Local structure of hyperkaehler singularities in O'Grady's examples

Hodge structure of O'Grady's singular moduli spaces

We investigate the Hodge structure of the singular O'Grady's six and ten dimensional examples of irreducible symplectic varieties. In particular, we compute some of their Betti numbers and their Euler characteristic. As consequence, we deduce that these varieties do not have finite quotient singularities answering a question of Bakker and Lehn.

La singularité de O'Grady

Let M be the moduli space of semistable sheaves with Mukai vector 2v on an abelian or K3 surface where v is primitive such that <v,v>=2. We show that the blow-up of the reduced singular locus of M provides a symplectic resolution of singularities. This gives a direct description of O'Grady's resolutions of M\_{K3}(2,0,4) and M\_{Ab}(2,0,2).

Desingularization of singular hyperkaehler varieties II

This is a second part of alg-geom/9611015. We construct a natural hyperkaehler desingularization for all singular hyperkaehler varieties. The desingularization theorem was proven in alg-geom/9611015 under additional assumption of local homogeneity. Here we show that local homogeneity is redundant: every singular hyperkaehler variety has locally homogeneous singularities.

高维 O'Grady 解消及其例外轨迹

Higher-dimensional O'Grady's resolutions and their exceptional locus

本文研究 K3 曲面上 Mukai 向量为 $2(1,0,1-n)$ 的半稳定层模空间 $M_n$，分析 O'Grady 构造的显式奇点解消，并全局刻画其例外轨迹，为探讨范畴 crepant 解消奠定基础。

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…

关于具有平凡局部基本群的孤立辛奇点的注记

A remark on isolated symplectic singularities with trivial local fundamental group

本文提出了一种简单方法构造任意维度的孤立辛奇点，其局部基本群为平凡群，回应了Beauville的问题。