Equivariant Holomorphic Morse Inequalities II: Torus and Non-Abelian Group Actions
Siye Wu
We extend the equivariant holomorphic Morse inequalities of circle actions to cases with torus and non-Abelian group action. For torus actions, there is a set of inequalities for each choice of action chambers specifying directions in the Lie algebra of the torus. 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 (after incooperating the character of the isotropy representation on the fiber) and over its orbits. We apply the results to invariant line bundles over toric manifolds and to homogeneous vector bundles over flag manifolds. In the latter case, the theorems of Borel-Weil-Bott and of Griffiths are recovered.