Deformation Finiteness for Real Hyperkahler Manifolds

Isometries of ideal lattices and hyperkähler manifolds

We prove that there exists a holomorphic symplectic manifold deformation equivalent to the Hilbert scheme of two points on a K3 surface that admits a non-symplectic automorphism of order 23, that is the maximal possible prime order in this deformation family. The proof uses the theory of ideal lattices in cyclomotic fields.

三十类紧致超kähler orbifold 的变形类

Thirty-three deformation classes of compact hyperkähler orbifolds

本文研究了通过商映射得到的不可约辛流形的变形类，探讨了在不同维度下通过终端化得到的orbifold例子，并计算了其奇异性质。

MBM loci in families of hyperkahler manifolds and centers of birational contractions

An MBM class on a hyperkahler manifold M is a second cohomology class such that its orthogonal complement in H^2(M) contains a maximal dimensional face of the boundary of the Kahler cone for some hyperkahler deformation of M. An MBM curve is a rational curve in an MBM class and such that its local deformation space has minimal possible dimension 2n-2, where 2n is the complex dimension of M. We study the MBM loci, defined as the subvarieties covered by deformations of an MBM curve within M. When M is projective, MBM loci are centers of birational contractions. For each MBM class z, we consider …

Families of hyperkähler manifolds

We show the density of the jumping loci of the Picard number of a hyperkähler manifold under small deformation and provide several applications. In particular, we apply this to reveal the structure of hierarchy among all the narrow Mordell-Weil lattices of Jacobian K3 surfaces.

Deformations of trianalytic subvarieties of hyperkähler manifolds

Let $M$ be a compact complex manifold equipped with a hyperkähler metric, and $X$ be a closed complex analytic subvariety of $M$. In alg-geom/9403006, we proved that $X$ is trianalytic, i. e., complex analytic with respect to all complex structures induced by the hyperkähler structure, provided that $M$ is generic in its deformation class. Here we study the complex analytic deformations of trianalytic subvarieties. We prove that all deformations of $X$ are trianalytic and naturally isomorphic to $X$ as complex analytic varieties. We show that this isomorphism is compatible with the metric indu…

Picard numbers in a family of hyperkähler manifolds - A supplement to the article of R. Borcherds, L. Katzarkov, T. Pantev, N. I. Shepherd-Barron

We remark the density of the jumping loci of the Picard number of a hyperkähler manifold under small one-dimensional deformation and provide some applications for the Mordell-Weil groups of Jacobian K3 surfaces.

Automorphisms of hyperkähler manifolds

The present paper provides several results on automorphisms of hyperkähler (or irreducible holomorphic symplectic) manifolds. In particular it focuses on the symplectic case and contains a classification of prime order symplectic automorphisms on deformations of the Hilbert scheme of two points on a K3 surface and, with additional hypothesis, also on Hilbert schemes of n points on a K3 surface. We also give numerical conditions to establish whether a symplectic automorfism of order 2,3 and 5 on a deformation of the Hilbert scheme of two points on a K3 surface is standard. Moreover we also give…

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…

稳定 Lagrangian 纤维化的有限性

Finiteness of stable Lagrangian fibrations

该研究综述 hyperkähler Lagrangian 纤维化形变类的有限性，并证明给定判别除子下稳定 Lagrangian 纤维化的有限性。

Rational curves and MBM classes on hyperkahler manifolds: a survey

This paper deals with rational curves and birational contractions on irreducible holomorphically symplectic manifold. We survey some recent results about minimal rational curves, their deformations, extremal rays associated with these curves, and the geometry of the Kähler cone.

Geometry of moduli spaces of meromorphic connections on curves, Stokes data, wild nonabelian Hodge theory, hyperkahler manifolds, isomonodromic deformations, Painleve equations, and relations to Lie theory

Summary of main work 1999-2012

Rational curves on hyperkahler manifolds

Let $M$ be an irreducible holomorphically symplectic manifold. We show that all faces of the Kahler cone of $M$ are hyperplanes $H_i$ orthogonal to certain homology classes, called monodromy birationally minimal (MBM) classes. Moreover, the Kahler cone is a connected component of a complement of the positive cone to the union of all $H_i$. We provide several characterizations of the MBM-classes. We show the invariance of MBM property by deformations, as long as the class in question stays of type (1,1). For hyperkahler manifolds with Picard group generated by a negative class $z$, we prove tha…

反射格与超凯勒流形

Reflective lattices and hyperkahler manifolds

利用Nikulin和Vinberg关于反射生成的等距群的结果，在极化超凯勒流形的模空间中构造了称为Nikulin-Vinberg轨迹的子簇，并证明了具有b_2(M)≥6的超凯勒流形的任何非平凡射影变形族具有稠密的纤维集，其双有理自同构群是无限的。

