Holomorphic Morse Inequalities and Symplectic Reduction

TL;DR

This paper develops Morse inequalities for holomorphic circle actions on vector bundles over compact Kaehler manifolds, linking cohomology weight multiplicities to fixed points and symplectic reduction, generalizing previous inequalities.

Contribution

It introduces a unified set of Morse inequalities that extend prior results to a broader context involving holomorphic actions and symplectic reduction.

Findings

Provides bounds on weight multiplicities in cohomology

Generalizes Wu-Zhang and Tian-Zhang inequalities

Offers a new proof of the Tian-Zhang index theorem

Abstract

We introduce Morse-type inequalities for a holomorphic circle action on a holomorphic vector bundle over a compact Kaehler manifold. Our inequalities produce bounds on the multiplicities of weights occurring in the twisted Dolbeault cohomology in terms of the data of the fixed points and of the symplectic reduction. This result generalizes both Wu-Zhang extension of Witten's holomorphic Morse inequalities and Tian-Zhang Morse-type inequalities for symplectic reduction. As an application we get a new proof of the Tian-Zhang relative index theorem for symplectic quotients.