2022/04/21 by Chen Yuhang, Chen, Yuhang
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2204.09824
We study the orbifold Hirzebruch-Riemann-Roch (HRR) theorem for quotient Deligne-Mumford stacks, explore its relation with the representation theory of finite groups, and derive a new orbifold HRR formula via an orbifold Mukai pairing. As a first application, we use this formula to compute the dimensions of G -equivariant moduli spaces of stable sheaves on a K3 surface X under the action of a finite subgroup G of its symplectic automorphism group. We 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. We 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.