超凯勒流形的第二陈类与Fujiki常数

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。

Scalar curvature and an infinite-dimensional hyperkähler reduction

We discuss a natural extension of the Kähler reduction of Fujiki and Donaldson, which realises the scalar curvature of Kähler metrics as a moment map, to a hyperkähler reduction. Our approach is based on an explicit construction of hyperkähler metrics due to Biquard and Gauduchon. This extension is reminiscent of how one derives Hitchin's equations for harmonic bundles, and yields real and complex moment map equations which deform the constant scalar curvature Kähler (cscK) condition. In the special case of complex curves we recover previous results of Donaldson. We focus on the case of comple…

OG6 型簇的强全局 Torelli 定理

A Strong Global Torelli theorem for OG6 type varieties

本文证明 OG6 型超凯勒流形由其次积分上同调及其 Hodge 结构和 Beauville--Bogomolov--Fujiki 形式唯一决定，并由此推出双亚纯等价即双全纯等价。

Symmetric Powers of Symmetric Bilinear Forms, Homogeneous Orthogonal Polynomials on the Sphere and an Application to Compact Hyperkähler Manifolds

The Beauville-Fujiki relation for a compact Hyperkähler manifold $X$ of dimension $2k$ allows to equip the symmetric power $\text{Sym}^kH^2(X)$ with a symmetric bilinear form induced by the Beauville-Bogomolov form. We study some of its properties and compare it to the form given by the Poincaré pairing. The construction generalizes to a definition for an induced symmetric bilinear form on the symmetric power of any free module equipped with a symmetric bilinear form. We point out how the situation is related to the theory of orthogonal polynomials in several variables. Finally, we construct a…

反辛对合的几何学，I

The geometry of antisymplectic involutions, I

本文研究K3^[n]型射影超凯勒流形上反辛对合的不动轨迹，证明由平方为2的丰富类诱导时，其连通分量数等于该类的可除性（1或2）。

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…

The HcscK equations in symplectic coordinates

The Donaldson-Fujiki Kähler reduction of the space of compatible almost complex structures, leading to the interpretation of the scalar curvature of Kähler metrics as a moment map, can be lifted canonically to a hyperkähler reduction. Donaldson proposed to consider the corresponding vanishing moment map conditions as (fully nonlinear) analogues of Hitchin's equations, for which the underlying bundle is replaced by a polarised manifold. However this construction is well understood only in the case of complex curves. In this paper we study Donaldson's hyperkähler reduction on abelian varieties a…

广义Kummer流形中Lagrange平面的分类

Classification of Lagrangian planes in generalised Kummer manifolds

本文证明Kummer型超Kähler流形中Lagrange平面内直线的类具有特定的Beauville-Bogomolov-Fujiki平方和判别群阶，并证明满足这些条件的极原始射线对应某Lagrange平面中的直线。

Riemann-Roch多项式的正性与超凯勒流形的Todd类

Positivity of Riemann-Roch polynomials and Todd classes of hyperkähler manifolds

本文证明了超凯勒流形的Riemann-Roch多项式系数全为正，证实了Cao-作者的猜想和Riess的严格单调性问题，并蕴含Kawamata有效非消失猜想；方法上利用Rozansky-Witten理论构造Todd类平方根的Lefschetz型分解。