Levi Problem Under the Negativity of the Canonical Bundle near the Boundary

Abstract

A theorem asserting the existence of proper holomorphic maps with connected fibers to an open subset of ${\mathbb{C}}^N$ from a locally pseudoconvex bounded domain in a complex manifold will be proved under the negativity of the canonical bundle on the boundary. Related results by Takayama on the holomorphic embeddability and holomorphic convexity of pseudoconvex manifolds will be extended under similar curvature conditions.