Holomorphic Witten instanton complexes on stratified pseudomanifolds with K\"ahler wedge metrics

TL;DR

This paper extends Witten's holomorphic Morse inequalities and instanton complexes to stratified pseudomanifolds with K"ahler wedge metrics, providing new tools for analyzing $L^2$ cohomology and related invariants in singular spaces.

Contribution

It introduces Witten instanton complexes on stratified pseudomanifolds with wedge K"ahler metrics, extending Morse inequalities and cohomology analysis to singular spaces for the first time.

Findings

Extended Witten's holomorphic Morse inequalities to singular spaces.

Proved rigidity of $L^2$ de Rham cohomology under circle actions.

Derived formulas for Rarita Schwinger operators and generalized gravitational instanton charges.

Abstract

We construct Witten instanton complexes for K\"ahler Hamiltonian Morse functions on stratified pseudomanifolds with wedge K\"ahler metrics satisfying a local conformally totally geodesic condition. We use this to extend Witten's holomorphic Morse inequalities for the $L^2$ cohomology of Dolbeault complexes, deriving versions for Poincar\'e Hodge polynomials, the spin Dirac and signature complexes for which we prove rigidity results, in particular establishing the rigidity of $L^2$ de Rham cohomology for these circle actions. We study formulas for Rarita Schwinger operators, generalize formulas studied by Witten and Gibbons-Hawking for the equivariant signature and extend formulas used to compute NUT charges of gravitational instantons. We discuss conjectural inequalities extending known Lefschetz-Riemann-Roch formulas for other cohomology theories including those of Baum-Fulton-Quart. This article contains the first extension of Witten's holomorphic Morse inequalities and instanton complexes to singular spaces.