Narrow Results

The purpose of this paper is to construct a solution with L^p-estimates, 1≤p≤∞, to the equation on strongly q-convex domain of Kähler manifold. This is done for forms of type (n,s), s≥ max(q,k), with values in a holomorphic vector bundle which is Nakano semi-positive of type k and for forms of type (0,s), q≤s≤n-k, with values in a holomorphic vector bundle which is Nakano semi-negative of type k.

Let X be strongly q-convex domain of an n-dimensional Kähler manifold M and E be a holomorphic vector bundle over M. Then, if E satisfies certain positivity conditions, we prove vanishing theorems for the -cohomology groups of X with values in E.