On the infinite dimensionality of the Dolbeault cohomology groups

Abstract:

Let $M$ be an open subset of a Stein manifold without isolated points. Let ${\Omega ^p}$ be the sheaf of germs of holomorphic $p$-forms on $M$. Then ${H^q}(M,{\Omega ^p})$ is either $0$ or else infinite dimensional. ${H^q}(M,\mathcal {S})$ may be nonzero and finite dimensional if $M$ is the regular points of a Stein space or if $\mathcal {S}$ is an arbitrary coherent sheaf over $M$.