Characteristic classes and stability conditions for projective Kleinian orbisurfaces

Riemann-Roch coefficients for Kleinian orbisurfaces

Suppose $\mathcal{S}$ is a smooth, proper, and tame Deligne-Mumford stack. Toën's Grothendieck-Riemann-Roch theorem requires correction terms, involving components of the inertia stack, to the standard formula for schemes. We give a brief overview of Toën's Grothendieck-Riemann-Roch theorem, and explicitly compute the correction terms in the case of an orbifold surface with stabilizers of types ADE.

Pure sheaves and Kleinian singularities

Grothendieck proved that any locally free sheaf on a projective line over a field (uniquely) decomposes into a direct sum of line bundles. Ishii and Uehara construct an analogue of Grothendieck's theorem for pure sheaves on the fundamental cycle of the Kleinian singularity $A_n$. We first study the analogue for the other Kleinian singularities except for $A_n$. We also study the classification of rigid pure sheaves on the reduced scheme of the fundamental cycles. The classification is related to the classification of spherical objects in a certain Calabi-Yau $2$-dimensional category.

紧化Kleinian奇点上的框架层模空间

Moduli spaces of framed sheaves on compactified Kleinian singularities

该研究针对Kleinian奇点$\mathbb{C}^2/\Gamma$（$\Gamma$为$SL_2(\mathbb{C})$有限子群），构造了紧化该奇点的射影Deligne-Mumford栈上的框架层模空间，证明其为拟射影概形。

Moduli of twisted orbifold sheaves

We study stacks of slope-semistable twisted sheaves on orbisurfaces with projective coarse spaces and prove that in certain cases they have many of the asymptotic properties enjoyed by the moduli of slope-semistable sheaves on smooth projective surfaces.

双有理克莱因群

Birational Kleinian groups

该文聚焦双有理克莱因群的研究，明确其为复射影簇上满足自由、纯不连续且余紧作用的双有理变换群，并完成了二维维度下该类群的分类工作。

通过联络与留数的曲面等变黎曼-罗赫定理

Equivariant Riemann-Roch for Surfaces via Connections and Residues

本文通过联络与留数方法，结合除子剥离和双有理技巧，为有限群作用下的光滑射影曲面上的等变欧拉示性数提供了统一计算程序，并给出ADE奇点修正类及实例验证。