not KLH, since it has positive first Chern class. Remark 2.10. Note that the Kähler hypothesis was heavily used during the proof. So one could ask two questions in the non-Kähler setting. Does it follow that a compact complex THS manifold has holomorphic torsion canonical bundle, so that $$\omega $$ determines a holomorphic symplectic form on some finite unramified cover of $$M$$? Which are the compact complex manifolds admitting a holomorphic symplectic form? For the first question, the problem comes from cohomology. For a general compact complex manifold, one can define many cohomologies (de Rham, Dolbeault, Bott-Chern, Aeppli) that are not necessarily comparable. In particular, one does not always have a map from $$H^{1,1}_{\bar {\partial }}(M,\mathbb {C})$$ to $$H^2_{dR}(M,\mathbb {C})$$. Since what we actually show is that $$c_1(K_M)_{\bar {\partial }}=0$$, we cannot conclude that this Chern class vanishes in all other cohomologies (except for Aeppli). Moreover, even if it was the case, this would still not imply that $$K_M$$ is holomorphically torsion, see [10] for a detailed discussion and for examples showing the non-equivalence of the notions. For Fujiki's class $$\mathcal {C}$$ manifolds, the answer is yes though. Since these manifolds satisfy the $$\partial \bar {\partial }$$-lemma, we can conclude that the first Chern class of the manifold vanishes in all cohomologies. We then use the result of [10] stating that a Fujiki's class $$\mathcal {C}$$ manifold $$M$$ with $$c_1(M)_{BC}=0$$ has holomorphic torsion canonical bundle. Examples of THS manifolds that do not verify the $$\partial \bar {\partial }$$-lemma can be given as follows: let $$S$$ be a primary Kodaira surface. It admits a closed holomorphic symplectic form, so does $$S^m$$. Moreover, if $$\Gamma =\langle \gamma \rangle \subset {\rm Aut}(S)$$ is a finite cyclic group so that $$S/_{\Gamma }$$ is a secondary Kodaira surface, then $$S^m/_{\langle \gamma ,\ldots ,\gamma \rangle }$$ is THS by Theorem 3.1 in the next section. Note that this manifold still has holomorphic torsion canonical bundle. Regarding the second question, what we can say for sure is that the holomorphic symplectic class strictly contains the hyperkähler manifolds. A non-Kähler example is given as follows: start with a global complex contact manifold, the Iwasawa 3-fold for instance, that is a complex manifold $$M^{2m+1}$$ admitting a global holomorphic form $$\theta \in H^0(M,\Omega ^1_M)$$ such that $$\theta \wedge d\theta ^m$$ is nowhere zero. Let $$\mathbb {T}^1$$ be the one-dimensional complex torus, and take on $$X=M\times \mathbb {T}^1$$ the form $$\omega =d\theta +\theta \wedge \varphi $$, where $$\varphi $$ is a generator of $$H^0(\mathbb {T}^1, \Omega ^1_{\mathbb {T}^1})$$. Then $$\omega $$ is holomorphic symplectic and verifies $$0\neq d\omega =\varphi \wedge \omega $$. More examples can be constructed as complex ‘mapping tori’ over $$M$$: let $$f\in {\rm Aut}(M,\theta )$$ be a contactomorphism, that is, $$f^*\theta =\theta $$. Write $$\mathbb {T}^1=\mathbb {C}/_\Lambda $$, $$\Lambda =\mathbb {Z}\oplus \tau \mathbb {Z}$$, and let $$\Lambda $$ act on $$M$$ by $$1.x=x$$ and $$\tau .x=f(x)$$. Then $$\omega $$ descends to $$M_f:=M\times _{\Lambda }\mathbb {C}$$, which is again holomorphic symplectic. These examples are the holomorphic version of what is usually called locally conformally symplectic manifolds. 3. A characterization In this section, we want to investigate the converse problem. It is not true that all KLH manifolds are THS. Already we have seen that a product of strictly THS manifolds is never THS, but it turns out that being reducible is not the only obstruction. In what follows, we will give some description of THS manifolds and their fundamental groups. By a strictly THS manifold, we always mean a THS manifold $$(M,I,L,\omega )$$ such that the line bundle $$L$$ is not holomorphically trivial. Theorem 3.1. A compact Kähler manifold$$M$$of complex dimension greater than 2 is THS if and only if there exists a holomorphic symplectic form$$\omega _0$$on its universal cover$$\tilde M$$so that the action of$$\Gamma =\pi _1(M)$$on$$H^0(\tilde M,\Omega _{\tilde M}^2)$$preserves$$\mathbb {C}\omega _0$$. In particular, any THS manifold is a finite cyclic quotient of a hyperkähler manifold. Proof. Suppose first that $$M$$ admits a twisted-symplectic form   \[ \omega\in H^0(M,\Omega^2_M\otimes L). \] Then, by Theorem 2.5, $$L$$ is unitary flat, and thus given by a unitary representation $$\rho :\Gamma \rightarrow U(1)$$, that is, if we see $$\pi :\tilde M \rightarrow M$$ as a $$\Gamma $$-principal bundle over $$M$$, then we have $$L=\tilde M \times _\rho \mathbb {C}$$. Let $$s_i:U_i\rightarrow \tilde M$$ be local sections of $$\pi :\tilde M \rightarrow M$$ over a trivializing cover $$\{U_i\}$$. We then have $$s_i=\gamma _{ij}s_j$$ on $$U_i \cap U_j$$, where $$\gamma _{ij}: U_i\cap U_j \rightarrow \Gamma $$ are the transition functions for $$\tilde M$$. Then, $$\sigma _i:=[s_i,1]$$ are local frames for $$L$$, where $$[\cdot ,\cdot ]$$ denotes the orbit of an element of $$\tilde M\times \mathbb {C}$$ under the left action of $$\Gamma $$. The locally constant functions $$g_{ij}:=\rho (\gamma _{ij}^{-1})$$ are the transition functions for $$L$$ verifying   \[ \sigma_i=[\gamma_{ij}s_j,1]=[s_j,\rho(\gamma_{ij}^{-1})]=g_{ij}\sigma_j. \] Since $$\pi ^*L$$ is trivial, there exist $$f_i\in \mathcal {O}^*_{\tilde M}(\pi ^{-1}U_i)$$ such that $$\pi ^*g_{ij}={f_i}/{f_j}$$ on $$\pi ^{-1}U_i\cap \pi ^{-1}U_j$$. Also, the sections $$({\pi ^*\sigma _i}/{f_i})\in H^0(\pi ^{-1}U_i,\pi ^*L)$$ all coincide on intersections and are non-vanishing, thus giving a global frame for $$\pi ^*L$$ that we can suppose equal to 1, so that $$\pi ^*\sigma _i=f_i$$. Thus, if we write $$\omega =\omega _i\otimes \sigma _i$$ and define $$\omega _0:=\pi ^*\omega $$, then we get   \[ \omega_0|_{\pi^{-1}U_i}=\pi^*\omega_if_i \] and, for any $$\gamma \in \Gamma $$  \[ \gamma^*\omega_0|_{\pi^{-1}U_i}=\pi^*\omega_i\gamma^*f_i=\omega_0\frac{\gamma^*f_i}{f_i}. \] Moreover, for any $$\gamma $$, we have on $$\pi ^{-1}U_i\cap \pi ^{-1}U_j$$:   \[ \frac{f_j}{f_i}=\frac{f_j\circ\gamma}{f_i\circ\gamma} \Longleftrightarrow g_{ij}\circ\pi=g_{ij}\circ\pi\circ\gamma \] hence the constant function $$({f_i\circ \gamma }/{f_i})$$ does not depend on $$i$$. On the other hand, we have   \begin{equation} \frac{\gamma^*f_i}{f_i}=\frac{[s_i\circ\pi\circ\gamma,1]}{[s_i\circ\pi,1]}= \frac{[s_i\circ\pi,\rho(\gamma^{-1})]}{[s_i\circ\pi,1]}=\frac{1}{\rho(\gamma)}. \end{equation} (3.1) Hence $$\Gamma $$ preserves the subspace $$\mathbb {C}\omega _0\subset H^0(\tilde M,\Omega _{\tilde M}^2)$$ and $$\rho $$ is determined by the action of $$\Gamma $$ on the holomorphic symplectic form $$\omega _0$$ by   \[ \frac{1}{\rho(\gamma)}\cdot \omega_0=\gamma^*\omega_0. \] Conversely, suppose a holomorphic symplectic form $$\omega _0$$ is an eigenvector for $$\Gamma $$ acting on $$H^0(\tilde M,\Omega _{\tilde M}^2)$$. Define   \begin{align*} &\rho:\Gamma\longrightarrow \mathbb{C}^*,\\ &\gamma\longmapsto \frac{\omega_0}{\gamma^*\omega_0}. \end{align*} Let $$L:=\tilde {M}\times _{\rho }\mathbb {C}$$ and, with the same data for $$L$$ as before, define $$\omega \in H^0(M,\Omega _M^2\otimes L)$$ by $$\omega |_{U_i}=\omega _i\otimes \sigma _i$$, where $$\omega _i=s_i^*({\omega _0}/{f_i})$$. Then $$\omega $$ is THS and, seeing $$s_i\pi $$ as an element of $$\Gamma $$, we have, by (3.1), on $$\pi ^{-1}(U_i)$$:   \[ \pi^*\omega= \frac{\pi^*s_i^*\omega_0}{\pi^*s_i^*f_i}f_i= \frac{1}{\rho(s_i\pi)}\omega_0\frac{f_i}{(s_i\pi)^*f_i}=\omega_0. \] To prove the last part, suppose $$M$$ is THS and let, as in Theorem 2.5, $$\tilde M=\mathbb {C}^{2l}\times M_1\,{\times }\,{\cdots }\,{\times }\,M_k$$, with $$M_i$$ irreducible hyperkähler manifolds. The manifold $$M$$ has a finite étale cover $$M'=\mathbb {T}^{2l}\times M_1\times \cdots M_k$$ so that $$M=M'/ \Gamma '$$ and $$\Gamma \cong \mathbb {Z}^{4l}\ltimes \Gamma '$$. The symplectic form $$\omega _0$$ is preserved under the action of $$\mathbb {Z}^{4l}$$, so it descends to a holomorphic symplectic form on $$M'$$, which we will also denote by $$\omega _0$$. The group $$\Gamma '$$ preserves $$\mathbb {C}\omega _0$$. Let $$\rho ':\Gamma '\rightarrow U(1)$$ be the representation induced by $$\rho $$. Denote by $$N'$$ its kernel, and by $$N:=\mathbb {Z}^{4l}\ltimes N'$$. Then $$N$$ is normal inside $$\Gamma $$, so there exists a Galois covering $$M_N\rightarrow M$$ with $$\pi _1(M_N)=N$$. Moreover, since $$\pi _1(M')=\mathbb {Z}^{4l}$$ is normal in $$N$$, also $$M'\rightarrow M_N$$ is a covering whose deck transformation group is $$N/\mathbb {Z}^{4l}\cong N'$$. We thus have that $$M_N\cong M'/N'$$ and $$N'$$ preserves $$\omega _0$$, so $$\omega _0$$ descends to $$M_N$$. Since $$M_N$$ is compact holomorphic symplectic, it is hyperkähler. Finally, $$\rho (\Gamma )=\rho '(\Gamma ')$$ is a finite subgroup of U(1), so cyclic, and $$\Gamma /N\cong \Gamma '/N'\cong \rho (\Gamma )$$, so $$M_N$$ is a finite cyclic covering of $$M$$. This concludes the proof of the theorem. □ Corollary 3.2. A compact strictly THS manifold of dimension greater than 2 is de Rham irreducible. Proof. Suppose $$M\cong M_1\times M_2$$ is strictly THS. Let $$\tilde {M}\cong \tilde {M_1}\times \tilde {M_2}$$ be a finite étale cover of $$M$$ with holomorphic symplectic form $$\omega _0=\omega _1+\omega _2$$ preserved up to constants by $$\Gamma '\cong \Gamma _1'\times \Gamma _2'$$, where $$\pi _1(M_i)=\mathbb {Z}^{2l_i}\ltimes \Gamma _i'$$, $$i=1,2$$. Then we should have that $$\rho (\Gamma ')=\rho (\Gamma _1')\times \rho (\Gamma _2')$$ is a non-trivial cyclic group of the same order as $$\rho (\Gamma _1')$$, $$\rho (\Gamma _2')$$, which is impossible. □ Corollary 3.3. A compact locally irreducible Kähler manifold of dimension greater than 2 is KLH if and only if it is THS. In this case, the twisted-symplectic form is valued in the canonical bundle. Proof. Let $$M$$ be a locally irreducible KLH manifold and $$\tilde M$$ its universal cover endowed with a holomorphic symplectic form $$\omega _0$$. Since $$\tilde M$$ is irreducible, it is compact and $$H^0(\tilde M,\Omega _{\tilde M}^2)=\mathbb {C}\omega _0$$. Hence $$\Gamma =\pi _1(M)$$ preserves $$\mathbb {C}\omega _0$$ in a trivial way and $$M$$ is THS by the previous theorem. In particular, this implies that $$\Gamma $$ is cyclic. Let $$d$$ be its order. Then $$d|m+1$$, where dim$$M\,{=}\,2m$$. To see this, let $$\gamma \in \Gamma $$ be a generator, so that $$\gamma ^*\omega _0=\xi \cdot \omega _0$$, with $$\xi $$ a primitive $$d$$-root of unity. Since $$\gamma $$ has no fixed points, by the holomorphic Lefschetz fixed-point formula we must have that its Lefschetz number, which by definition is   \[ L(\gamma)=\sum_q (-1)^q\,{\rm tr}\gamma^*|_{H^q(\tilde M,\mathcal{O}_{\tilde M})} \] must vanish. On the other hand, we have   \[ \overline{H}^*(\tilde M,\mathcal{O}_{\tilde M})\cong H^0(\tilde M,\Omega^*_{\tilde M}) \cong\frac{\mathbb{C}[\omega_0]}{(\omega_0^{m+1})} \] so $$L(\gamma )=1+\xi +\cdots +\xi ^m$$. Thus, $$L(\gamma )=0$$ implies $$d|m+1$$. Let $$\rho :\Gamma \rightarrow $$U(1) be given by the action of $$\Gamma $$ on $$\omega _0$$ and $$L:=\tilde M\times _{\overline {\rho }}\mathbb {C}$$, so that the THS form is $$L$$-valued. Since the action of $$\Gamma $$ on $$K_{\tilde M}$$ is given by $$\rho ^m$$, we also have that $$K_M=\tilde M\times _{\rho ^m}\mathbb {C}$$. Now, $$\rho ^{m+1}=1$$ implies $$\overline {\rho }^m\cdot \overline {\rho }=1$$, or also $$K_M^*\otimes L=\underline {\mathbb {C}}$$, that is, $$L\cong K_M$$. □ Remark 3.4. For a THS manifold $$(M,I,L,\omega )$$, we always have, by Remark 2.3, that $$L$$ is a root of $$K_M^*$$. In the particular case when $$M$$ is locally irreducible, we obtain, moreover, that $$L$$ is precisely (up to isomorphism) $$K_M$$. It is difficult to give a nice criterion for being THS in the case of de Rham irreducible, locally reducible KLH manifolds. We can, though, give a somewhat more precise description of fundamental groups of THS manifolds. For this, we first give some lemmas concerning isometries of Riemannian products. Lemma 3.5. Let$$(M_i,g_i)$$be complete locally irreducible Riemannian manifolds of dimension bigger than 1 and let$$M_0=M_1\times \cdots \times M_k$$be endowed with the product metric. Let$$\gamma $$be an isometry of$$M_0$$and let$$\gamma _i:=p_i\gamma ,$$where$$p_i:M_0\rightarrow M_i$$are the canonical projections. Then$$\gamma _i$$is of the form$$\gamma _i=\tilde \gamma _ip_{\sigma (i)},$$where$$\tilde \gamma _i:M_{\sigma (i)}\rightarrow M_i$$is an isometry and$$\sigma $$a permutation of$$\{1,\ldots ,k\}$$. Proof. We have that $$\tilde g_i:=\gamma _i^* g_i$$ is a parallel section of $$S^2(T^*M_0)$$. On the other hand,   \[ S^2(T^*M_0)\cong \sum S^2(T^*M_i)\oplus \sum_{i<j}(T^*M_i\otimes T^*M_j). \] Now, $$T^*M_i\otimes T^*M_j$$ admits no parallel section for $$i<j$$, while the space of parallel sections of $$S^2(T^*M_i)$$ is exactly $$\mathbb {R} g_i$$. Indeed, by the holonomy principle, this is equivalent to saying that $$G_i\times G_j$$ has no fixed points when acting on $$T_x^*M_i\otimes T_y^*M_j$$, while the only $$G_i$$-invariant elements of $$S^2(T_x^*M_i)$$ are the multiples of $$(g_i)_x$$, where $$x\in M_i$$, $$y\in M_j$$ are any points and $$G_s$$ is the restricted holonomy group of $$M_s$$, $$s=1,\ldots ,k$$. The first assertion follows from the dimension hypothesis and the more general fact that if $$U$$ is a $$G$$-irreducible space and $$V$$ an $$H$$-irreducible space, then $$U\otimes V$$ is a $$G\times H$$-irreducible space. The second assertion is equivalent to Schur's lemma if we identify $$S^2(T_x^*M_i)$$ with the symmetric endomorphisms of $$T_x^*M_i$$ via $$g_i$$. Next, we want to show that for every $$i$$, there is exactly one $$j=j(i)$$ so that $$a_{ij}\neq 0$$. Thus, if we let $$A(i)=\{j|a_{ij}\neq 0\}$$, then we need to show that $$A(i)\neq \emptyset $$ for each $$i$$ and $$A(i)\cap A(j)\,{=}\,\emptyset $$ for all $$i\neq j$$. Now, since $$g_i$$ is definite and $$d\gamma _i$$ is surjective, we have that $$\ker \tilde g_i:=\{X \in TM_0| \tilde g_i(X,\cdot )=0\}=\ker d\gamma _i$$. Hence, since $$\ker d\gamma _i\neq TM$$, the first assertion follows. For the second assertion, first note that $$\ker \tilde g_i\cap TM_k\neq 0$$ if and only if $$a_{ik}=0$$, in which case $$TM_k\subset \ker \tilde g_i$$. Therefore, $$(\ker d\gamma _i)^\perp =\sum _{j\in A(i)} TM_j$$. Hence, for $$i\neq j$$, $$A(i)\cap A(j)=\emptyset $$ is equivalent to $$\{0\}=(\ker d\gamma _i)^\perp \cap (\ker d\gamma _j)^\perp =(\ker d\gamma _i + \ker d\gamma _j)^\perp $$. But we have   \[ \ker d\gamma_i + \ker d\gamma_j=d\gamma^{-1}(\ker d p_i+ \ker dp_j)=d\gamma^{-1}\left(\sum_{s\neq i} TM_s+\sum_{s\neq j}TM_s\right)=TM. \] It follows that there exists a permutation $$\sigma $$ of $$\{1,\ldots ,k\}$$ so that $$A(i)=\{\sigma (i)\}$$ for each $$i$$. Hence, since for any $$j$$, $$\sum _i a_{ij}=1$$, we have that $$a_{i\sigma (i)}=1$$ and $$\gamma _i=\tilde \gamma _ip_{\sigma (i)}$$ with $$\tilde \gamma _i:M_{\sigma (i)}\rightarrow M_i$$ an isometry. □ In what follows, we will omit writing the projections and identify $$\gamma _i$$ with $$\tilde \gamma _i$$. Lemma 3.6. Let$$M_i$$be irreducible compact hyperkähler manifolds, and$$M_0\,{=}\,M_1\,{\times }\,{\cdots }\,{\times }\,M_k$$be endowed with the product metric and a holomorphic symplectic form$$\omega _0$$. Then any isometry of$$M_0$$preserving$$\omega _0$$has fixed points. Proof. By Theorem 2.1, the manifolds $$M_i$$ are simply connected and admit unique holomorphic symplectic forms $$\omega _i$$ up to a scalar, so we have   \[ H^0(M_0,\Omega_{M_0}^2)=\mathbb{C}\omega_1\oplus\cdots\oplus\mathbb{C}\omega_k. \] Hence we can suppose, after rescaling each $$\omega _i$$, that $$\omega _0=\omega _1+\cdots +\omega _k$$. Let $$\gamma $$ be an isometry of $$M_0$$ with $$\gamma ^*\omega _0=\omega _0$$. Consider first the case where all $$M_i$$ are isometric, so that $$M_0\cong M_1^k$$. Let $$\sigma $$ be the permutation determined by $$\gamma $$ as in the previous lemma and let $$l$$ be the order of $$\sigma $$. If we define, for $$i=1,\ldots ,k,$$  \[ \gamma'_i=\gamma_i\gamma_{\sigma(i)}\cdots\gamma_{\sigma^{l-1}(i),} \] then $$\gamma ^l(x_1,\ldots ,x_k)=(\gamma '_1(x_1),\ldots ,\gamma '_k(x_k))$$. If $$\gamma $$ acts freely, then also $$\gamma ^l$$ acts freely. Otherwise, suppose $$\gamma ^l(y_1,\ldots ,y_k)=(y_1,\ldots ,y_k)$$. Let $$i_1,\ldots ,i_t \in \{1,\ldots ,k\}$$ represent the orbits of $$\langle \sigma \rangle $$ of cardinal $$l_1,\ldots ,l_t$$, and define $$(x_1,\ldots ,x_k)$$ by   \[ x_{i_\alpha}:=y_{i_\alpha}\quad \hbox{and} \quad x_{\sigma^j(i_\alpha)}:= \gamma_{\sigma^j(i_\alpha)}\cdots\gamma_{\sigma^{l_\alpha-1}(i_\alpha)}(y_{i_\alpha}) \] for $$\alpha =1,\ldots , t$$ and $$j=1,\ldots ,l_\alpha -1$$. The fact that $$\gamma _{i_\alpha }\gamma _{\sigma (i_\alpha )}\cdots \gamma _{\sigma ^{l_\alpha -1}(i_\alpha )}(y_{i_\alpha })=y_{i_\alpha }$$ implies that $$(x_1,\ldots ,x_k)$$ is a fixed point for $$\gamma $$, contradiction. Now, $$\gamma ^*\omega _0=\omega _0$$ implies $$(\gamma ^l)^*\omega _0=\sum _i(\gamma _i')^*\omega _1=\omega _0$$, or also $$(\gamma '_i)^*\omega _1=\omega _1$$ for any $$i=1,\ldots ,k$$. On the other hand, the fact that $$\gamma ^l$$ acts freely implies that some $$\gamma '_{i_0}$$ acts freely on $$M_1$$. By the holomorphic Lefschetz fixed-point formula, its