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

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.