Equivariant Holomorphic Morse Inequalities III: Non-Isolated Fixed Points

TL;DR

This paper extends equivariant holomorphic Morse inequalities to cases with non-isolated fixed points, providing bounds on cohomologies and applications to Hodge numbers and geometric quantization.

Contribution

It introduces new inequalities for non-isolated fixed points in holomorphic torus actions, broadening previous results to more general fixed-point sets.

Findings

Bounded twisted Dolbeault cohomologies in terms of fixed-point set cohomologies

Derived relations between Hodge numbers of fixed components and the entire manifold

Explored implications in geometric quantization and symplectic cutting

Abstract

We prove the equivariant holomorphic Morse inequalities for a holomorphic torus action on a holomorphic vector bundle over a compact Kahler manifold when the fixed-point set is not necessarily discrete. Such inequalities bound the twisted Dolbeault cohomologies of the Kahler 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.