Equivariant holomorphic morse inequalities II: Torus and non-Abelian group actions

Abstract

We extend the equivariant holomorphic Morse inequalities of circle actions to cases with torus and non-Abelian group actions on holomorphic vector bundles over Kähler manifolds and show the necessity of the Kähler condition. For torus actions, there is a set of inequalities for each choice of action chambers specifying directions in the Lie algebra of the torus. We apply the results to invariant line bundles over toric manifolds. If the group is non-Abelian, there is in addition an action of the Weyl group on the fixed-point set of its maximal torus. The sum over the fixed points can be rearranged into sums over the Weyl group (having incorporated the character of the isotropy representation on the fiber) and over the orbits of the Weyl group in the fixed-point set.