Vanishing theorems on manifolds with piecewise strictly pseudodonvex boundaris

Abstract

Let ω be a relatively compact subdomain of a complex manifold X with boundary ∂ ω strictly pseudoconvex and let there be a C2 strictly plurisubhannonic function defined in a neighborhood of the closure of ω in X. It is shown that the cohomology groups Hq (ωDp ) vanish for q ≥ 1, p ≥ 0, where Dp is the sheaf of germs of holomorphic p-forms on ω. This settles, in the negative, the question raised by I. Lieb concerning whether the vanishing of Hq (ωDp ) for q ≥ 1, p ≥. 0 implies that ω is Stein or not. When Ω has only a piecewise strictly pseudoconvex boundary, it also shows that Hq (ωVn) vanishes for q ≥ 1. This is applied to the Gleason problem, the weak corona problem and an approximation problem.