Equivariant Holomorphic Morse Inequalities II: Torus and Non-Abelian Group Actions

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. 0. Introduction Index theorems express analytical indices of elliptic complexes in terms of topological invariants; informations on the individual cohomology groups are usually obtained with the aid of some vanishing theorems. Taking the de Rham complex for example, the Euler number is not enough to determine the Betti numbers. However, if we consider a Morse function, then the Morse inequalities bound each Betti number by the data of the critical points. In this paper, we consider a holomorphic setting with group actions. The index theorem is the Atiyah-Bott fixed-point formula [2], which expresses the equivariant index, the alternating sum of the characters on Dolbeault cohomology groups, in terms of the fixed-point data. The corresponding equivariant holomorphic Morse inequalities when the group is the circle group was first obtained by Witten [26] and proved Current address. E-mail address: [email protected]