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Families of hypersurfaces of large degree

Christophe Mourougane — 2012

Journal of the European Mathematical Society

Grauert and Manin showed that a non-isotrivial family of compact complex hyperbolic curves has finitely many sections. We consider a generic moving enough family of high enough degree hypersurfaces in a complex projective space. We show the existence of a strict closed subset of its total space that contains the image of all its sections.

Hodge metrics and the curvature of higher direct images

Christophe MourouganeShigeharu Takayama — 2008

Annales scientifiques de l'École Normale Supérieure

Using the harmonic theory developed by Takegoshi for representation of relative cohomology and the framework of computation of curvature of direct image bundles by Berndtsson, we prove that the higher direct images by a smooth morphism of the relative canonical bundle twisted by a semi-positive vector bundle are locally free and semi-positively curved, when endowed with a suitable Hodge type metric.

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