Equivariant Holomorphic Morse Inequalities I: A Heat Kernel Proof

TL;DR

This paper provides a heat kernel proof of equivariant holomorphic Morse inequalities on compact Kähler manifolds with circle group actions, offering bounds on cohomology weight multiplicities based on fixed point data.

Contribution

It introduces a heat kernel approach to prove equivariant holomorphic Morse inequalities, extending previous results by Witten with a new analytical method.

Findings

Bounds on multiplicities of weights in twisted Dolbeault cohomologies

Application of Bismut and Lebeau techniques

Extension of Morse inequalities to equivariant setting

Abstract

Assume that the circle group acts holomorphically on a compact K\"ahler manifold with isolated fixed points and that the action can be lifted holomorphically to a holomorphic Hermitian vector bundle. We give a heat kernel proof of the equivariant holomorphic Morse inequalities. We use some techniques developed by Bismut and Lebeau. These inequalities, first obtained by Witten using a different argument, produce bounds on the multiplicities of weights occurring in the twisted Dolbeault cohomologies in terms of the data of the fixed points.