HyperKähler Manifolds and Birational Transformations in dimension 4

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.

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.

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.

Mordell-Weil groups of a hyperkähler manifold - a question of F. Campana

Mordell-Weil groups of different abelian fibrations of a hyperkähler manifold may have non-trivial relation even among elements of infinite order, but have essentially no relation, as its birational transformation. Precise definition of the terms "essentially no relation" will be given in Introduction.

关于Calabi-Yau流形的超越无基点猜想的一个注记

A note on the transcendental basepoint-free conjecture for Calabi-Yau manifolds

本文证明Calabi-Yau流形的超越无基点猜想在其Beauville-Bogomolov分解的超Kähler因子上成立时即成立，并利用Bakker-Lehn收缩定理在超Kähler流形上给出该猜想成立的一个充分条件。

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.

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 …

On birational transformations of Hilbert schemes of points on K3 surfaces

We classify the group of birational automorphisms of Hilbert schemes of points on algebraic K3 surfaces of Picard rank one. We study whether these automorphisms are symplectic or non-symplectic and if there exists a hyperkähler birational model on which they become biregular. We also present new geometrical constructions of these automorphisms.

Manifolds of low dimension with trivial canonical bundle in Grassmannians

We study fourfolds with trivial canonical bundle which are zero loci of sections of homogeneous, completely reducible bundles over ordinary and classical complex Grassmannians. We prove that the only HyperK{ä}hler fourfolds among them are the example of Beauville and Donagi, and the example of Debarre and Voisin. In doing so, we give a complete classification of those varieties. We include also the analogous classification for surfaces and threefolds.

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.

Lagrangian fibrations on hyperkähler fourfolds

Let X be a projective hyperkähler manifold containing a Lagrangian subtorus L. We study intersections of deformations of L, defining a canonical almost holomorphic map called L-reduction, which is not birational if and only if X admits an almost holomorphic Lagrangian fibration with (strong) fibre L. In dimension four we prove that in the above situation there is always a holomorphic Lagrangian fibration with fibre L, thus answering a question of Beauville in this particular case.

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 $…

关于超凯勒流形的超越格

On the transcendental lattices of Hyperkähler manifolds

本研究引入由K3型霍奇结构诱导的超凯勒流形概念，针对已知形变类型给出判断其与K3/阿贝尔曲面层模空间双有理等价的格论准则，并揭示O'Grady型超凯勒流形的特殊行为。

超凯勒六维流形、韦伊型阿贝尔四维流形与一个霍奇类

Hyperkähler sixfolds, abelian fourfolds of Weil type and a Hodge class

本文针对具有平凡判别式的一般韦伊型阿贝尔四维流形的霍奇猜想给出新证明。借助其到\(K3^{[3]}\)型超凯勒六维流形的映射，通过超凯勒流形双有理几何及形变理论成果，得出六维流形切丛第二陈类拉回相关代数类结论。

Hyperbolic geometry of the ample cone of a hyperkahler manifold

Let $M$ be a compact hyperkahler manifold with maximal holonomy (IHS). The group $H^2(M, R)$ is equipped with a quadratic form of signature $(3, b_2-3)$, called Bogomolov-Beauville-Fujiki (BBF) form. This form restricted to the rational Hodge lattice $H^{1,1}(M,Q)$, has signature $(1,k)$. This gives a hyperbolic Riemannian metric on the projectivisation of the positive cone in $H^{1,1}(M,Q)$, denoted by $H$. Torelli theorem implies that the Hodge monodromy group $Γ$ acts on $H$ with finite covolume, giving a hyperbolic orbifold $X=H/Γ$. We show that there are finitely many geodesic hypersurfac…

Cubic threefolds and hyperkähler manifolds uniformized by the 10-dimensional complex ball

We first prove an isomorphism between the moduli space of smooth cubic threefolds and the moduli space of hyperkaehler fourfolds of K3^{[2]}-type with a non-symplectic automorphism of order three, whose invariant lattice has rank one and is generated by a class of square 6, both these spaces are uniformized by the same 10-dimensional arithmetic complex ball quotient. We then study the degeneration of the automorphism along the loci of nodal or chordal degenerations of the cubic threefold, showing the birationality of these loci with some moduli spaces of hyperkaehler fourfolds of K3^{[2]}-type…

A global Torelli theorem for hyperkahler manifolds

A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. We compute a mapping class group of a hypekahler manifold $M$, showing that it is commensurable to an arithmetic subgroup in SO(3, b_2-3). A Teichmuller space of $M$ is a space of complex structures on $M$ up to isotopies. We define a birational Teichmuller space by identifying certain points corresponding to bimeromorphically equivalent manifolds, and show that the period map gives an isomorphism of the birational Teichmuller space and the corresponding period space $SO(b_2-3, 3)/SO(2)\ti…

On the period map for polarized hyperkähler fourfolds

This is an improved version of the eprint previously entitled "Unexpected isomorphisms between hyperkähler fourfolds." We study smooth projective hyperkähler fourfolds that are deformations of Hilbert squares of K3 surfaces and are equipped with a polarization of fixed degree and divisibility. They are parametrized by a quasi-projective irreducible 20-dimensional moduli space and Verbitksy's Torelli theorem implies that their period map is an open embedding. Our main result is that the complement of the image of the period map is a finite union of explicit Heegner divisors that we describe. We…

