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Description

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 talk is to establish holomorphic Morse inequalities, analogous to Demailly's one, 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 then 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, in the spirit of "quantization commutes with reduction"