Lefschetz number must then vanish. But $$L(\gamma '_{i_0})=m+1$$, where dim$$M_1=2m$$, contradiction. In the general situation, write $$M_0=(M_1)^{k_1}\times \cdots \times (M_s)^{k_s}$$, with $$M_i$$ irreducible and $$M_i\ncong M_j$$ for all $$i\neq j$$. By the previous lemma, $$\gamma =(\gamma _1,\ldots , \gamma _s)$$, with $$\gamma _i$$ an isometry of $$(M_i)^{k_i}$$. Again, $$\gamma ^*\omega _0=\omega _0$$ implies $$\gamma _i^*\tilde \omega _{i}=\tilde \omega _{i}$$, where $$\tilde {\omega }_i$$ is the induced symplectic form on $$(M_i)^{k_i}$$, $$i=1,\ldots ,s$$. Also, if $$\gamma $$ acts freely on $$M_0$$, then some $$\gamma _i$$ acts freely on $$(M_i)^{k_i}$$ and we already showed that this is impossible. □ Remark 3.7. We can now say slightly more about THS manifolds $$M$$ with compact universal cover $$\tilde M$$. In this case, with the notations of Theorem 3.1, $$\Gamma =\Gamma '$$, the representation $$\rho $$ is faithful by the previous lemma, so $$\Gamma =\rho (\Gamma )$$ is cyclic. Thus, if $$\gamma $$ is a generator of $$\Gamma $$ of order $$d$$ and $$\gamma ^*\omega _0=\xi \omega _0$$, then $$\xi $$ is necessarily a primitive $$d$$-root of unity. Moreover, if we write $$\gamma =(\gamma _1,\ldots ,\gamma _k)$$ just as in Lemma 3.5, then each $$\gamma _i$$ must have the same order $$d$$. To see this, let $$d_i={\rm ord}\gamma _i$$. Then $$d_i|d={\rm lcm}(d_i)_i$$. Since $$\gamma ^*\omega _0=\sum _i\gamma _i^*\omega _i=\xi \sum _ip^*_i\omega _i$$, we have, for all $$i$$, $$\gamma _i^*\omega _i=\xi \omega _{\sigma (i)}$$, hence $$\xi ^{d_i}=1$$. But $$\xi $$ was primitive, so $$d_i=d$$. We can conclude the following. Corollary 3.8. If the fundamental group of a compact THS manifold is finite, then it is cyclic and of the form$$\Gamma =\langle \gamma =(\gamma _1,\ldots ,\gamma _k)\rangle ,$$with$$\gamma _i$$isometries of the irreducible components of the universal cover, all of the same order. Remark 3.9. When $$M$$ is THS but $$\tilde M$$ is not compact, it is not necessarily the case for $$\Gamma '$$ to be cyclic, that is, $$\rho ':\Gamma '\rightarrow $$U(1) need not be faithful. By the same type of arguments as in Lemma 3.5, it can be seen that an element of $$\Gamma '$$ is of the form $$\gamma =(\gamma _T,\gamma _0)$$, with $$\gamma _T\in {\rm Aut}(\mathbb {T}^{2l})$$ and $$\gamma _0\in {\rm Aut}(M_0)$$. There exist fixed point free complex symplectomorphisms of $$\mathbb {T}^{2l}$$ of finite order (for instance, translation by a torsion element $$a \in \mathbb {T}^{2l}$$). So, if $$\gamma _T$$ is one and $$\gamma _0$$ is a symplectomorphism of $$M_0$$ of the same order as $$\gamma _T$$, then $$(\gamma _T,\gamma _0)$$ is an element in the kernel of $$\rho '$$. Corollary 3.10. A compact strictly THS manifold$$M$$of dimension greater than 2 with finite fundamental group is projective. Proof. Let $$\pi :\tilde M\rightarrow M$$ be the compact universal covering, where, by Theorem 2.5, $$\tilde M=M_1\times \cdots \times M_k$$ with $$M_i$$ irreducible hyperkähler manifolds. Then, by Lemma 3.6, each $$M_i$$ admits an automorphism that is not symplectic. By a result of [1], such manifolds are necessarily projective, so is $$\tilde M$$. But it is a well-known fact that a compact Kähler manifold is projective if and only if any of its finite unramified covering is projective, thus the conclusion follows. □ 4. Final remarks Remark 4.1. Concerning examples of Kähler type, finding locally irreducible KLH manifolds is equivalent to finding a fixed point free automorphism $$\gamma $$ of an irreducible symplectic manifold, so that all powers of $$\gamma $$ also act freely. In complex dimension 2, by Remark 2.4, all manifolds are THS. On the other hand, the only finite cyclic quotients of hyperkähler surfaces are the Enriques surfaces and some bielliptic surfaces. The first ones are quotients $$K/_{\langle \iota \rangle }$$, with $$K$$ a K3 surface admitting a fixed point free involution $$\iota $$, and they are locally irreducible. The bielliptic surfaces are quotients of products of two elliptic curves by some groups of order 2, 3, 4 or 6. This shows that Theorem 3.1 does not hold in complex dimension 2. Next, one can easily construct locally reducible THS manifolds of any dimension by iterating the KLH examples from above. For instance, if $$(K, \iota )$$ is a K3 surface as before, then $${K^{m}}/_{\langle \iota ,\ldots ,\iota \rangle }$$ is THS. In the same way, if $$A=\mathbb {C}/_{(\mathbb {Z}\oplus i\mathbb {Z})}$$ and $$P$$ is a 4-torsion point on $$A$$, defining $$f\in {\rm Aut}(A^{2m})$$ by $$f(X_1,\ldots ,X_{2m}):=(iX_1,\ldots ,iX_m, X_{m+1}+P,\ldots ,X_{2m}+P)$$ gives the THS manifold $$A^{2m}/_{\langle f\rangle }$$. More generally, if $$M_1$$ and $$M_2$$ are hyperkähler and $$\gamma _i\in {\rm Aut}(M_i)$$, $$i=1,2$$, have the same order, with $$\langle \gamma _1\rangle $$ acting freely but $$\langle \gamma _2\rangle $$ possibly not, then $$(M_1\times M_2)/_{\langle \gamma _1,\gamma _2\rangle }$$ is again a smooth THS manifold. Finding locally irreducible THS manifolds of higher dimension is more difficult. For the Hilbert schemes of points on K3 surfaces, see [2] for the construction, all known automorphisms have fixed points, so we have no hope of constructing examples out of them. On the other hand, there is hope with the generalized Kummer varieties $$K_r$$, see again [2] for the definition. In [3, 9], the authors find fixed point free cyclic groups of automorphisms $$\Gamma $$ of order 3 for the manifolds $$K_2$$ and $$K_5$$, and of order 4 for $$K_3$$. The corresponding quotients give the desired examples of dimension 4, 10 and 6, respectively. Remark 4.2. In order to actually classify Kählerian THS manifolds, one should be able to classify fixed point free groups of automorphisms of irreducible hyperkähler manifolds. The problem is clear in low dimension. It is also clear that if the Hilbert schemes of points on K3 surfaces admit such groups, then the corresponding automorphisms are not natural, that is, do not arise from automorphisms of the K3 surface. On the other hand, for the moment the only known non-natural automorphisms have fixed points. For the generalized Kummer varieties, there exist some examples formed out of natural automorphisms, but we do not know a classification of such groups. Remark 4.3. As already stated, the natural generalizations of the subject studied here are widely open problems. On the one hand, one could study compact Kähler manifolds with non-degenerate $$(2,0)$$ forms, or equivalently with $${\rm Sp}(m)U(1)$$ topological structure, see Remark 2.9. On the other hand, one could drop the Kähler assumption, and a number of new questions arise, as seen in Remark 2.10. We will address these in a forthcoming paper. Acknowledgments I thank my PhD advisor Andrei Moroianu for introducing me to the topic and for the enlightening remarks, and my co-advisor Simone Diverio for showing interest in the subject and for the valuable comments. I also thank Alexandra Otiman for the fruitful non-Kählerian discussions, and the anonymous referee for useful suggestions. References 1 A. Beauville, ‘ Some remarks on Kähler manifolds with $$c_1 = 0$$’, Prog. Math.  39 ( 1983) 1– 26. 2 A. Beauville, ‘ Variétés Kähleriennes dont la première classe de Chern est nulle’, J. Differential Geom  18 ( 1983) 755– 782. 3 S. Boissiére, M. Nieper-Wißkirchen and A. Sarti, ‘ Higher-dimensional Enriques varieties and automorphisms of generalized Kummer varieties’, J. Math. Pures Appl.  95 ( 2011) 553– 563. Google Scholar CrossRef Search ADS   4 M. Inoue, S. Kobayashi and T. Ochiai, ‘ Holomorphic affine connections on compact complex surfaces’, J. Fac. Sci. Univ. Tokyo Sect. IA Math.  27 ( 1980) 247– 264. 5 P. Jahnke and I. Radloff, ‘ Projective threefolds with holomorphic conformal structure’, Int. J. Math.  16 ( 2005) 595– 607. Google Scholar CrossRef Search ADS   6 S. Kobayashi and T. Ochiai, ‘ Holomorphic structures modeled after hyperquadrics’, Tôhoku Math. J.  ( 2) 34 ( 1982) 587– 629. Google Scholar CrossRef Search ADS   7 A. Moroianu, Lectures on Kähler geometry , London Mathematical Society Student Texts 69 ( Cambridge University Press, Cambridge, 2007). 8 A. Moroianu, M. Pilca and U. Semmelmann, ‘ Homogeneous almost quaternion-Hermitian manifolds’, Math. Ann.  357 ( 2013) 1205– 1216. Google Scholar CrossRef Search ADS   9 K. Oguiso and S. Schröer, ‘ Enriques manifolds’, J. reine angew. Math.  661 ( 2011) 215– 235. Google Scholar CrossRef Search ADS   10 V. Tosatti, ‘ Non-Kähler Calabi–Yau manifolds’, Contemp. Math.  644 ( 2015) 261– 277. © 2016 London Mathematical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Oxford University Press

Twisted holomorphic symplectic forms

Istrati, N.
