  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.
