Non-locally-free locus of O'Grady's ten dimensional example

Automorphisms of O'Grady's Manifolds Acting Trivially on Cohomology

We determine the subgroup of automorphisms acting trivially on the second integral cohomology for hyperkaehler manifolds which are deformation equivalent to O'Grady's sporadic examples. In particular, we prove that this subgroup is trivial in the ten dimensional case and isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{\times 8}$ in the six dimensional case. Furthermore we study the fixed loci of the automorphisms in the six dimensional case.

关于O'Grady六维超Kähler簇的动机

On the motive of O'Grady's six dimensional hyper-Kähler varieties

本文证明了O'Grady类型六维超Kähler簇的动机可由其基底表面的动机生成，并给出了其动机的公式，从而验证了多个OG6型超Kähler簇的Hodge和Tate猜想。

On the motive of O'Grady's ten-dimensional hyper-Kähler varieties

We investigate how the motive of hyper-Kähler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular moduli spaces of semistable sheaves on K3 and abelian surfaces as well as the Chow motive of their crepant resolutions, when they exist. We show that these motives are in the tensor subcategory generated by the motive of the surface, provided that a crepant resolution exists. This extends a recent result of Bülles to the O'Grady-10 situation. In the non-commutative setting, similar results are proved for …

Moduli spaces of polarised symplectic O'Grady varieties and Borcherds products

We study moduli spaces of O'Grady's ten-dimensional irreducible symplectic manifolds. These moduli spaces are covers of modular varieties of dimension 21, namely quotients of hermitian symmetric domains by a suitable arithmetic group. The interesting and new aspect of this case is that the group in question is strictly bigger than the stable orthogonal group. This makes it different from both the K3 and the K3^[n] case, which are of dimension 19 and 20 respectively.

On the Beauville form of the known irreducible symplectic varieties

We study the global geometry of the ten dimensional O'Grady irreducible symplectic variety. We determine its second Betti number, its Beauville form and its Fujiki constant.

Fano manifolds of degree ten and EPW sextics

O'Grady showed that certain special sextics in $\mathbb{P}^5$ called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.

包含在Torelli轨迹中的Mumford型Shimura曲线

Mumford-type Shimura curves contained in the Torelli locus

该研究构造了两类亏格4的Shimura曲线族的显式方程，其雅可比簇为定义在有理数域上、自同态代数为整数环且带额外Hodge张量的Mumford型阿贝尔四维簇，是该类阿贝尔簇的首批显式实例。

三角谱中Fourier-Mukai伙伴的粘合与双有理几何

Gluing of Fourier-Mukai partners in a triangular spectrum and birational geometry

本文在三角谱中构造Fourier-Mukai轨迹以粘合FM伙伴的张量三角谱，结合实例揭示其与代数簇几何、双有理性质的关联，并对比相关轨迹提出关于Serre不变轨迹的精确猜想。

张量极限的一致性

Uniformity for limits of tensors

本文研究多线性代数中各类秩的边界秩问题，证明去边界化定理并揭示实现边界秩的极限复杂度不随维数增长，基于GL-簇理论建立统一框架。

椭圆面的高度模数：动机高度ζ函数的有理性和Kudla-Millson模性与Mordell-Weil秩跃迁

Analytically dense Mordell-Weil rank jumps on elliptic surfaces via transverse realization

本文研究椭圆面的高度模数，通过动机高度ζ函数的有理性与Kudla-Millson模性，探讨Mordell-Weil秩跃迁问题。