Equivariant holomorphic Morse inequalities .1. A heat kernel proof

Assume that the circle group acts holomorphically on a compact Kahler manifold with isolated fixed points and that the action can be lifted holomorphically to a holomorphic vector bundle. We use some techniques developed by Bismut and Lebeau to give a heat kernel proof of the equivariant holomorphic Morse inequalities, which, first obtained by Witten using a different argument, produce bounds on the multiplicities of weights occurring in the twisted Dolbeault cohomologies in terms of the data of the fixed points.