Bulletin of the London Mathematical Society , Volume 48 (5) – Jul 14, 2016
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Oxford University Press
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© 2016 London Mathematical Society
ISSN
0024-6093
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1469-2120
DOI
10.1112/blms/bdw039
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Abstract

Abstract We show that a compact Kähler manifold admitting a non-degenerate holomorphic 2-form valued in a line bundle is a finite cyclic cover of a hyperkähler manifold. With respect to the connection induced by the locally hyperkähler metric, the form is parallel. We then describe the structure of the fundamental group of such manifolds and derive some consequences. 1. Introduction In this note, we are concerned with compact complex manifolds that admit a particular kind of structure: holomorphic non-degenerate 2-forms valued in a line bundle. Manifolds admitting such a structure will be called twisted holomorphic symplectic (THS). The problem has different analogues that have been intensively studied. On the one hand, there is the non-twisted problem concerning holomorphic symplectic forms. On the other hand, its symmetric avatar consists in the study of holomorphic (conformal) metrics. In the compact setting, the class of complex manifolds of Kähler type admitting holomorphic symplectic forms coincides with the class of hyperkähler manifolds, as shown in [2]. There is a rich literature concerning this subject, and its study is ongoing. Turning to the symmetric counterpart, the situation is somewhat different. Although the class of compact Kähler manifolds admitting a holomorphic metric is rather small (it consists in all the finite coverings of tori, as shown in [4]), as soon as one allows the structure to be twisted, thus studying holomorphic conformal structures, one enters a very rich class of manifolds. A complete classification of these has been reached only in dimension 2 and 3, in [5, 6]. Although one could expect that the class of THS manifolds is also wide, it turns out that the situation is not much different from the non-twisted case. More precisely, we show in Theorem 2.5 that compact THS manifolds of Kähler type are locally hyperkähler. In particular, the presence of such a structure ensures the existence of a Ricci-flat Kähler metric, and with respect to the connection induced by this metric the form is parallel. Roughly speaking, the proof goes as follows: we first note that the THS form induces local Lefschetz-type operators acting on the sheaves of holomorphic forms $$\Omega ^*$$, which then determine a local splitting of $$\Omega ^3$$ into $$\Omega ^1$$ and some other summand. This, in turn, allows us to find local holomorphic 1-forms that behave like connection forms on the line bundle where the twisted form takes its values. Finally, this means that the bundle admits a holomorphic connection, thus also a flat one, and that the manifold is Ricci-flat locally holomorphic symplectic, thus locally hyperkähler. In the next section, we give a more precise description of THS manifolds. In Theorem 3.1, we show that they are finite cyclic quotients of hyperkähler manifolds. Then we investigate under which conditions a locally hyperkähler manifold admits a THS form. The two classes do not coincide, and this is essentially because locally hyperkähler manifolds behave well to products, while THS manifolds never do, as shown in Corollary 3.2. Still, for locally irreducible manifolds, the two classes coincide by Corollary 3.3. Finally, for the intermediate case of irreducible, locally reducible manifolds, a discussion depending on the compactness of the universal cover is carried out in the remaining part of Section 3. As a consequence, we also obtain that strict THS manifolds with finite fundamental group are necessarily projective. 2. THS manifolds We start by discussing the complex symplectic case. For this, let us first define the objects we will be interested in the following definition. Definition 1. A Riemannian manifold $$(M,g)$$ is called hyperkähler if it admits three complex structures $$I,J$$ and $$K$$ that are compatible with the metric, that is,   \[ g(\cdot,\cdot)=g(I\cdot,I\cdot)=g(J\cdot,J\cdot)=g(K\cdot,K\cdot); \] verify the quaternionic relations:   \[ IJ=-JI=K; \] are parallel with respect to the Levi-Civita connection given by $$g$$. In particular, a hyperkähler manifold is Kähler with respect to its fixed metric and any complex structure $$aI+bJ+cK$$, with $$a, b$$ and $$c$$ real constants verifying $$a^2+b^2+c^2=1$$. Equivalently, we could say that a $$4n$$-dimensional Riemannian manifold $$(M,g)$$ is hyperkähler if and only if its holonomy group is a subgroup of Sp$$(n)$$. Definition 2. A holomorphic 2-form on a complex manifold $$M,$$$$\omega \in H^0(M,\Omega _M^2),$$ is called a holomorphic symplectic form if it is non-degenerate in the following sense:   \[ i_v\omega_x=0 \Rightarrow v=0\quad \forall x\in M,\enspace \forall v\in T^{1,0}_xM,\quad \hbox{where}\ i_v\ \hbox{is the contraction with}\ v. \] We call a manifold admitting such a form a holomorphic symplectic manifold. In particular, a holomorphic symplectic manifold $$(M,\omega )$$ has even complex dimension $$2m$$ and $$\omega ^m$$ is a nowhere vanishing holomorphic section of the canonical bundle $$K_M={\rm det} \Omega _M^1$$. Thus, $$K_M$$ is holomorphically trivial and $$c_1(M)=0$$. It can be easily seen that, once we fix a complex structure on a hyperkähler manifold $$M$$, say $$I$$, there exists a holomorphic symplectic form $$\omega $$ on $$(M,I)$$ defined by   \[ \omega(\cdot,\cdot)=g(J\cdot,\cdot)+ig(K\cdot,\cdot) \] Thus, a hyperkähler manifold is a holomorphic symplectic manifold (but not in a canonical way). In the compact case, the converse is also true. Theorem 2.1 (Beauville [2]). Let$$(M,I)$$be a compact complex manifold of Kähler type admitting a holomorphic symplectic form. Then, for any Kähler class$$\alpha \in H^2(M,\mathbb {R}),$$there exists a unique metric$$g$$on M that is Kähler with respect to$$I,$$representing$$\alpha ,$$so that$$(M,g)$$is hyperkähler. Moreover, the manifold$$(M,I)$$admits a metric with holonomy exactly$${\rm Sp}(m)$$if and only if it is simply connected and admits a unique holomorphic symplectic form up to multiplication by a scalar. Remark 2.2. The existence and uniqueness of the Kähler metric representing the given Kähler class comes from Yau's theorem: it is exactly the unique representative in the class that has vanishing Ricci curvature. Consequently, the holomorphic symplectic form in the theorem is parallel with respect to the Levi-Civita connection given by this Ricci-flat metric. We will now concentrate on the twisted case, and see that the situation is similar to the non-twisted one. Specifically, we will show that a Kähler manifold admitting a non-degenerate twisted holomorphic form admits a locally hyperkähler metric that is moreover Kähler for the given complex structure. With respect to the connection induced by this metric, the form will be parallel. Definition 3. A Riemannian manifold $$(M^{4m},g)$$ is called locally hyperkähler if its universal cover with the pullback metric is hyperkähler or, equivalently, if the restricted holonomy group $$\hbox {Hol}^0(g)$$ is a subgroup of Sp$$(m)$$. If, moreover, the manifold admits a global complex structure $$I$$ that is parallel with respect to the Levi-Civita connection induced by $$g,$$ then we will call it Kähler locally hyperkähler, or KLH for short. Hence, a locally hyperkähler manifold is the one that admits locally three orthogonal complex structures parallel for the Levi-Civita connection and verifies the quaternionic relations. It can be shown that in the case of a KLH manifold $$(M,g,I)$$, one of these complex structures can be taken to be $$I$$, so that an equivalent definition for KLH is a Kähler manifold that admits two local parallel complex structures preserved by $$g$$ that verify the quaternionic relations together with $$I$$. Definition 4. A compact manifold $$(M,I,L,\omega )$$ is called twisted holomorphic symplectic, or THS, if $$I$$ is a complex structure on $$M$$ and there exists a holomorphic line bundle $$L$$ over $$M$$ and a non-degenerate $$L$$-valued holomorphic form:   \[ \omega\in H^0(M,\Omega^2_M\otimes L). \] Remark 2.3. The existence of a THS form implies that $$M$$ is of even complex dimension $$2m$$. Moreover, $$\omega ^m$$ is a nowhere vanishing holomorphic section of the line bundle $$K_M\otimes L^m$$. Thus, we have that $$L^m\cong K_M^*$$. In particular, any metric on $$M$$ naturally induces one on $$L$$, and we also have   \[ c_1(M)=mc_1(L). \] Remark 2.4. Any complex surface $$M$$, of Kähler type or not, is THS in a tautological way. Simply