ORBIFOLD HIRZEBRUCH-RIEMANN-ROCH
Dr. Yuhang Chen
2023-06-13 15:30-16:30
MCM110
Speaker: Dr. Yuhang Chen (Ohio State University)Time: 15:30-16:30 June 13, 2023 (Tuesday)
Place: MCM110
Title: ORBIFOLD HIRZEBRUCH-RIEMANN-ROCH
Abstract: This talk 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 e ch(E) and the orbifold Todd class e td(E) of a coherent sheaf E on X when X is smooth. I will derive explicit formulas for e ch(E) and e td(E). 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.
