Strict and non strict positivity of direct image bundles

Abstract

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 Kähler 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.

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

References in corpus (3)

• Positivity of relative canonical bundles for families of canonically polarized manifolds
• Understanding Weil-Petersson curvature
• Recent Development on the Geometry of the Teichmuller and Moduli Spaces of Riemann Surfaces and Polarized Calabi-Yau Manifolds

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• A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry
• Positivity and vanishing theorems for ample vector bundles
• New proofs of the Torelli theorems for Riemann surfaces