$G$-invariant 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 à 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.