紧超凯勒流形在诱导辛自同构下的商空间的终化

Terminalizations of quotients of compact hyperkähler manifolds by induced symplectic automorphisms

通过分类K3曲面或广义Kummer簇的Hilbert概形在有限辛自同构群作用下的商空间的终化，确定其第二Betti数和正则基本群，并得到至少九个新的不可约辛四维簇的形变类型。

Kobayashi pseudometric on hyperkahler manifolds

The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaré disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture for any hyperkähler manifold that admits a deformation with two Lagrangian fibrations and whose Picard rank is not maximal. The Strominger-Yau-Zaslow (SYZ) conjecture claims that parabolic nef line bundles on hyperkähler manifolds are semi-ample. We prove that the Kobayashi pseudom…

Contraction centers in families of hyperkahler manifolds

We study the exceptional loci of birational (bimeromorphic) contractions of a hyperkähler manifold $M$. Such a contraction locus is the union of all minimal rational curves in a collection of cohomology classes which are orthogonal to a wall of the Kähler cone. Homology classes which can possibly be orthogonal to a wall of the Kähler cone of some deformation of $M$ are called MBM classes. We prove that all MBM classes of type (1,1) can be represented by rational curves, called MBM curves. All MBM curves can be contracted on an appropriate birational model of $M$, unless $b_2(M) \leq 5$. When $…

Automorphism groups of Calabi-Yau manifolds of Picard number two

We prove that the automorphism group of an odd dimensional Calabi-Yau manifold of Picard number two is always a finite group. This makes a sharp contrast to the automorphism groups of K3 surfaces and hyperkähler manifolds and birational automorphism groups, as we shall see. We also clarify the relation between finiteness of the automorphism group (resp. birational automorphism group) and the rationality of the nef cone (resp. movable cone) for a hyperkähler manifold of Picard number two. We will also discuss a similar conjectual relation together with exsistence of rational curve, expected by …

Birational symplectic manifolds and their deformations

The known counterexamples to the global Torelli theorem for higher-dimensional hyperkahler manifolds are provided by birational manifolds. We address the question whether two birational hyperkahler manifolds (i.e. irreducible symplectic) manifolds always define non-separated points in the moduli space of marked manifolds. An affirmative answer is given for the cases of Mukai's elementary transformations and birational correspondences which are isomorphic in codimension two. The techniques are applied to show that the moduli spaces of rank two sheaves on a K3 surface are deformation equivalent …

齐性空间中的抛物线轨道集合、齐性动力学与超凯勒几何

Collections of parabolic orbits in homogeneous spaces, homogeneous dynamics and hyperkahler geometry

该研究针对满足$b_2(M)\u003e=5$的超凯勒流形，运用齐性空间遍历理论改进Morrison-Kawamata锥猜想相关结果，证明本原MBM类的Beauville-Bogomolov平方绝对值有仅依赖流形形变类的上界。

超凯勒流形族中的丰度与SYZ猜想

The abundance and SYZ conjectures in families of hyperkahler manifolds

研究超凯勒流形上具有nef且非大第一陈类的全纯线丛的半丰性，通过引入Teichmuller空间并证明全局Torelli定理，建立半丰性的形变不变性。

Finiteness and Quasi-Simplicity for Symmetric K3-Surfaces

We compare the smooth and deformation equivalence of actions of finite groups on K3-surfaces by holomorphic and anti-holomorphic transformations. We prove that the number of deformation classes is finite and, in a number of cases, establish the expected coincidence of the two equivalence relations. More precisely, in these cases we show that an action is determined by the induced action in the homology. On the other hand, we construct two examples to show that, first, in general the homological type of an action does not even determine its topological type, and second, that K3-surfaces $X$ and…

Integral constraints on the monodromy group of the hyperkahler resolution of a symmetric product of a K3 surface

Let M be a 2n-dimensional Kahler manifold deformation equivalent to the Hilbert scheme of length n subschemes of a K3 surface S. Let Mon be the group of automorphisms of the cohomology ring of M, which are induced by monodromy operators. The second integral cohomology of M is endowed with the Beauville-Bogomolov bilinear form. We prove that the restriction homomorphism from Mon to the isometry group O[H^2(M)] is injective, for infinitely many n, and its kernel has order at most 2, in the remaining cases. For all n, the image of Mon in O[H^2(M)] is the subgroup generated by reflections with res…

