Finiteness results for hyperkaehler manifolds

Subvarieties in non-compact hyperkaehler manifolds

Let M be a hyperkaehler manifold, not necessarily compact, and $S\cong CP^1$ the set of complex structures induced by the quaternionic action. Trianalytic subvariety of M is a subvariety which is complex analytic with respect to all $I \in CP^1$. We show that for all $I \in S$ outside of a countable set, all compact complex subvarieties $Z \subset (M,I)$ are trianalytic. For M compact, this result was proven in alg-geom/9403006 using Hodge theory.

On minimal models of projective Hyperkaehler manifolds

Any minimal model of a projective Hyperkaehler manifold is a projective Hyperkaehler manifold. As a consequence, moduli spaces of sheaves on a k3 that don't admit a symplectic resolution are not birational to Hyperkaehler manifolds.

Families of Lagrangian fibrations on hyperkaehler manifolds

A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known that a given compact manifold admits only finitely many holomorphic symplectic structures, up to deformation. We prove that a given compact manifold with $b_2 \geq 7$ admits only finitely many deformation types of holomorphic Lagrangian fibrations. We also prove that all known hyperkahler manifolds are never Kobayashi hyperbolic.

Erratum to the paper: Compact hyperkaehler manifolds: basic results

This is an Erratum to the paper: Compact hyperkaehler manifolds: basic results. (alg-geom/9705025, Inv. math. 135). We give a correct proof of the projectivity criterion for hyperkaehler manifolds. We use a recent result of Demailly and Paun math.AG/0105176.

Hyperkaehler manifolds with torsion, supersymmetry and Hodge theory

Let M be a hypercomplex Hermitian manifold, (M,I) the same manifold considered as a complex Hermitian with a complex structure I induced by the quaternions. The standard linear-algebraic construction produces a canonical nowhere degenerate (2,0)-form on (M,I). It is well known that M is hyperkaehler if and only if the form Ωis closed with respect to the de Rham differential. The M is called HKT (hyperkaehler with torsion) if this form is closed with respect to the Dolbeault differential (this condition is weaker. Conjecturally, all compact hypercomplex manifolds admit an HKT-metrics. We exploi…

The Kaehler cone of a compact hyperkaehler manifold

This is an attempt towards the understanding of the (birational) Kaehler cone of a compact hyperkaehler manifold in terms of the Beauville-Bogomolov form on its second cohomology. We discuss birational correspondences between hyperkaehler manifolds and their effects on the cohomology. Many of the results are conjectural in as much as they depend on a projectivity criterion for compact hyperkaehler manifolds contained in this paper's predecessor, but in which a serious mistake has oocured. An erratum is given in Sect. 6 and a way to rescue the approach is proposed in Sect. 7.

Complex Lagrangians in a hyperKaehler manifold and the relative Albanese

Let $M$ be the moduli space of complex Lagrangian submanifolds of a hyperKähler manifold $X$, and let $\varpi : \widehat{\mathcal{A}} \rightarrow M$ be the relative Albanese over $M$. We prove that $\widehat{\mathcal{A}}$ has a natural holomorphic symplectic structure. The projection $\varpi$ defines a completely integrable structure on the symplectic manifold $\widehat{\mathcal{A}}$. In particular, the fibers of $\varpi$ are complex Lagrangians with respect to the symplectic form on $\widehat{\mathcal{A}}$. We also prove analogous results for the relative Picard over $M$.

Parabolic nef currents on hyperkaehler manifolds

Let M be a compact, holomorphically symplectic Kahler manifold, and $η$ a (1,1)-current which is nef (a limit of Kahler forms). Assume that the cohomology class of $η$ is parabolic, that is, its top power vanishes. We prove that all Lelong sets of $η$ are coisotropic. When M is generic, this is used to show that all Lelong numbers of $η$ vanish. We prove that any hyperkahler manifold with Pic(M) of rank 1 has non-trivial coisotropic subvarieties, if a generator of Pic(M) is parabolic.

