Rational curves and MBM classes on hyperkahler manifolds: a survey

On the existence of rational curves on projective hyperkähler fourfolds

We show that if $X$ is a projective hyperkähler fourfold and there exists a nonzero effective divisor $D$ which is not of bi-elliptic type and contained in the boundary of the nef cone of $X$, then $X$ contains a rational curve. This is a very special case of Oguiso's conjecture for projective hyperkähler fourfolds.

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。

Hyperkähler geometry of rational curves in twistor spaces

We investigate the pseudo-hyperkähler geometry of higher degree rational curves in the twistor space of a hyperkähler $4$-manifold.

Hyperkahler manifolds of Jacobian type

In this paper we define the notion of a hyperkähler manifold (potentially) of Jacobian type. If we view hyperkähler manifolds as "abelian varieties", then those of Jacobian type should be viewed as "Jacobian varieties". Under a minor assumption on the polarization, we show that a very general polarized hyperkähler fourfold $F$ of $K3^{[2]}$-type is not of Jacobian type. As a potential application, we conjecture that if a cubic fourfold is rational then its variety of lines is of Jacobian type. Under some technical assumption, it is proved that the variety of lines on a rational cubic fourfold …

On $k$-gonal loci in Severi varieties on general $K3$ surfaces and rational curves on hyperkähler manifolds

In this paper we study the gonality of the normalizations of curves in the linear system $|H|$ of a general primitively polarized complex $K3$ surface $(S,H)$ of genus $p$. We prove two main results. First we give a necessary condition on $p, g, r, d$ for the existence of a curve in $ |H|$ with geometric genus $g$ whose normalization has a $g^ r_d$. Secondly we prove that for all numerical cases compatible with the above necessary condition, there is a family of \emph{nodal} curves in $|H|$ of genus $g$ carrying a $g^1_k$ and of dimension equal to the \emph{expected dimension} $\min\{2(k-1),g\…

Riemann-Roch多项式、MBM类与贫乏IHS流形

Riemann-Roch Polynomials, MBM Classes and Poor IHS Manifolds

本文研究贫乏不可约全纯辛流形，证明其自然锥重合且无全纯叶理，并利用Riemann-Roch多项式根给出检测MBM类及有理曲线存在的统一准则，同时刻画了此类流形的形变空间。

On $k$-gonal loci in Severi varieties on general $K3$ surfaces and rational curves on hyperkähler manifolds (first version, superseded by arXiv:1204.4838)

In this paper we study the gonality of the normalizations of curves in the linear system $|H|$ of a general primitively polarized $K3$ surface $(S,H)$ of genus $p$. We prove two main results. First we give a necessary condition on $p, g, k$ for the existence of a curve in $|H|$ with geometric genus $g$ whose normalization has a $g^ 1_k$. Secondly we prove that for $p$ even and all numerical cases compatible with the above necessary condition, there is a family of \emph{nodal} curves $|H|$ with the given $g,k$ and of dimension equal to the \emph{expected dimension} $\min\{2(k-1),g\}$. For odd $…

The Looijenga-Lunts-Verbitsky algebra and Verbitsky's Theorem

In these notes we review some basic facts about the LLV Lie algebra. It is a rational Lie algebra, introduced by Looijenga-Lunts and Verbitsky, acting on the rational cohomology of a compact Kähler manifold. We study its structure and describe one irreducible component of the rational cohomology in the case of a compact hyperkähler manifold.

Maps from K-trivial varieties and connectedness problems

In this paper we study varieties covered by rational or elliptic curves. First, we show that images of Calabi-Yau or irreducible symplectic varieties under rational maps are almost always rationally connected. Second, we investigate elliptically connected and elliptically chain connected varieties, and varieties swept out by a family of elliptic curves. Among other things, we show that Calabi-Yau or hyperkähler manifolds which are covered by a family of elliptic curves contain uniruled divisors and that elliptically chain connected varieties of dimension at least two contain a rational curve, …

不可约全纯辛流形nef锥的初等描述

An elementary description of nef cone for irreducible holomorphic symplectic manifolds

该研究描述K3型和Kummer型不可约全纯辛流形的MBM类，分析其与相关理论的联系，并将低维数值方法应用于这两类情况，为nef锥研究提供基础结果。

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…

Ergodic complex structures on hyperkahler manifolds: an erratum

Let $M$ be a hyperkahler manifold, $Γ$ its mapping class group, and $Teich$ the Teichmuller space of complex structures of hyperkahler type. After we glue together birationally equivalent points, we obtain the so-called birational Teichmuller space $Teich_b$. Every connected component of $Teich_b$ is identified with its period space $P$ by global Torelli theorem. The mapping class group of $M$ acts on $P$ as a finite index subgroup of the group of isometries of the integer cohomology lattice, that is, satisfies assumptions of Ratner theorem. We prove that there are three classes of orbits, clo…

紧致超kähler流形上的刚性电流

Rigid currents on compact hyperkahler manifolds

研究超kähler流形上抛物(1,1)类的刚性性质，证明一般抛物类为刚性类

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…

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…

全纯辛4-流形的Gopakumar-Vafa型不变量

Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds

本文利用约化Gromov-Witten理论为全纯辛4-流形定义了新的Gopakumar-Vafa型不变量，猜想其整数性并给出层论解释，在多个例子中验证猜想，并得到K3^{[2]}型4-流形上亏格2曲线计数的猜想公式及Fujiki常数的新闭公式。