Equivariant holomorphic morse inequalities III: Non-isolated fixed points

Abstract

We prove the equivariant holomorphic Morse inequalities for a holomorphic torus action on a holomorphic vector bundle over a compact Kähler manifold when the fixed-point set is not necessarily discrete. Such inequalities bound the twisted Dolbeault cohomologies of the Kähler manifold in terms of those of the fixed-point set. We apply the inequalities to obtain relations of Hodge numbers of the connected components of the fixed-point set and the whole manifold. We also investigate the consequences in geometric quantization, especially in the context of symplectic cutting.