Moduli spaces of hyperkaehler manifolds and mirror symmetry

These are notes of my lectures given at the school on intersection theory and moduli at the ICTP, Trieste. We construct moduli spaces of K3 surfaces and higherdimensional hyperkaehler manifolds, including moduli spaces of (2,2)-conformal field theories associated to hyperkaehler manifolds, and describe their period domains. Recent results on these manifolds are discussed from the moduli space point of view. The second goal is to present a detailed account of mirror symmetry for K3 surfaces. Arguments due to Aspinwall and Morrison allow one to explain various versions of mirror symmetry (e.g. f…

Mirror Symmetry for hyperkaehler manifolds

We prove the Mirror Conjecture for Calabi-Yau manifolds equipped with a holomorphic symplectic form. Such manifolda are also known as complex manifolds of hyperkaehler type. We obtain that a complex manifold of hyperkaehler type is Mirror dual to itself. The Mirror Conjecture is stated (following Kontsevich, ICM talk) as the equivalence of certain algebraic structures related to variations of Hodge structures. We compute the canonical flat coordinates on the moduli space of Calabi-Yau manifolds of hyperkaehler type, introduced to Mirror Symmetry by Bershadsky, Cecotti, Ooguri and Vafa.

Picard number of the generic fiber of an abelian fibered hyperkaehler manifold

We shall show that the Picard number of the generic fiber, in the sense of scheme, of an abelian fibered hyperkähler manifold over the projective space is always one. We then give a few applications for the Mordell-Weil group. In particular, by deforming O'Grady's 10-dimensional manifold, we construct an abelian fibered hyperkähler manifold of Mordell-Weil rank 20, which is the maximum possible among all known ones.

On families of lagrangian tori on hyperkaehler manifolds

This is a note on Beauville's problem (solved by Greb, Lehn and Rollenske in the non-algebraic case and by Hwang and Weiss in general) whether a lagrangian torus on an irreducible holomorphic symplectic manifold is a fiber of a lagrangian fibration. We provide a different, very short solution in the non-algebraic case and make some observations suggesting a different approach in the algebraic case.

Algebraic structures on hyperkaehler manifold

Let $M$ be a compact hyperkaehler manifold. The hyperkaehler structure equips $M$ with a set $R$ of complex structures parametrized by $CP^1$, called "the set of induced complex structures". It was known previously that induced complex structures are non-algebraic, except may be a countable set. We prove that a countable set of induced complex structures is algebraic, and this set is dense in $R$. A more general version of this theorem was proven by Fujiki.

Cohomology of compact hyperkaehler manifolds

Let M be a compact simply connected hyperkähler (or holomorphically symplectic) manifold, \dim H^2(M)=n. Assume that M is not a product of hyperkaehler manifolds. We prove that the Lie algebra so(n-3,3) acts by automorphisms on the cohomology ring H^*(M). Under this action, the space H^2(M) is isomorphic to the fundamental representation of so(n-3,3). Let A^r be the subring of H^*(M) generated by H^2(M). We construct an action of the Lie algebra so(n-2,4) on the space A, which preserves A^r. The space A^r is an irreducible representation of so(n-2,4). This makes it possible to compute the ring…

Any component of moduli of polarized hyperkaehler manifolds is dense in its deformation space

Let M be a compact hyperkaehler manifold, and W the coarse moduli of complex deformations of M. Every positive integer class v in $H^2(M)$ defines a divisor $D_v$ in W consisting of all algebraic manifolds polarized by v. We prove that every connected component of this divisor is dense in W.