Modular sheaves on hyperkähler varieties

A torsion free sheaf on a hyperkähler variety $X$ is modular if the discriminant satisfies a certain condition, for example if it is a multiple of $c_2(X)$ the sheaf is modular. The definition is taylor made for torsion-free sheaves on a polarized hyperkähler variety (X,h) which deform to all small deformations of (X,h). For hyperkählers deformation equivalent to $K3^{[2]}$ we prove an existence and uniqueness result for slope-stable modular vector bundles with certain ranks, $c_1$ and $c_2$. As a consequence we get uniqueness up to isomorphism of the tautological quotient rank $4$ vector bund…

Geometric Invariant Theory and Birational Geometry

In this paper we will survey some recent developments in the last decade or so on variation of Geometric Invariant Theory and its applications to Birational Geometry such as the weak Factorization Theorems of nonsingular projective varieties and more generally projective varieties with finite quotient singularities. Along the way, we will also mention some progresses on birational geometry of hyperKähler manifolds as well as certain open problems and conjectures.

关于 double EPW-cube 的双有理自同构

On birational automorphisms of double EPW-cubes

本文分类了 K3^[3]-型流形上的有限辛双有理自同构群，描述了光滑 double EPW-cube 的极化自同构群，并据此构造了具有大群辛作用且 Picard 秩极大的射影 hyperkaehler 流形实例。

有限阶辛理性自映射在Kummer型 manifold 上

Finite order symplectic birational self-maps on Kummer-type manifolds

研究Kummer型射影超kähler manifold 的有限阶辛理性自映射性质，证明除某些Picard rank 3情况外，此类映射均属于扭曲模形式，且通过Néron-Severi lattices完全刻画例外情况。

Lagrangian submanifolds in Hyperkahler manifolds, Legendre transformation

We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperkahler manifold. We establish a characterization, a Chern number inequality, topological and geometrical properties of Lagrangian submanifolds. We discuss a category of Lagrangian subvarieties and its relationship with the theory of Lagrangian intersection. We also introduce and study extensively a normalized Legendre transformation of Lagrangian subvarieties under a birational transformation of projective hyperkahler manifolds. We give a Plucker type formula for Lagrangian intersections under thi…

关于具有大皮卡德数的K3$^{[n]}$型超凯勒流形

On Hyperkähler manifolds of K3$^{[n]}$-type with large Picard number

本文证明皮卡德数至少为4的K3$^{[n]}$型超凯勒流形必同构于K3曲面上的扭曲稳定层模空间，并对皮卡德数为3等较低秩的非双有理等价情形给出显式刻画。

Tits alternative in hypekähler manifolds

We show an analogous result of the famous Tits alternative for a group G of birational automorphisms of a projective hyperkäher manifold: Either G contains a non-commutative free group or G is an almost abelian group of finite rank. As an application, we show that the automorphism groups of the so-called singular K3 surfaces contain non-commutative free groups.

Elliptic quintics on cubic fourfolds, O'Grady 10, and Lagrangian fibrations

For a smooth cubic fourfold Y, we study the moduli space M of semistable objects of Mukai vector $2λ_1+2λ_2$ in the Kuznetsov component of Y. We show that with a certain choice of stability conditions, M admits a symplectic resolution $\tilde M$, which is a smooth projective hyperkähler manifold, deformation equivalent to the 10-dimensional examples constructed by O'Grady. As applications, we show that a birational model of $\tilde M$ provides a hyperkähler compactification of the twisted family of intermediate Jacobians associated to Y. This generalizes the previous result of Voisin arXiv:161…

Lagrangian fibrations on hyperkähler manifolds - On a question of Beauville

Let X be a compact hyperkähler manifold containing a complex torus L as a Lagrangian subvariety. Beauville posed the question whether X admits a Lagrangian fibration with fibre L. We show that this is indeed the case if X is not projective. If X is projective we find an almost holomorphic Lagrangian fibration with fibre L under additional assumptions on the pair (X, L), which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic Lagrangian fibration there exists a smooth good minimal model, i.e., a hyperkähler manifold birationa…

超Kähler收缩下K3曲面上Brauer类的几何实现

Geometric realizations of Brauer classes on K3 surfaces from hyperkähler contractions

本文通过扭曲层模空间的双有理几何，为Picard秩1的K3曲面上的Brauer类构造几何实现，统一了已知构造并新证周期-指数定理。

GIT 稳定三次三维簇与某些 $K3^{[2]}$ 型四维簇

GIT stable cubic threefolds and certain fourfolds of $K3^{[2]}$-type

本文研究光滑与奇异节点三次三维簇与 $K3^{[2]}$ 型超凯勒四维簇模空间之间的同构在节点超平面上的退化行为，推广至 $A_i$ 型奇点情形，并引入 $K$ 型凯勒锥截面概念以处理 $A_2$ 情形。

关于超凯勒簇的反双曲性

On anti-hyperbolicity for hyperkähler varieties

本文研究超凯勒流形被同维数复仿射空间全纯或亚纯支配的反双曲性判据与例子，推广了 K3 曲面的相关已知结果。

Kähler 簇的非消没结果

Nonvanishing results for Kähler varieties

本文研究紧 Kähler 簇的非消没定理，分别证明 klt 对与 K-平凡簇中数值维数一线丛的结果，并给出超 Kähler 流形上 nef 非大线丛的二分及四维更强丰饶性结论。