take $$L$$ to be $$K_M^*$$, so that $$\Omega ^{2}_M\otimes L=K_M\otimes K_M^*$$ is holomorphically trivial. Thus, any non-zero section of this bundle is a twisted-symplectic form. Therefore, the class of THS manifolds is interesting only starting from complex dimension 4. Our main result in this section is the following theorem. Theorem 2.5. Let$$(M^{2m},I,L,\omega ),$$$$m>1,$$be a compact THS manifold of Kähler type, and let$$\alpha \in H^2(M,\mathbb {R})$$be a Kähler class. Then there exists a unique Kähler metric$$g$$with respect to$$I$$representing$$\alpha $$so that$$(M,g,I)$$is KLH. Moreover,$$L$$is unitary flat and$$\omega $$is parallel with respect to the natural connection induced by$$g$$on$$L$$. Proof. Let $$\{U_i\}$$ be a trivializing open cover for the line bundle $$L$$ and $$\sigma _i \in H^0(U_i,L)$$ be holomorphic frames, so that the holomorphic transition functions $$\{g_{ij}\}$$ are given by $$\sigma _i = g_{ij}\sigma _j$$. Then, if we write over $$U_i$$  \[ \omega=\omega_i\otimes\sigma_i \] we get local holomorphic symplectic forms $$\omega _i$$ that verify, on $$U_i \cap U_j$$, $$\omega _i = g_{ji}\omega _j$$. The forms $$\omega _i$$, being holomorphic, induce the morphisms of sheaves of $$\mathcal {O}_{U_i}$$-modules over $$U_i$$:   \begin{align*} &L_k:\Omega^k_{U_i}\longrightarrow\Omega^{k+2}_{U_i},\\ &L_k=\omega_i\wedge\cdot \end{align*} Lemma 2.6. For$$m>1,$$we have an isomorphism of sheaves of$$\mathcal {O}_{U_i}$$-modules:  \[ \Omega^3_{U_i}\cong\Omega^1_{U_i}\oplus\Omega^3_{0,U_i}, \]where$$\Omega ^3_{0,U_i}$$is the sheaf$${\rm Ker} (L_3^{m-2}:\Omega ^3_{U_i}\rightarrow \Omega ^{n-1}_{U_i})$$and$$n=2m$$. Proof. We claim that $$L_1^{m-1}:\Omega ^1_{U_i}\rightarrow \Omega ^{n-1}_{U_i}$$ is an isomorphism of sheaves over $$U_i$$. We inspect this at the germ level, so we fix $$z\in U_i$$. Since the corresponding free $$\mathcal {O}_z$$-modules have the same rank, it suffices to prove the injectivity of $$L^{m-1}_{1,z}$$. But this becomes a trivial linear algebra problem, noting that we can always find a basis over $$\mathbb {C}$$ in $$T^{1,0}M_z^*$$$$\{e_1,\ldots ,e_m,f_1,\ldots , f_m\}$$ so that   \[ \omega_i(z)=\sum_{s=1}^me_s\wedge f_s. \] Next, since $$L_1^{m-1}=L_3^{m-2}\circ L_1$$ we get that $$L_1$$ is injective and $$L_3^{m-2}$$ is surjective. Hence, we have an exact sequence of sheaves:   \[ 0 \longrightarrow \Omega^3_{0,U_i} \longrightarrow \Omega^3_{U_i} \xrightarrow{T} \Omega^{1}_{U_i}\longrightarrow 0 \] where $$T:=(L_1^{m-1})^{-1}\circ L_3^{m-2}$$ admits as a section $$L_1:\Omega _{U_i}^1\rightarrow \Omega ^3_{U_i}$$. Thus, the sequence splits and we get the desired isomorphism. This ends the proof of the lemma. □ Now, we have $$d\omega _i\in \Omega ^3_M(U_i)$$, so we can write:   \begin{equation} d\omega_i=\omega_i\wedge\theta_i+\xi_i \end{equation} (2.1) with $$\theta _i\in \Omega ^1_M(U_i)$$ and $$\xi _i\in \Omega _{0,M}^3(U_i)$$ holomorphic sections uniquely determined by the previous lemma. Since $$\omega _i=g_{ji}\omega _j$$, we get the following:   \[ dg_{ji}\wedge\omega_j+g_{ji} d\omega_j=g_{ji} \omega_j \wedge \theta_i+\xi_i \] whence   \[ \omega_j \wedge \theta_j+ \xi_j=d\omega_j = \omega_j\wedge\theta_i+ \frac{1}{g_{ji}}\xi_i-\frac{dg_{ji}}{g_{ji}}\wedge\omega_j \] Thus the forms $$\theta _i$$ change by the rule:   \begin{equation} \theta_i=\theta_j+d\log g_{ji}. \end{equation} (2.2) Hence, the differential operator $$D:C^\infty (M,L) \rightarrow C^\infty (M, T^*M\otimes L)$$ given over $$U_i$$ by $$D(f\otimes \sigma _i)=(df-\theta _i)\otimes \sigma _i$$ is a well-defined connection on $$L$$. On the other hand, given some hermitian metric $$h$$ on $$L$$, its Chern connection $$D^h$$ must differ from $$D$$ by a linear operator:   \[ D^h=D+A, \quad A\in C^\infty(T^*M\otimes \hbox{End}\,L). \] Moreover, since $$D^{0,1}=(D^h)^{0,1}=\bar {\partial }_L$$, $$A$$ must be a global $$(1,0)$$ form on $$M$$. Now $$\Theta (D^h)=\Theta (D)+dA$$, and since $$\Theta (D)_{U_i}=-d\theta _i$$ is of type (2,0) and $$i\Theta (D^h)$$ is a real (1,1)-form, we have that $$i\Theta (D^h)=i\bar {\partial } A$$ is exact in $$H^{1,1}(M,\mathbb {R})$$. But on a compact Kähler manifold, $$H^{1,1}(M,\mathbb {R})\subset H^2_{dR}(M,\mathbb {R})$$, so $$c_1(L)=({i}/{2\pi })[\Theta (D^h)]=0\in H^2_{dR}(M,\mathbb {R})$$. Thus we also get $$c_1(M)=mc_1(L)=0$$. So, by Yau's theorem, there exists a unique Ricci-flat Kähler metric $$g$$ whose fundamental form $$\omega _g$$ represents the given class $$\alpha $$. Now, on $$\Omega ^{2,0}_M\otimes L$$ we have the Weitzenböck formula (see, for instance, [7]):   \[ 2\bar{\partial}^*\bar{\partial} =\nabla^*\nabla+\mathcal{R,} \] where $$\nabla $$ is the naturally induced connection by $$g$$ on $$\Omega ^{2,0}_M\otimes L$$ and $$\mathcal {R}$$ is a curvature operator that on decomposable sections is given by   \[ \mathcal{R}(\beta\otimes s)=i\rho_g\beta\otimes s + \beta\otimes \hbox{Tr}_{\omega_g}(i\Theta(L))s \] with $$\rho _g:\Omega ^{2,0}_M\rightarrow \Omega ^{2,0}_M$$ the induced action of the Ricci form on $$\Omega ^{2,0}_M$$. Now, since $$g$$ is Ricci-flat, $$\rho _g\equiv 0$$. Also, if we consider the curvatures induced by $$g$$, then we have   \[ 0=-i\rho=\Theta(K^*_M)=\Theta(L^m) \] so the induced connection on $$L$$ is flat and $$\mathcal {R}$$ vanishes. Hence, applying the Weitzenböck formula to $$\omega $$, we get $$0=\nabla ^*\nabla \omega $$ or also, after integrating over $$M$$, $$\|\nabla \omega \|^2_{L^2}=0$$. Thus $$\nabla \omega =0$$. Finally, if we let $$\pi :(\tilde {M},\tilde {g},\tilde {I})\rightarrow (M,g,I)$$ be the universal cover with the pullback metric and complex structure, then we have that $$\pi ^*L$$ is holomorphically trivial and $$\tilde \omega =\pi ^*\omega \in H^0(\tilde {M},\Omega _{\tilde M}^2)$$ is a holomorphic symplectic form. By the Cheeger–Gromoll theorem, $$\tilde M\cong \mathbb {C}^l\times M_0$$, where $$M_0$$ is compact, simply connected, Kähler, Ricci-flat and $$\mathbb {C}^l$$ has the standard Kähler metric. Moreover, by the theorems of de Rham and Berger, the holonomy of $$M_0$$ is a product of groups of type Sp$$(k)$$ and SU$$(k)$$. We have that $$\tilde \omega $$ is a parallel section of   \[ \bigwedge^2T^*\tilde M= \bigwedge^2 \hbox{pr}_1^*T^*\mathbb{C}^l\oplus (\hbox{pr}_1^*T^*\mathbb{C}^l\otimes \hbox{pr}_2^* T^*M_0)\oplus \bigwedge^2 \hbox{pr}_2^*T^*M_0. \] But $$\hbox {pr}_1^*T^*\mathbb {C}^l\otimes \hbox {pr}_2^* T^*M_0\cong (T^*M_0)^{\oplus l}$$ has no parallel sections by the holonomy principle, so $$\tilde \omega $$ is of the form $$\omega _c+\omega _0$$, with $$\omega _c$$, $$\omega _0$$ holomorphic symplectic forms on $$\mathbb {C}^l$$, $$M_0$$, respectively. Thus, $$l$$ is even, so $$\mathbb {C}^l$$ is hyperkähler, and also, by Theorem 2.1, $$M_0$$ is hyperkähler. It follows that $$(M,g,I)$$ is KLH. This concludes the proof of the theorem. □ Remark 2.7. Note that $$D^{1,0}$$ is actually a holomorphic connection on $$L$$, so this gives another reason of why $$L$$ must be unitary flat. Remark 2.8. The flat connection induced by $$g$$ on $$L$$ does not depend on the Kähler class $$\alpha $$. It is uniquely determined by $$\omega $$ and is equal to the connection $$D$$ given in the above proof. To see this, let $$D^g$$ be the Chern connection on $$L$$ induced by $$g$$ and write $$D^g\sigma _i=\tau _i\otimes \sigma _i$$. Then we have   \[ 0=\nabla\omega=\nabla\omega_i\otimes\sigma_i +\omega_i\otimes\tau_i \otimes\sigma_i. \] So, denoting by $$a: \Omega _M^2\otimes T^*M^c\otimes L \rightarrow (\Omega _M^{3,0}\oplus \Omega _M^{2,1})\otimes L$$ the antisymmetrization map, we get   \[ d\omega_i=a(\nabla\omega_i)= -\omega_i\wedge\tau_i. \] Thus, by (2.1) we deduce that $$\xi _i=0$$ and $$\tau _i=-\theta _i$$, that is, $$D^g=D$$. Remark 2.9. If we suppose only that $$\omega $$ is a non-degenerate $$(2,0)$$ twisted form, not necessarily holomorphic, then $$\omega $$ still induces a connection on $$L$$ in the same manner. This time, we have the morphisms of sheaves of $$\mathcal {E}_{U_i}$$-modules $$L_k:\Omega ^{k,0}_{U_i}\rightarrow \Omega ^{k+2,0}_{U_i}$$ that induce isomorphisms $$\Omega ^{3,0}_M(U_i)\cong \Omega ^{1,0}_M(U_i)\oplus \Omega ^{3,0}_{0,M}(U_i)$$. Writing   \[ \Omega^{3,0}_M(U_i)\ni\partial \omega_i=\omega_i\wedge\theta_i+\xi_i, \] we get the $$(1,0)$$ forms $$\theta _i$$ that define a connection $$D$$ just as before. It is only at this point that the holomorphicity of $$\omega $$ becomes essential to have that $$D$$ defines a holomorphic connection on $$L$$. Actually, the complex manifolds that admit a non-degenerate (2,0)-form valued in a complex line bundle are exactly those that have a topological Sp($$m$$)U(1) structure. As expected, these are not necessarily locally hyperkähler : a counterexample is given by the quadric $$\mathbb {Q}_6={\rm SO}7)/{\rm