Dolbeault cohomology of compact complex manifolds with an action of a complex Lie group

Abstract

Let G be a complex Lie group acting on a compact complex Hermitian manifold M by holomorphic isometries. We prove that the induced action on the Dolbeault cohomology and on the Bott–Chern cohomology is trivial. We also apply this result to compute the Dolbeault cohomology of Vaisman manifolds.