Singular oscillatory integrals in equivariant cohomology. Residue formulae for basic differential forms on general symplectic manifolds

TL;DR

This paper develops residue formulae connecting equivariant cohomology of symplectic manifolds with the cohomology of reduced spaces, using advanced techniques like resolution of singularities and stationary phase, extending previous results to more general settings.

Contribution

It introduces new residue formulae for basic differential forms on general symplectic manifolds, broadening the scope of prior work on equivariant cohomology.

Findings

Residue formulae relate equivariant cohomology to reduced space cohomology.

Extension of Jeffrey, Kirwan, and Ramacher's results to general symplectic manifolds.

Application of resolution of singularities and stationary phase in this context.

Abstract

Let $M$ be a symplectic manifold and $G$ a connected, compact Lie group acting on $M$ in a Hamiltonian way. In this paper, we study the equivariant cohomology of $M$ represented by basic differential forms, and relate it to the cohomology of the Marsden-Weinstein reduced space via certain residue formulae using resolution of singularities and the stationary phase principle. In case that $ M $ is a compact, symplectic manifold or the co-tangent bundle of a $G$-manifold, similar residue formulae were derived by Jeffrey, Kirwan et al. for general equivariantly closed forms and by Ramacher for basic differential forms, respectively.