Background cohomology of a non-compact Kähler 𝐺-manifold

Abstract

For a compact Lie group G G we define a regularized version of the Dolbeault cohomology of a G G -equivariant holomorphic vector bundle over non-compact Kähler manifolds. The new cohomology is infinite dimensional, but as a representation of G 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.