Deformations and BBF form on non-Kahler holomorphically symplectic manifolds

In 1995, Dan Guan constructed examples of non-Kahler, simply-connected holomorphically symplectic manifolds. An alternative construction, using the Hilbert scheme of Kodaira-Thurston surface, was given by F. Bogomolov. We investigate topology and deformation theory of Bogomolov-Guan manifolds and show that it is similar to that of hyperkahler manifolds. We prove the local Torelli theorem, showing that holomorphically symplectic deformations of BG-manifolds are unobstructed, and the corresponding period map is locally a diffeomorphism. Using the local Torelli theorem, we prove the Fujiki formul…

Deformation quantization modules I:Finiteness and duality

We introduce the notion of being cohomologically complete for objects of the derived category of sheaves of $Z[\hbar]$-modules on a topological space. Then we consider a $Z[\hbar]$-algebra satisfying some suitable conditions and prove coherency results by using the property of being cohomologically complete. We apply these results to the study of modules over deformation quantization algebroids on complex Poisson manifolds. We prove in particular that under a natural properness condition, the convolution of two coherent kernels over such algebroids is coherent. We also construct the dualizing …

ample锥的圆度与超凯勒流形上双拉格朗日纤维化的存在性

Roundness of the ample cone and existence of double Lagrangian fibrations on hyperkahler manifolds

本文证明任何满足 $b_2 > 4$ 的最大全纯超凯勒流形存在具有圆凯勒锥和符号 $(1,1)$ 的 Picard 格的形变，由此推出所有已知超凯勒流形具有双横截拉格朗日纤维化的形变，且 Kobayashi 度量在 Picard 秩非最大时消失。

霍奇相似性、代数类与 Kuga-Satake 簇

Hodge similarities, algebraic classes, and Kuga-Satake varieties

本文引入超凯勒流形超越格的霍奇相似性概念，证明特定 K3 曲面族平方的霍奇猜想，并利用 Kuga-Satake 构造的函子性推出广义 Kummer 形变型超凯勒流形的霍奇相似性均为代数的。

Monodromy and birational geometry of O'Grady's sixfolds

We prove that the bimeromorphic class of a hyperkähler manifold deformation equivalent to O'Grady's six dimensional one is determined by the Hodge structure of its Beauville-Bogomolov lattice by showing that the monodromy group is maximal. As applications, we give the structure for the Kähler and the birational Kähler cones in this deformation class and we prove that the existence of a square zero divisor implies the existence a rational lagrangian fibration with fixed fibre types.

Teichmuller space for hyperkahler and symplectic structures

Let S be an infinite-dimensional manifold of all symplectic, or hyperkahler, structures on a compact manifold M, and $Diff_0$ the connected component of its diffeomorphism group. The quotient $S/\Diff_0$ is called the Teichmuller space of symplectic (or hyperkahler) structures on M. MBM classes on a hyperkahler manifold M are cohomology classes which can be represented by a minimal rational curve on a deformation of M. We determine the Teichmuller space of hyperkahler structures on a hyperkahler manifold, identifying any of its connected components with an open subset of the Grassmannian $SO(b…

Finiteness results for hyperkaehler manifolds

In this short note we prove that the number of deformation types of compact hyperkaehler manifolds with prescribed second cohomology and second Chern class is finite. The proof uses the finiteness result of Kollar and Matsusaka, a formula by Hitchin and Sawon and the surjectivity of the period map.

Orbifold resolution via hyperkahler quotients: the $D_2$ ALF manifold

We propose an infinite-dimensional generalization of Kronheimer's construction of families of hyperkahler manifolds resolving flat orbifold quotients of $\mathbb{R}^4$. As in [Kro89], these manifolds are constructed as hyperkahler quotients of affine spaces. This leads to a study of \emph{singular equivariant instantons} in various dimensions. In this paper, we study singular equivariant Nahm data to produce the family of $D_2$ asymptotically locally flat (ALF) manifolds as a deformation of the flat orbifold $(\mathbb{R}^3 \times S^1)/Z_2$. We furthermore introduce a notion of stability for Na…

$K3^{[2]}$ 型超凯勒流形的辛自同构有限群

Finite groups of symplectic automorphisms of hyperkähler manifolds of type $K3^{[2]}$

本文确定了与K3曲面第二希尔伯特方案形变等价的超凯勒流形的所有辛自同构有限群，刻画了极大群并给出了形变等价类，且与马蒂厄月光预测进行了比较。

Unobstructed symplectic packing by ellipsoids for tori and hyperkahler manifolds

Let M be a closed symplectic manifold of volume V. We say that the symplectic packings of M by ellipsoids are unobstructed if any collection of disjoint symplectic ellipsoids (possibly of different sizes) of total volume less than V admits a symplectic embedding to M. We show that the symplectic packings by ellipsoids are unobstructed for all even-dimensional tori equipped with Kahler symplectic forms and all closed hyperkahler manifolds of maximal holonomy, or, more generally, for closed Campana simple manifolds (that is, Kahler manifolds that are not unions of their complex subvarieties), as…

