题目：Orbifold Hirzebruch-Riemann-Roch and Equivariant Moduli Theory on K3 Surfaces

报告人：陈宇航 讲师 （俄亥俄州立大学）

时间：2023年7月3日 周一 9:30-11:00，14：00-15:30

地点：宁静楼104室

内摘要：The talks contains two lectures. This first lecture is mainly expository with the purpose of demystifying the Riemann-Roch theorem for Deligne-Mumford (DM) stacks. In this talk, I will restrict DM stacks to quotient stacks. I will review some basic facts about the K-theory of quotient stacks and derive an orbifold Hirzebruch-Riemann-Roch (HRR) formula for connected proper smooth quotient stacks. Let X be a connected separated quotient stack. I will study the inertia stack IX of X in details. In particular, we will decompose IX into connected components. Such a decomposition is essential in the constructions of the orbifold Chern character and the orbifold Todd class of a coherent sheaf on X when X is smooth. I will derive explicit formulas for the Chern character and the Todd class. I will define an orbifold Mukai pairing and derive an orbifold HRR formula for X when it is proper and smooth. I will show that the orbifold HRR formula for the classifying stack BG when G = Z/nZ recovers Parseval’s theorem for the discrete Fourier transform.

In the second lecture, I will define equivatiant moduli spaces of stable sheaves on a projective scheme under an action of a finite group. Let X be a K3 surface, and let G be a finite subgroup G of the symplectic automorphism group of X. I will use the orbifold HRR formula to compute the dimensions of G-equivariant moduli spaces of stable sheaves on X under the action of the finite group G. I will then apply the orbifold HRR formula to reproduce the number of fixed points on X when G is cyclic without using the Lefschetz fixed point formula. I will prove that under some mild conditions, equivariant moduli spaces of stable sheaves on X are irreducible symplectic manifolds deformation equivalent to Hilbert schemes of points on X via a connection between Gieseker and Bridgeland moduli spaces, as well as the derived McKay correspondence. As a corollary, these moduli spaces are also deformation equivalent to G-equivariant Hilbert schemes of points on X.

欢迎各位参加！