Hyperholomorphic sheaves and new examples of hyperkaehler manifolds

Given a compact hyperkaehler manifold $M$ and a holomorphic bundle B over $M$, we consider a Hermitian connection $\nabla$ on B which is compatible with all complex structures on $M$ induced by the hyperkaehler structure. Such a connection is unique, because it is Yang-Mills. We call the bundles admitting such connections hyperholomorphic bundles. A stable bundle is hyperholomorphic if and only if its Chern classes $c_1$, $c_2$ are SU(2)-invariant, with respect to the natural SU(2)-action on the cohomology. For several years, it was known that the moduli space of stable hyperholomorphic bundle…

Intersection forms of toric hyperkaehler varieties

This note proves combinatorially that the intersection pairing on the middle dimensional compactly supported cohomology of a smooth toric hyperkaehler variety is always definite, providing a large number of non-trivial L^2 harmonic forms for toric hyperkaehler metrics on these varieties. This is motivated by a result of Hitchin about the definiteness of the pairing of L^2 harmonic forms on complete hyperkaehler manifolds of linear growth.

Compact Hyperkaehler Manifolds: Basic Results

The paper generalizes some of the well-known results for K3 surfaces to higher-dimensional irreducible symplectic (or, equivalently, compact irreducible hyperkaehler) manifolds. In particular, we discuss the projectivity of such manifolds and their ample resp. Kaehler cones. It is proved that the period map surjects any non-empty connected component of the moduli space of marked manifolds onto the corresponding period domain. We also establish an anlogue of the transitivity of the Weyl-action on the set of chambers of a K3 surface. As a converse of a higher-dimensional version of the `Main Lem…

Cohomology of compact hyperkaehler manifolds and its applications

This article contains a compression of results from alg-geom/9501001, with most proofs omitted. We prove that every two points of the connected moduli space of holomorphically symplectic manifolds can be connected with so-called ``twistor lines'' -- projective lines holomorphically embedded to the moduli space and corresponding to the hyperkähler structures. This has interesting implications for the geometry of compact hyperkähler manifolds and of holomorphic vector bundles over such manifolds.

On the Kobayashi pseudometric, complex automorphisms and hyperkaehler manifolds

We define the Kobayashi quotient of a complex variety by identifying points with vanishing Kobayashi pseudodistance between them and show that if a compact complex manifold has an automorphism whose order is infinite, then the fibers of this quotient map are nontrivial. We prove that the Kobayashi quotients associated to ergodic complex structures on a compact manifold are isomorphic. We also give a proof of Kobayashi's conjecture on the vanishing of the pseudodistance for hyperkähler manifolds having Lagrangian fibrations without multiple fibers in codimension one. For a hyperbolic automorphi…

Hyperkaehler and holomorphic symplectic geometry

We prove that all complex analytic subvarieties of a generic compact hyperkaehler manifold are even-dimensional. Moreover, these subvarieties are holomorphically symplectic.

Trianalytic subvarieties of generalized Kummer varieties

Let $X$ be a hyperkaehler manifold. Trianalytic subvarieties of $X$ are subvarieties which are complex analytic with respect to all complex structures induced by the hyperkaehler structure. Given a 2-dimensional complex torus $T$, the Hilbert scheme $T^{[n]}$ classifying zero-dimensional subschemes of $T$ admits a hyperkaehler structure. A finite cover of $T^{[n]}$ is a product of $T$ and a simply connected hyperkaehler manifold $K^{[n-1]}$, called generalized Kummer variety. We show that for $T$ generic, the corresponding generalized Kummer variety has no trianalytic subvarieties. This implie…

Hyperkaehler Embeddings II

In the first part, Hyperkaehler Embeddings and Holomorphic symplectic Geometry I, we prove the following. Let $N$ be a closed analytic subvariety of a generic deformation of a holomorphically symplectic compact manifold $M$. Then the restriction of a holomorphic symplectic form is non-degenerate on $N$. In particular, $N$ is even-dimensional. In present paper, we prove that there exist a hyperkaehler metric on $M$, such that the embedding of $N$ to $M$ is hyperkaehler.

