Strict and non strict positivity of direct image bundles

Details Strict and non strict positivity of direct image bundles Strict and non strict positivity of direct image bundles

2010-02-25

arxiv.org/abs/1002.4797v2

Mathematics

Algebraic Geometry

This version is revised following suggestions of a referee, and also slightly expanded. To appear in Math Zeitschrift

Scientific paper

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\"ahler 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.

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