G-invariant Holomorphic Morse inequalities

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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 μ by the Kostant formula and we define the reduction of M under a natural hypothesis on μ⁻¹(0). Our inequalities are given in term of the curvature of the bundle induced by L on this reduction.