Trihyperkahler reduction and instanton bundles on CP^3

A trisymplectic structure on a complex 2n-manifold is a triple of holomorphic symplectic forms such that any linear combination of these forms has constant rank 2n, n or 0, and degenerate forms in $Ω$ belong to a non-degenerate quadric hypersurface. We show that a trisymplectic manifold is equipped with a holomorphic 3-web and the Chern connection of this 3-web is holomorphic, torsion-free, and preserves the three symplectic forms. We construct a trisymplectic structure on the moduli of regular rational curves in the twistor space of a hyperkaehler manifold, and define a trisymplectic reductio…

Characteristic foliation on non-uniruled smooth divisors on hyperkaehler manifolds

We prove that the characteristic foliation $F$ on a non-singular divisor $D$ in an irreducible projective hyperkaehler manifold $X$ cannot be algebraic, unless the leaves of $F$ are rational curves or $X$ is a surface. More generally, we show that if $X$ is an arbitrary projective manifold carrying a holomorphic symplectic $2$-form, and $D$ and $F$ are as above, then $F$ can be algebraic with non-rational leaves only when, up to a finite étale cover, $X$ is the product of a symplectic projective manifold $Y$ with a symplectic surface and $D$ is the pull-back of a curve on this surface. When $D…

Positive forms on hyperkahler manifolds

Let $(M,I,J,K)$ be a hyperkaehler manifold, $\dim_\R M =4n$. We study positive, Dolbeault-closed $(2p,0)$-forms on $(M,I)$. These forms are quaternionic analogues of the positive $(p,p)$-forms. We construct an injective homomorphism mapping Dolbeault-closed $(2p,0)$-forms to closed $(n+p,n+p)$-forms, and positive $(2p,0)$-forms to positive $(n+p,n+p)$-forms. This construction is used to prove a hyperkaehler version of the classical Skoda-El Mir theorem, which says that a trivial extension of a closed, positive current over a pluripolar set is again closed. We also prove the hyperkaehler versio…

Compact tori associated to hyperkaehler manifolds of Kummer type

Let X be a hyperkaehler manifold of Kummer type. We study the 4 dimensional intermediate Jacobian J^3(X) constructed out of the 3rd cohomology of X, motivated by the desire to understand the Kuga-Satake torus associated to X. We prove that there is an embedding of Hodge structures of H^2(X) into H^2(J^3(X)). It follows that if X is projective with polarization L, the projective Kuga-Satake torus of (X,L) is isogenous to the fourth power of J^3(X). By studying the cohomology ring of X, we find that there is a natural smooth quadric Q(X) in the projectivization of H^3(X), with a natural choice o…

Hyperholomorphic bundles

Hyperholomorphic bundle is a bundle with connection defined over a hyperkaehler manifold such that this connection is holomorphic with respect to all complex structures induced by a hyperkaehler structure. A hyperholomorphic connection is Yang-Mills. If a stable bundle has first two Chern classes invariant with respect to ${\Bbb H}$, then it admits a hyperholomorphic connection. Deformation spaces of hyperholomorphic bundles are hyperkaehler. There are no higher obstructions to a deformation besides Ioneda product. This deformation space does not depend on a base manifold.

HyperKähler Manifolds and Birational Transformations

In this paper we study the birational geometry of HyperKaehler manifolds by combining the method of minimal model program and the traditional approach of symplectic geometry.

Model theory and Kaehler geometry

We survey and explain some recent work at the intersection of model theory and bimeromorphic geometry (classification of compact complex manifolds). Included here are the essential saturation of the many sorted structure $\mathcal{C}$ of compact Kaehler manifolds, the conjectural role of hyperkaehler manifolds in the description of strongly minimal sets in $\mathcal{C}$ and Campana's work on the isotriviality of hyperkaehler families and its connection with the nonmultidimensionality conjecture.

Deformation Finiteness for Real Hyperkahler Manifolds

We show that the number of equivariant deformation classes of real structures in a given deformation class of compact hyperkahler manifolds is finite.

