Strict and nonstrict positivity of direct image bundles

• DOI: 10.1007/s00209-010-0783-5
• Publication Date: 2010-10-05T17:11:14Z
• Authors: Bo Berndtsson
• Keywords: Morphism; Tautological line bundle; Canonical bundle; Line (geometry)

This paper is a sequel to \cite{Berndtsson}. In that paper we studied the vector bundle associated to the direct image of the relative canonical bundle of a smooth K��hler morphism, twisted with a semipositive line bundle. We proved that the curvature of a such vector bundles is always semipositive (in the sense of Nakano). Here we adress the question if the curvature is strictly positive when the Kodaira-Spencer class does not vanish. We prove that this is so provided the twisting line bundle is stricty positive along fibers, but not in general.<br/>This version is revised following suggestions of a referee, and also slightly expanded. To appear in Math Zeitschrift<br/>