Title: Holomorphic Lefschetz fixed point theorems on stratified pseudomanifolds with wedge metrics.
Abstract: The Atiyah Bott Lefschetz fixed point theorem relates indices of elliptic complexes to local contributions at fixed points of self maps of spaces, which are more easily computable.
It's generalizations are key to both computing and defining various quantities of interest in many areas of mathematics and physics. When there are complex structures near fixed points, most computations can actually be reduced to the holomorphic Lefschetz fixed point theorem.
I will discuss a generalization of the holomorphic Lefschetz fixed point theorem in $L^2$ cohomology of stratified pseudomanifolds with wedge (roughly, iterated conic) metrics, focusing mainly on the case of algebraic varieties. I will touch on how different choices of domains for the Dolbeault operator gives different formulae, and compare these with the Lefschetz Riemann Roch numbers of Baum-Fulton-MacPherson-Quart. I will also discuss how different types of singularities give rise to different phenomena in these different cohomology theories.
I'll also explain how to compute equivariant indices of spin Dirac complexes, Hirzebruch $\chi_y$ genera on Hermitian spaces, explaining how important dualities in the smooth setting, (which fail in the case of BFMQ) go through in the L2 theory, and how this can be used in various applications including equivariant index computations including those of the self dual and anti-self dual complexes, with applications to more general instanton counting.
The talk is based on part of the work in https://arxiv.org/abs/2309.15845.
