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 μ by the Kostant formula and then 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, in the spirit of "quantization commutes with reduction".