A finiteness theorem for holonomic DQ-modules on Poisson manifolds

On a complex symplectic manifold we prove a finiteness result for the global sections of solutions of holonomic DQ-modules in two cases: (a) by assuming that there exists a Poisson compactification (b) in the algebraic case. This extends our previous results in which the symplectic manifold was compact. The main tool is a finiteness theorem for R-constructible sheaves on a real analytic manifold in a non proper situation.

Effective finiteness theorems for maps between canonically polarized compact complex manifolds

Effective bounds for the finite number of surjective holomorphic maps between canonically polarized compact complex manifolds of any dimension with fixed domain are proven. Both the case of a fixed target and the case of varying targets are treated. In the case of varying targets, bounds on the complexity of Chow varieties are used.

Survey of finiteness results for hyperkähler manifolds

This paper is a survey of finiteness results in hyperkähler geometry. We review some classical theorems by Sullivan, Kollár-Matsusaka, Huybrechts, as well as theorems in the recent literature by Charles, Sawon, and joint results of the author with Verbitsky. We also strengthen a finiteness theorem of the author. These are extended notes of the author's talk during the closing conference of the Simons Semester in the Banach Center in Bȩdlewo, Poland.

Finiteness of polarized K3 surfaces and hyperkähler manifolds

In the moduli space of polarized varieties the same unpolarized variety can occur multiple times However, for K3 surfaces, compact hyperkähler manifolds, and abelian varieties the number is finite. This may be viewed as a consequence of the Kawamata-Morrison cone conjecture. In this note we provide a proof of this finiteness not relying on the cone conjecture and, in fact, not even on the global Torelli theorem. Instead, it uses the geometry of the moduli space of polarized varieties to conclude the finiteness by means of Baily-Borel type arguments. We also address related questions concerning…

Rozansky-Witten invariants via formal geometry

We show that recently constructed invariants of 3-dimensional manifolds and of hyperkaehler manifolds (L.Rozansky and E.Witten, hep-th/9612216) come from characteristic classes of foliations and from Gelfand-Fuks cohomology. In particular, any symplectic foliation gives invariants of 3-manifolds. Our preprint has many intersections with the preprint alg-geom/9704009 by M.Kapranov.

A Finiteness Theorem for Quaternionic-Kaehler Manifolds with Positive Scalar Curvature

We study the topology and geometry of those compact Riemannian (4n)-manifolds (M,g), n > 1, with positive scalar curvature and holonomy in Sp(n)Sp(1). Up to homothety, we show that there are only finitely many such manifolds of any dimension 4n.

Supersymmetry and the cohomology of (hyper)Kaehler manifolds

The cohomology of a compact Kaehler (resp. hyperKaehler) manifold admits the action of the Lie algebra so(2,1) (resp. so(4,1)). In this paper we show, following an idea of Witten, how this action follows from supersymmetry, in particular from the symmetries of certain supersymmetric sigma models. In addition, many of the fundamental identities in Hodge-Lefschetz theory are also naturally derived from supersymmetry.

Higher-dimensional analogues of K3 surfaces

In the first part we survey some of the known results and conjectures on compact Hyperkaehler (HK) manifolds. In the second part we presents a program which aims to show that HK four-folds whose second cohomology (with 4-tuple cup-product) is isomorphic to that of the Hilbert square of a K3 enjoy many of the beautiful properties of K3 surfaces, in particular they should form a single deformations class.

Finiteness of Klein actions and real structures on compact hyperkähler manifolds

One central problem in real algebraic geometry is to classify the real structures of a given complex manifold. We address this problem for compact hyperkähler manifolds by showing that any such manifold admits only finitely many real structures up to equivalence. We actually prove more generally that there are only finitely many, up to conjugacy, faithful finite group actions by holomorphic or anti-holomorphic automorphisms (the so-called Klein actions). In other words, the automorphism group and the Klein automorphism group of a compact hyperkähler manifold contain only finitely many conjugac…

Hyperkahler SYZ conjecture and semipositive line bundles

Let $M$ be a compact, holomorphic symplectic Kaehler manifold, and $L$ a non-trivial line bundle admitting a metric of semi-positive curvature. We show that some power of $L$ is effective. This result is related to the hyperkaehler SYZ conjecture, which states that such a manifold admits a holomorphic Lagrangian fibration, if $L$ is not big.

