Background cohomology of a non-compact Kahler G-manifold

TL;DR

This paper introduces a regularized G-equivariant Dolbeault cohomology for non-compact Kähler manifolds, enabling well-defined equivariant Betti numbers and establishing a Kodaira-type vanishing theorem.

Contribution

It defines a new infinite-dimensional cohomology theory for non-compact Kähler G-manifolds that decomposes into finite multiplicity irreducible components, extending classical results.

Findings

Cohomology decomposes into irreducible G-representations with finite multiplicities

Equivariant Betti numbers are well-defined in this framework

A Kodaira-type vanishing theorem is proved for the new cohomology

Abstract

For a compact Lie group G we define a regularized version of the Dolbeault cohomology of a G-equivariant holomorphic vector bundles over non-compact Kahler manifolds. The new cohomology is infinite-dimensional, but as a representation of G it decomposes into a sum of irreducible components, each of which appears in it with finite multiplicity. Thus equivariant Betti numbers are well defined. We study various properties of the new cohomology and prove that it satisfies a Kodaira-type vanishing theorem.