U}(3)\subset \mathbb {P}^7\mathbb {C}$$, which is a Kähler manifold with topological Sp(3)U(1) structure, see [8], but is not KLH, since it has positive first Chern class. Remark 2.10. Note that the Kähler hypothesis was heavily used during the proof. So one could ask two questions in the non-Kähler setting. Does it follow that a compact complex THS manifold has holomorphic torsion canonical bundle, so that $$\omega $$ determines a holomorphic symplectic form on some finite unramified cover of $$M$$? Which are the compact complex manifolds admitting a holomorphic symplectic form? For the first question, the problem comes from cohomology. For a general compact complex manifold, one can define many cohomologies (de Rham, Dolbeault, Bott-Chern, Aeppli) that are not necessarily comparable. In particular, one does not always have a map from $$H^{1,1}_{\bar {\partial }}(M,\mathbb {C})$$ to $$H^2_{dR}(M,\mathbb {C})$$. Since what we actually show is that $$c_1(K_M)_{\bar {\partial }}=0$$, we cannot conclude that this Chern class vanishes in all other cohomologies (except for Aeppli). Moreover, even if it was the case, this would still not imply that $$K_M$$ is holomorphically torsion, see [10] for a detailed discussion and for examples showing the non-equivalence of the notions. For Fujiki's class $$\mathcal {C}$$ manifolds, the answer is yes though. Since these manifolds satisfy the $$\partial \bar {\partial }$$-lemma, we can conclude that the first Chern class of the manifold vanishes in all cohomologies. We then use the result of [10] stating that a Fujiki's class $$\mathcal {C}$$ manifold $$M$$ with $$c_1(M)_{BC}=0$$ has holomorphic torsion canonical bundle. Examples of THS manifolds that do not verify the $$\partial \bar {\partial }$$-lemma can be given as follows: let $$S$$ be a primary Kodaira surface. It admits a closed holomorphic symplectic form, so does $$S^m$$. Moreover, if $$\Gamma =\langle \gamma \rangle \subset {\rm Aut}(S)$$ is a finite cyclic group so that $$S/_{\Gamma }$$ is a secondary Kodaira surface, then $$S^m/_{\langle \gamma ,\ldots ,\gamma \rangle }$$ is THS by Theorem 3.1 in the next section. Note that this manifold still has holomorphic torsion canonical bundle. Regarding the second question, what we can say for sure is that the holomorphic symplectic class strictly contains the hyperkähler manifolds. A non-Kähler example is given as follows: start with a global complex contact manifold, the Iwasawa 3-fold for instance, that is a complex manifold $$M^{2m+1}$$ admitting a global holomorphic form $$\theta \in H^0(M,\Omega ^1_M)$$ such that $$\theta \wedge d\theta ^m$$ is nowhere zero. Let $$\mathbb {T}^1$$ be the one-dimensional complex torus, and take on $$X=M\times \mathbb {T}^1$$ the form $$\omega =d\theta +\theta \wedge \varphi $$, where $$\varphi $$ is a generator of $$H^0(\mathbb {T}^1, \Omega ^1_{\mathbb {T}^1})$$. Then $$\omega $$ is holomorphic symplectic and verifies $$0\neq d\omega =\varphi \wedge \omega $$. More examples can be constructed as complex ‘mapping tori’ over $$M$$: let $$f\in {\rm Aut}(M,\theta )$$ be a contactomorphism, that is, $$f^*\theta =\theta $$. Write $$\mathbb {T}^1=\mathbb {C}/_\Lambda $$, $$\Lambda =\mathbb {Z}\oplus \tau \mathbb {Z}$$, and let $$\Lambda $$ act on $$M$$ by $$1.x=x$$ and $$\tau .x=f(x)$$. Then $$\omega $$ descends to $$M_f:=M\times _{\Lambda }\mathbb {C}$$, which is again holomorphic symplectic. These examples are the holomorphic version of what is usually called locally conformally symplectic manifolds. 3. A characterization In this section, we want to investigate the converse problem. It is not true that all KLH manifolds are THS. Already we have seen that a product of strictly THS manifolds is never THS, but it turns out that being reducible is not the only obstruction. In what follows, we will give some description of THS manifolds and their fundamental groups. By a strictly THS manifold, we always mean a THS manifold $$(M,I,L,\omega )$$ such that the line bundle $$L$$ is not holomorphically trivial. Theorem 3.1. A compact Kähler manifold$$M$$of complex dimension greater than 2 is THS if and only if there exists a holomorphic symplectic form$$\omega _0$$on its universal cover$$\tilde M$$so that the action of$$\Gamma =\pi _1(M)$$on$$H^0(\tilde M,\Omega _{\tilde M}^2)$$preserves$$\mathbb {C}\omega _0$$. In particular, any THS manifold is a finite cyclic quotient of a hyperkähler manifold. Proof. Suppose first that $$M$$ admits a twisted-symplectic form   \[ \omega\in H^0(M,\Omega^2_M\otimes L). \] Then, by Theorem 2.5, $$L$$ is unitary flat, and thus given by a unitary representation $$\rho :\Gamma \rightarrow U(1)$$, that is, if we see $$\pi :\tilde M \rightarrow M$$ as a $$\Gamma $$-principal bundle over $$M$$, then we have $$L=\tilde M \times _\rho \mathbb {C}$$. Let $$s_i:U_i\rightarrow \tilde M$$ be local sections of $$\pi :\tilde M \rightarrow M$$ over a trivializing cover $$\{U_i\}$$. We then have $$s_i=\gamma _{ij}s_j$$ on $$U_i \cap U_j$$, where $$\gamma _{ij}: U_i\cap U_j \rightarrow \Gamma $$ are the transition functions for $$\tilde M$$. Then, $$\sigma _i:=[s_i,1]$$ are local frames for $$L$$, where $$[\cdot ,\cdot ]$$ denotes the orbit of an element of $$\tilde M\times \mathbb {C}$$ under the left action of $$\Gamma $$. The locally constant functions $$g_{ij}:=\rho (\gamma _{ij}^{-1})$$ are the transition functions for $$L$$ verifying   \[ \sigma_i=[\gamma_{ij}s_j,1]=[s_j,\rho(\gamma_{ij}^{-1})]=g_{ij}\sigma_j. \] Since $$\pi ^*L$$ is trivial, there exist $$f_i\in \mathcal {O}^*_{\tilde M}(\pi ^{-1}U_i)$$ such that $$\pi ^*g_{ij}={f_i}/{f_j}$$ on $$\pi ^{-1}U_i\cap \pi ^{-1}U_j$$. Also, the sections $$({\pi ^*\sigma _i}/{f_i})\in H^0(\pi ^{-1}U_i,\pi ^*L)$$ all coincide on intersections and are non-vanishing, thus giving a global frame for $$\pi ^*L$$ that we can suppose equal to 1, so that $$\pi ^*\sigma _i=f_i$$. Thus, if we write $$\omega =\omega _i\otimes \sigma _i$$ and define $$\omega _0:=\pi ^*\omega $$, then we get   \[ \omega_0|_{\pi^{-1}U_i}=\pi^*\omega_if_i \] and, for any $$\gamma \in \Gamma $$  \[ \gamma^*\omega_0|_{\pi^{-1}U_i}=\pi^*\omega_i\gamma^*f_i=\omega_0\frac{\gamma^*f_i}{f_i}. \] Moreover, for any $$\gamma $$, we have on $$\pi ^{-1}U_i\cap \pi ^{-1}U_j$$:   \[ \frac{f_j}{f_i}=\frac{f_j\circ\gamma}{f_i\circ\gamma} \Longleftrightarrow g_{ij}\circ\pi=g_{ij}\circ\pi\circ\gamma \] hence the constant function $$({f_i\circ \gamma }/{f_i})$$ does not depend on $$i$$. On the other hand, we have   \begin{equation} \frac{\gamma^*f_i}{f_i}=\frac{[s_i\circ\pi\circ\gamma,1]}{[s_i\circ\pi,1]}= \frac{[s_i\circ\pi,\rho(\gamma^{-1})]}{[s_i\circ\pi,1]}=\frac{1}{\rho(\gamma)}. \end{equation} (3.1) Hence $$\Gamma $$ preserves the subspace $$\mathbb {C}\omega _0\subset H^0(\tilde M,\Omega _{\tilde M}^2)$$ and $$\rho $$ is determined by the action of $$\Gamma $$ on the holomorphic symplectic form $$\omega _0$$ by   \[ \frac{1}{\rho(\gamma)}\cdot \omega_0=\gamma^*\omega_0. \] Conversely, suppose a holomorphic symplectic form $$\omega _0$$ is an eigenvector for $$\Gamma $$ acting on $$H^0(\tilde M,\Omega _{\tilde M}^2)$$. Define   \begin{align*} &\rho:\Gamma\longrightarrow \mathbb{C}^*,\\ &\gamma\longmapsto \frac{\omega_0}{\gamma^*\omega_0}. \end{align*} Let $$L:=\tilde {M}\times _{\rho }\mathbb {C}$$ and, with the same data for $$L$$ as before, define $$\omega \in H^0(M,\Omega _M^2\otimes L)$$ by $$\omega |_{U_i}=\omega _i\otimes \sigma _i$$, where $$\omega _i=s_i^*({\omega _0}/{f_i})$$. Then $$\omega $$ is THS and, seeing $$s_i\pi $$ as an element of $$\Gamma $$, we have, by (3.1), on $$\pi ^{-1}(U_i)$$:   \[ \pi^*\omega= \frac{\pi^*s_i^*\omega_0}{\pi^*s_i^*f_i}f_i= \frac{1}{\rho(s_i\pi)}\omega_0\frac{f_i}{(s_i\pi)^*f_i}=\omega_0. \] To prove the last part, suppose $$M$$ is THS and let, as in Theorem 2.5, $$\tilde M=\mathbb {C}^{2l}\times M_1\,{\times }\,{\cdots }\,{\times }\,M_k$$, with $$M_i$$ irreducible hyperkähler manifolds. The manifold $$M$$ has a finite étale cover $$M'=\mathbb {T}^{2l}\times M_1\times \cdots M_k$$ so that $$M=M'/ \Gamma '$$ and $$\Gamma \cong \mathbb {Z}^{4l}\ltimes \Gamma '$$. The symplectic form $$\omega _0$$ is preserved under the action of $$\mathbb {Z}^{4l}$$, so it descends to a holomorphic symplectic form on $$M'$$, which we will also denote by $$\omega _0$$. The group $$\Gamma '$$ preserves $$\mathbb {C}\omega _0$$. Let $$\rho ':\Gamma '\rightarrow U(1)$$ be the representation induced by $$\rho $$. Denote by $$N'$$ its kernel, and by $$N:=\mathbb {Z}^{4l}\ltimes N'$$. Then $$N$$ is normal inside $$\Gamma $$, so there exists a Galois covering $$M_N\rightarrow M$$ with $$\pi _1(M_N)=N$$. Moreover, since $$\pi _1(M')=\mathbb {Z}^{4l}$$ is normal in $$N$$, also $$M'\rightarrow M_N$$ is a covering whose deck transformation group is $$N/\mathbb {Z}^{4l}\cong