Complex-analytic intermediate hyperbolicity, and finiteness properties

Motivated by the finiteness of the set of automorphisms Aut(X) of a projective manifold X, and by Kobayashi-Ochiai's conjecture that a projective manifold dim(X)-analytically hyperbolic (also known as strongly measure hyperbolic) should be of general type, we investigate the finiteness properties of Aut(X) for a complex manifold satisfying a (pseudo-) intermediate hyperbolicity property. We first show that a complex manifold X which is (dim(X) -- 1)-analytically hyperbolic has indeed finite automorphisms group. We then obtain a similar statement for a pseudo-(dim(X) -- 1)-analytically hyperbol…

Finiteness of simple homotopy type up to s-cobordism of aspherical 4-manifolds

In this paper we show that for a large class C of 4-manifolds each member of C has only finitely many simple homotopy type upto s-cobordism. This result generalizes a similar result of Hillman for certain complex surfaces. We also present a correction in the proof of Hillman's result.

Finiteness and infiniteness results for Torelli groups of (hyper-)Kähler manifolds

The Torelli group $\mathcal T(X)$ of a closed smooth manifold $X$ is the subgroup of the mapping class group $π_0(\mathrm{Diff}^+(X))$ consisting of elements which act trivially on the integral cohomology of $X$. In this note we give counterexamples to Theorem 3.4 of Verbitsky's paper "Mapping class group and a global Torelli theorem for hyperkähler manifolds" (Duke Math.~J.~162 (2013), no.~15, 2929-2986) which states that the Torelli group of simply connected Kähler manifolds of complex dimension $\ge 3$ is finite. This is done by constructing under some mild conditions homomorphisms $J: \mat…

Wirtinger numbers and holomorphic symplectic immersions

For any subvariety of a compact holomorphic symplectic Kaehler manifold, we define the number W(X), which we call Wirtinger number. We show that $W(X)\leq 1$, and the equality is reached if and only if the subvariety $X\subset M$ is trianalytic, i. e. compactible with the hyperkaehler structure on M. For a sequence $X_1 \arrow X_2 \arrow ... X_n\arrow M$ of immersions of simple holomorphic symplectic manifolds, we show that $W(X_1) \leq W(X_2) \leq >... \leq W(X_n)$.

极化Calabi--Yau流形有限性的多势理论方法

A Pluripotential-Theoretic Approach to Finiteness of Polarized Calabi--Yau Manifolds

本文提出多势理论框架，通过m-Hessian容量等估计连接复势理论与几何代数有界性，以研究极化Calabi--Yau流形的有限性，并指出关键待发展步骤。

On the quantum K-ring of the flag manifold

We establish a finiteness property of the quantum K-ring of the complete flag manifold.

复流形自同构的射影多重典范表示的有限性

Finiteness of projective pluricanonical representation for automorphisms of complex manifolds

本文研究紧复流形双亚纯自同构群在多重典范映射像上的射影多重典范表示，在 Kodaira 维数等于维数减一的条件下证明其像为有限群，并建立相对维数 1 的典范丛公式，从而验证 Prokhorov–Shokurov 猜想的解析版本。

自覆盖、有限性、交换性与环面上的纤维化

Self-covering, finiteness, commutativity, and fibering over tori

该研究探讨自覆盖流形的纤维化性质，证明满足温和假设、基本群为阿贝尔群的闭自覆盖流形可在环面上纤维化，给出高维自由阿贝尔基本群情形的推论，并构造非交换基本群下的非纤维化反例。