G-invariant Holomorphic Morse inequalities

TL;DR

This paper extends holomorphic Morse inequalities to the setting of group actions on complex manifolds, relating invariant Dolbeault cohomology to curvature on a reduced space.

Contribution

It establishes G-invariant holomorphic Morse inequalities for Dolbeault cohomology, introducing a moment map and reduction technique under group symmetry.

Findings

Inequalities relate invariant cohomology to curvature on the reduced space.

Defines a moment map using the Kostant formula.

Provides a new framework for equivariant holomorphic Morse inequalities.

Abstract

Consider an action of a connected compact Lie group on a compact complex manifold $M$, and two equivariant vector bundles $L$ and $E$ on $M$, with $L$ of rank 1. The purpose of this paper is to establish holomorphic Morse inequalities \`{a} la Demailly for the invariant part of the Dolbeault cohomology of tensor powers of $L$ twisted by $E$. To do so, we define a moment map $\mu$ by the Kostant formula and we define the reduction of $M$ under a natural hypothesis on $\mu^{-1}(0)$. Our inequalities are given in term of the curvature of the bundle induced by $L$ on this reduction.