N'$$. We thus have that $$M_N\cong M'/N'$$ and $$N'$$ preserves $$\omega _0$$, so $$\omega _0$$ descends to $$M_N$$. Since $$M_N$$ is compact holomorphic symplectic, it is hyperkähler. Finally, $$\rho (\Gamma )=\rho '(\Gamma ')$$ is a finite subgroup of U(1), so cyclic, and $$\Gamma /N\cong \Gamma '/N'\cong \rho (\Gamma )$$, so $$M_N$$ is a finite cyclic covering of $$M$$. This concludes the proof of the theorem. □ Corollary 3.2. A compact strictly THS manifold of dimension greater than 2 is de Rham irreducible. Proof. Suppose $$M\cong M_1\times M_2$$ is strictly THS. Let $$\tilde {M}\cong \tilde {M_1}\times \tilde {M_2}$$ be a finite étale cover of $$M$$ with holomorphic symplectic form $$\omega _0=\omega _1+\omega _2$$ preserved up to constants by $$\Gamma '\cong \Gamma _1'\times \Gamma _2'$$, where $$\pi _1(M_i)=\mathbb {Z}^{2l_i}\ltimes \Gamma _i'$$, $$i=1,2$$. Then we should have that $$\rho (\Gamma ')=\rho (\Gamma _1')\times \rho (\Gamma _2')$$ is a non-trivial cyclic group of the same order as $$\rho (\Gamma _1')$$, $$\rho (\Gamma _2')$$, which is impossible. □ Corollary 3.3. A compact locally irreducible Kähler manifold of dimension greater than 2 is KLH if and only if it is THS. In this case, the twisted-symplectic form is valued in the canonical bundle. Proof. Let $$M$$ be a locally irreducible KLH manifold and $$\tilde M$$ its universal cover endowed with a holomorphic symplectic form $$\omega _0$$. Since $$\tilde M$$ is irreducible, it is compact and $$H^0(\tilde M,\Omega _{\tilde M}^2)=\mathbb {C}\omega _0$$. Hence $$\Gamma =\pi _1(M)$$ preserves $$\mathbb {C}\omega _0$$ in a trivial way and $$M$$ is THS by the previous theorem. In particular, this implies that $$\Gamma $$ is cyclic. Let $$d$$ be its order. Then $$d|m+1$$, where dim$$M\,{=}\,2m$$. To see this, let $$\gamma \in \Gamma $$ be a generator, so that $$\gamma ^*\omega _0=\xi \cdot \omega _0$$, with $$\xi $$ a primitive $$d$$-root of unity. Since $$\gamma $$ has no fixed points, by the holomorphic Lefschetz fixed-point formula we must have that its Lefschetz number, which by definition is   \[ L(\gamma)=\sum_q (-1)^q\,{\rm tr}\gamma^*|_{H^q(\tilde M,\mathcal{O}_{\tilde M})} \] must vanish. On the other hand, we have   \[ \overline{H}^*(\tilde M,\mathcal{O}_{\tilde M})\cong H^0(\tilde M,\Omega^*_{\tilde M}) \cong\frac{\mathbb{C}[\omega_0]}{(\omega_0^{m+1})} \] so $$L(\gamma )=1+\xi +\cdots +\xi ^m$$. Thus, $$L(\gamma )=0$$ implies $$d|m+1$$. Let $$\rho :\Gamma \rightarrow $$U(1) be given by the action of $$\Gamma $$ on $$\omega _0$$ and $$L:=\tilde M\times _{\overline {\rho }}\mathbb {C}$$, so that the THS form is $$L$$-valued. Since the action of $$\Gamma $$ on $$K_{\tilde M}$$ is given by $$\rho ^m$$, we also have that $$K_M=\tilde M\times _{\rho ^m}\mathbb {C}$$. Now, $$\rho ^{m+1}=1$$ implies $$\overline {\rho }^m\cdot \overline {\rho }=1$$, or also $$K_M^*\otimes L=\underline {\mathbb {C}}$$, that is, $$L\cong K_M$$. □ Remark 3.4. For a THS manifold $$(M,I,L,\omega )$$, we always have, by Remark 2.3, that $$L$$ is a root of $$K_M^*$$. In the particular case when $$M$$ is locally irreducible, we obtain, moreover, that $$L$$ is precisely (up to isomorphism) $$K_M$$. It is difficult to give a nice criterion for being THS in the case of de Rham irreducible, locally reducible KLH manifolds. We can, though, give a somewhat more precise description of fundamental groups of THS manifolds. For this, we first give some lemmas concerning isometries of Riemannian products. Lemma 3.5. Let$$(M_i,g_i)$$be complete locally irreducible Riemannian manifolds of dimension bigger than 1 and let$$M_0=M_1\times \cdots \times M_k$$be endowed with the product metric. Let$$\gamma $$be an isometry of$$M_0$$and let$$\gamma _i:=p_i\gamma ,$$where$$p_i:M_0\rightarrow M_i$$are the canonical projections. Then$$\gamma _i$$is of the form$$\gamma _i=\tilde \gamma _ip_{\sigma (i)},$$where$$\tilde \gamma _i:M_{\sigma (i)}\rightarrow M_i$$is an isometry and$$\sigma $$a permutation of$$\{1,\ldots ,k\}$$. Proof. We have that $$\tilde g_i:=\gamma _i^* g_i$$ is a parallel section of $$S^2(T^*M_0)$$. On the other hand,   \[ S^2(T^*M_0)\cong \sum S^2(T^*M_i)\oplus \sum_{i<j}(T^*M_i\otimes T^*M_j). \] Now, $$T^*M_i\otimes T^*M_j$$ admits no parallel section for $$i<j$$, while the space of parallel sections of $$S^2(T^*M_i)$$ is exactly $$\mathbb {R} g_i$$. Indeed, by the holonomy principle, this is equivalent to saying that $$G_i\times G_j$$ has no fixed points when acting on $$T_x^*M_i\otimes T_y^*M_j$$, while the only $$G_i$$-invariant elements of $$S^2(T_x^*M_i)$$ are the multiples of $$(g_i)_x$$, where $$x\in M_i$$, $$y\in M_j$$ are any points and $$G_s$$ is the restricted holonomy group of $$M_s$$, $$s=1,\ldots ,k$$. The first assertion follows from the dimension hypothesis and the more general fact that if $$U$$ is a $$G$$-irreducible space and $$V$$ an $$H$$-irreducible space, then $$U\otimes V$$ is a $$G\times H$$-irreducible space. The second assertion is equivalent to Schur's lemma if we identify $$S^2(T_x^*M_i)$$ with the symmetric endomorphisms of $$T_x^*M_i$$ via $$g_i$$. Next, we want to show that for every $$i$$, there is exactly one $$j=j(i)$$ so that $$a_{ij}\neq 0$$. Thus, if we let $$A(i)=\{j|a_{ij}\neq 0\}$$, then we need to show that $$A(i)\neq \emptyset $$ for each $$i$$ and $$A(i)\cap A(j)\,{=}\,\emptyset $$ for all $$i\neq j$$. Now, since $$g_i$$ is definite and $$d\gamma _i$$ is surjective, we have that $$\ker \tilde g_i:=\{X \in TM_0| \tilde g_i(X,\cdot )=0\}=\ker d\gamma _i$$. Hence, since $$\ker d\gamma _i\neq TM$$, the first assertion follows. For the second assertion, first note that $$\ker \tilde g_i\cap TM_k\neq 0$$ if and only if $$a_{ik}=0$$, in which case $$TM_k\subset \ker \tilde g_i$$. Therefore, $$(\ker d\gamma _i)^\perp =\sum _{j\in A(i)} TM_j$$. Hence, for $$i\neq j$$, $$A(i)\cap A(j)=\emptyset $$ is equivalent to $$\{0\}=(\ker d\gamma _i)^\perp \cap (\ker d\gamma _j)^\perp =(\ker d\gamma _i + \ker d\gamma _j)^\perp $$. But we have   \[ \ker d\gamma_i + \ker d\gamma_j=d\gamma^{-1}(\ker d p_i+ \ker dp_j)=d\gamma^{-1}\left(\sum_{s\neq i} TM_s+\sum_{s\neq j}TM_s\right)=TM. \] It follows that there exists a permutation $$\sigma $$ of $$\{1,\ldots ,k\}$$ so that $$A(i)=\{\sigma (i)\}$$ for each $$i$$. Hence, since for any $$j$$, $$\sum _i a_{ij}=1$$, we have that $$a_{i\sigma (i)}=1$$ and $$\gamma _i=\tilde \gamma _ip_{\sigma (i)}$$ with $$\tilde \gamma _i:M_{\sigma (i)}\rightarrow M_i$$ an isometry. □ In what follows, we will omit writing the projections and identify $$\gamma _i$$ with $$\tilde \gamma _i$$. Lemma 3.6. Let$$M_i$$be irreducible compact hyperkähler manifolds, and$$M_0\,{=}\,M_1\,{\times }\,{\cdots }\,{\times }\,M_k$$be endowed with the product metric and a holomorphic symplectic form$$\omega _0$$. Then any isometry of$$M_0$$preserving$$\omega _0$$has fixed points. Proof. By Theorem 2.1, the manifolds $$M_i$$ are simply connected and admit unique holomorphic symplectic forms $$\omega _i$$ up to a scalar, so we have   \[ H^0(M_0,\Omega_{M_0}^2)=\mathbb{C}\omega_1\oplus\cdots\oplus\mathbb{C}\omega_k. \] Hence we can suppose, after rescaling each $$\omega _i$$, that $$\omega _0=\omega _1+\cdots +\omega _k$$. Let $$\gamma $$ be an isometry of $$M_0$$ with $$\gamma ^*\omega _0=\omega _0$$. Consider first the case where all $$M_i$$ are isometric, so that $$M_0\cong M_1^k$$. Let $$\sigma $$ be the permutation determined by $$\gamma $$ as in the previous lemma and let $$l$$ be the order of $$\sigma $$. If we define, for $$i=1,\ldots ,k,$$  \[ \gamma'_i=\gamma_i\gamma_{\sigma(i)}\cdots\gamma_{\sigma^{l-1}(i),} \] then $$\gamma ^l(x_1,\ldots ,x_k)=(\gamma '_1(x_1),\ldots ,\gamma '_k(x_k))$$. If $$\gamma $$ acts freely, then also $$\gamma ^l$$ acts freely. Otherwise, suppose $$\gamma ^l(y_1,\ldots ,y_k)=(y_1,\ldots ,y_k)$$. Let $$i_1,\ldots ,i_t \in \{1,\ldots ,k\}$$ represent the orbits of $$\langle \sigma \rangle $$ of cardinal $$l_1,\ldots ,l_t$$, and define $$(x_1,\ldots ,x_k)$$ by   \[ x_{i_\alpha}:=y_{i_\alpha}\quad \hbox{and} \quad x_{\sigma^j(i_\alpha)}:= \gamma_{\sigma^j(i_\alpha)}\cdots\gamma_{\sigma^{l_\alpha-1}(i_\alpha)}(y_{i_\alpha}) \] for $$\alpha =1,\ldots , t$$ and $$j=1,\ldots ,l_\alpha -1$$. The fact that $$\gamma _{i_\alpha }\gamma _{\sigma (i_\alpha )}\cdots \gamma _{\sigma ^{l_\alpha -1}(i_\alpha )}(y_{i_\alpha })=y_{i_\alpha }$$ implies that $$(x_1,\ldots ,x_k)$$ is a fixed point for $$\gamma $$, contradiction. Now, $$\gamma ^*\omega _0=\omega _0$$ implies $$(\gamma ^l)^*\omega _0=\sum _i(\gamma _i')^*\omega _1=\omega _0$$, or also $$(\gamma '_i)^*\omega _1=\omega _1$$ for any $$i=1,\ldots ,k$$. On the other hand, the fact that $$\gamma ^l$$ acts freely implies that some $$\gamma '_{i_0}$$ acts freely on $$M_1$$. By the holomorphic Lefschetz fixed-point formula, its Lefschetz number