Deformation of singularities and additive invariants

In this survey on local additive invariants of real and complex definable singular germs we systematically present classical or more recent invariants of different nature as emerging from a tame degeneracy principle. For this goal, we associate to a given singular germ a specific deformation family whose geometry degenerates in such a way that it eventually gives rise to a list of invariants attached to this germ. Complex analytic invariants, real curvature invariants and motivic type invariants are encompassed under this point of view. We then explain how all these invariants are related to e…

Unobstructed symplectic packing for tori and hyperkahler manifolds

Let M be a closed symplectic manifold of volume V. We say that M admits an unobstructed symplectic packing by balls if any collection of symplectic balls (of possibly different radii) of total volume less than V admits a symplectic embedding to M. In 1994 McDuff and Polterovich proved that symplectic packings of Kahler manifolds can be characterized in terms of the Kahler cones of their blow-ups. When M is a Kahler manifold which is not a union of its proper subvarieties (such a manifold is called Campana simple) these Kahler cones can be described explicitly using the Demailly and Paun struct…

Transcendental Hodge algebra

The transcendental Hodge lattice of a projective manifold $M$ is the smallest Hodge substructure in $p$-th cohomology which contains all holomorphic $p$-forms. We prove that the direct sum of all transcendental Hodge lattices has a natural algebraic structure, and compute this algebra explicitly for a hyperkahler manifold. As an application, we obtain a theorem about dimension of a compact torus $T$ admitting a symplectic embedding to a hyperkahler manifold $M$. If $M$ is generic in a $d$-dimensional family of deformations, then $\dim T\geq 2^{[(d+1)/2]}$.

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.

Unpolarized Shafarevich conjectures for hyper-Kähler varieties

Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was proved by Y. She. For hyper-Kähler varieties, which are higher-dimensional analogs of K3 surfaces, Y. André has verified the Shafarevich conjecture for hyper-Kähler varieties of a given dimension and admitting a very ample polarization of bounded degree. In this paper, we provide a unification of both results by proving the (unpolarized) Shafarevich conjecture f…

Holography principle for twistor spaces

Let $S$ be a smooth rational curve on a complex manifold $M$. It is called ample if its normal bundle is positive. We assume that $M$ is covered by smooth holomorphic deformations of $S$. The basic example of such a manifold is a twistor space of a hyperkahler or a 4-dimensional anti-selfdual Riemannian manifold $X$ (not necessarily compact). We prove "a holography principle" for such a manifold: any meromorphic function defined in a neighbourhood $U$ of $S$ can be extended to $M$, and any section of a holomorphic line bundle can be extended from $U$ to $M$. This is used to define the notion o…

Deformation quantization modules

We study modules over stacks of deformation quantization algebroids on complex Poisson manifolds. We prove finiteness and duality theorems in the relative case and construct the Hochschild class of coherent modules. We prove that this class commutes with composition of kernels, a kind of Riemann-Roch theorem in the non-commutative setting. Finally we study holonomic modules on complex symplectic manifolds and we prove in particular a constructibility theorem.

Deformation quantization modules on complex symplectic manifolds

We study modules over the algebroid stack $\W[\stx]$ of deformation quantization on a complex symplectic manifold $\stx$ and recall some results: construction of an algebra for $\star$-products, existence of (twisted) simple modules along smooth Lagrangian submanifolds, perversity of the complex of solutions for regular holonomic $\W[\stx]$-modules, finiteness and duality for the composition of ``good'' kernels. As a corollary, we get that the derived category of good $\W[\stx]$-modules with compact support is a Calabi-Yau category. We also give a conjectural Riemann-Roch type formula in this …

无极化超Kähler流形的沙法雷维奇猜想

Unpolarized Shafarevich conjectures for hyper-Kähler varieties

本文统一了K3表面和超Kähler流形的沙法雷维奇猜想，证明了在给定变形类型下无极化超Kähler流形的有限性，并讨论了同调泛化猜想。

Topological strings on noncommutative manifolds

We identify a deformation of the N=2 supersymmetric sigma model on a Calabi-Yau manifold X which has the same effect on B-branes as a noncommutative deformation of X. We show that for hyperkahler X such deformations allow one to interpolate continuously between the A-model and the B-model. For generic values of the noncommutativity and the B-field, properties of the topologically twisted sigma-models can be described in terms of generalized complex structures introduced by N. Hitchin. For example, we show that the path integral for the deformed sigma-model is localized on generalized holomorph…