must then vanish. But $$L(\gamma '_{i_0})=m+1$$, where dim$$M_1=2m$$, contradiction. In the general situation, write $$M_0=(M_1)^{k_1}\times \cdots \times (M_s)^{k_s}$$, with $$M_i$$ irreducible and $$M_i\ncong M_j$$ for all $$i\neq j$$. By the previous lemma, $$\gamma =(\gamma _1,\ldots , \gamma _s)$$, with $$\gamma _i$$ an isometry of $$(M_i)^{k_i}$$. Again, $$\gamma ^*\omega _0=\omega _0$$ implies $$\gamma _i^*\tilde \omega _{i}=\tilde \omega _{i}$$, where $$\tilde {\omega }_i$$ is the induced symplectic form on $$(M_i)^{k_i}$$, $$i=1,\ldots ,s$$. Also, if $$\gamma $$ acts freely on $$M_0$$, then some $$\gamma _i$$ acts freely on $$(M_i)^{k_i}$$ and we already showed that this is impossible. □ Remark 3.7. We can now say slightly more about THS manifolds $$M$$ with compact universal cover $$\tilde M$$. In this case, with the notations of Theorem 3.1, $$\Gamma =\Gamma '$$, the representation $$\rho $$ is faithful by the previous lemma, so $$\Gamma =\rho (\Gamma )$$ is cyclic. Thus, if $$\gamma $$ is a generator of $$\Gamma $$ of order $$d$$ and $$\gamma ^*\omega _0=\xi \omega _0$$, then $$\xi $$ is necessarily a primitive $$d$$-root of unity. Moreover, if we write $$\gamma =(\gamma _1,\ldots ,\gamma _k)$$ just as in Lemma 3.5, then each $$\gamma _i$$ must have the same order $$d$$. To see this, let $$d_i={\rm ord}\gamma _i$$. Then $$d_i|d={\rm lcm}(d_i)_i$$. Since $$\gamma ^*\omega _0=\sum _i\gamma _i^*\omega _i=\xi \sum _ip^*_i\omega _i$$, we have, for all $$i$$, $$\gamma _i^*\omega _i=\xi \omega _{\sigma (i)}$$, hence $$\xi ^{d_i}=1$$. But $$\xi $$ was primitive, so $$d_i=d$$. We can conclude the following. Corollary 3.8. If the fundamental group of a compact THS manifold is finite, then it is cyclic and of the form$$\Gamma =\langle \gamma =(\gamma _1,\ldots ,\gamma _k)\rangle ,$$with$$\gamma _i$$isometries of the irreducible components of the universal cover, all of the same order. Remark 3.9. When $$M$$ is THS but $$\tilde M$$ is not compact, it is not necessarily the case for $$\Gamma '$$ to be cyclic, that is, $$\rho ':\Gamma '\rightarrow $$U(1) need not be faithful. By the same type of arguments as in Lemma 3.5, it can be seen that an element of $$\Gamma '$$ is of the form $$\gamma =(\gamma _T,\gamma _0)$$, with $$\gamma _T\in {\rm Aut}(\mathbb {T}^{2l})$$ and $$\gamma _0\in {\rm Aut}(M_0)$$. There exist fixed point free complex symplectomorphisms of $$\mathbb {T}^{2l}$$ of finite order (for instance, translation by a torsion element $$a \in \mathbb {T}^{2l}$$). So, if $$\gamma _T$$ is one and $$\gamma _0$$ is a symplectomorphism of $$M_0$$ of the same order as $$\gamma _T$$, then $$(\gamma _T,\gamma _0)$$ is an element in the kernel of $$\rho '$$. Corollary 3.10. A compact strictly THS manifold$$M$$of dimension greater than 2 with finite fundamental group is projective. Proof. Let $$\pi :\tilde M\rightarrow M$$ be the compact universal covering, where, by Theorem 2.5, $$\tilde M=M_1\times \cdots \times M_k$$ with $$M_i$$ irreducible hyperkähler manifolds. Then, by Lemma 3.6, each $$M_i$$ admits an automorphism that is not symplectic. By a result of [1], such manifolds are necessarily projective, so is $$\tilde M$$. But it is a well-known fact that a compact Kähler manifold is projective if and only if any of its finite unramified covering is projective, thus the conclusion follows. □ 4. Final remarks Remark 4.1. Concerning examples of Kähler type, finding locally irreducible KLH manifolds is equivalent to finding a fixed point free automorphism $$\gamma $$ of an irreducible symplectic manifold, so that all powers of $$\gamma $$ also act freely. In complex dimension 2, by Remark 2.4, all manifolds are THS. On the other hand, the only finite cyclic quotients of hyperkähler surfaces are the Enriques surfaces and some bielliptic surfaces. The first ones are quotients $$K/_{\langle \iota \rangle }$$, with $$K$$ a K3 surface admitting a fixed point free involution $$\iota $$, and they are locally irreducible. The bielliptic surfaces are quotients of products of two elliptic curves by some groups of order 2, 3, 4 or 6. This shows that Theorem 3.1 does not hold in complex dimension 2. Next, one can easily construct locally reducible THS manifolds of any dimension by iterating the KLH examples from above. For instance, if $$(K, \iota )$$ is a K3 surface as before, then $${K^{m}}/_{\langle \iota ,\ldots ,\iota \rangle }$$ is THS. In the same way, if $$A=\mathbb {C}/_{(\mathbb {Z}\oplus i\mathbb {Z})}$$ and $$P$$ is a 4-torsion point on $$A$$, defining $$f\in {\rm Aut}(A^{2m})$$ by $$f(X_1,\ldots ,X_{2m}):=(iX_1,\ldots ,iX_m, X_{m+1}+P,\ldots ,X_{2m}+P)$$ gives the THS manifold $$A^{2m}/_{\langle f\rangle }$$. More generally, if $$M_1$$ and $$M_2$$ are hyperkähler and $$\gamma _i\in {\rm Aut}(M_i)$$, $$i=1,2$$, have the same order, with $$\langle \gamma _1\rangle $$ acting freely but $$\langle \gamma _2\rangle $$ possibly not, then $$(M_1\times M_2)/_{\langle \gamma _1,\gamma _2\rangle }$$ is again a smooth THS manifold. Finding locally irreducible THS manifolds of higher dimension is more difficult. For the Hilbert schemes of points on K3 surfaces, see [2] for the construction, all known automorphisms have fixed points, so we have no hope of constructing examples out of them. On the other hand, there is hope with the generalized Kummer varieties $$K_r$$, see again [2] for the definition. In [3, 9], the authors find fixed point free cyclic groups of automorphisms $$\Gamma $$ of order 3 for the manifolds $$K_2$$ and $$K_5$$, and of order 4 for $$K_3$$. The corresponding quotients give the desired examples of dimension 4, 10 and 6, respectively. Remark 4.2. In order to actually classify Kählerian THS manifolds, one should be able to classify fixed point free groups of automorphisms of irreducible hyperkähler manifolds. The problem is clear in low dimension. It is also clear that if the Hilbert schemes of points on K3 surfaces admit such groups, then the corresponding automorphisms are not natural, that is, do not arise from automorphisms of the K3 surface. On the other hand, for the moment the only known non-natural automorphisms have fixed points. For the generalized Kummer varieties, there exist some examples formed out of natural automorphisms, but we do not know a classification of such groups. Remark 4.3. As already stated, the natural generalizations of the subject studied here are widely open problems. On the one hand, one could study compact Kähler manifolds with non-degenerate $$(2,0)$$ forms, or equivalently with $${\rm Sp}(m)U(1)$$ topological structure, see Remark 2.9. On the other hand, one could drop the Kähler assumption, and a number of new questions arise, as seen in Remark 2.10. We will address these in a forthcoming paper. Acknowledgments I thank my PhD advisor Andrei Moroianu for introducing me to the topic and for the enlightening remarks, and my co-advisor Simone Diverio for showing interest in the subject and for the valuable comments. I also thank Alexandra Otiman for the fruitful non-Kählerian discussions, and the anonymous referee for useful suggestions. References 1 A. Beauville, ‘ Some remarks on Kähler manifolds with $$c_1 = 0$$’, Prog. Math.  39 ( 1983) 1– 26. 2 A. Beauville, ‘ Variétés Kähleriennes dont la première classe de Chern est nulle’, J. Differential Geom  18 ( 1983) 755– 782. 3 S. Boissiére, M. Nieper-Wißkirchen and A. Sarti, ‘ Higher-dimensional Enriques varieties and automorphisms of generalized Kummer varieties’, J. Math. Pures Appl.  95 ( 2011) 553– 563. Google Scholar CrossRef Search ADS   4 M. Inoue, S. Kobayashi and T. Ochiai, ‘ Holomorphic affine connections on compact complex surfaces’, J. Fac. Sci. Univ. Tokyo Sect. IA Math.  27 ( 1980) 247– 264. 5 P. Jahnke and I. Radloff, ‘ Projective threefolds with holomorphic conformal structure’, Int. J. Math.  16 ( 2005) 595– 607. Google Scholar CrossRef Search ADS   6 S. Kobayashi and T. Ochiai, ‘ Holomorphic structures modeled after hyperquadrics’, Tôhoku Math. J.  ( 2) 34 ( 1982) 587– 629. Google Scholar CrossRef Search ADS   7 A. Moroianu, Lectures on Kähler geometry , London Mathematical Society Student Texts 69 ( Cambridge University Press, Cambridge, 2007). 8 A. Moroianu, M. Pilca and U. Semmelmann, ‘ Homogeneous almost quaternion-Hermitian manifolds’, Math. Ann.  357 ( 2013) 1205– 1216. Google Scholar CrossRef Search ADS   9 K. Oguiso and S. Schröer, ‘ Enriques manifolds’, J. reine angew. Math.  661 ( 2011) 215– 235. Google Scholar CrossRef Search ADS   10 V. Tosatti, ‘ Non-Kähler Calabi–Yau manifolds’, Contemp. Math.  644 ( 2015) 261– 277. © 2016 London Mathematical Society

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Bulletin of the London Mathematical Society – Oxford University Press

Published: Jul 14, 2016

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