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Hodge metrics and the curvature of higher direct images

Christophe Mourougane, Shigeharu 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.

Hodge-Tate and de Rham representations in the imperfect residue field case

Kazuma Morita (2010)

Annales scientifiques de l'École Normale Supérieure

Let K be a p -adic local field with residue field k such that [ k : k p ] = p e < + and V be a p -adic representation of Gal ( K ¯ / K ) . Then, by using the theory of p -adic differential modules, we show that V is a Hodge-Tate (resp. de Rham) representation of Gal ( K ¯ / K ) if and only if V is a Hodge-Tate (resp. de Rham) representation of Gal ( K pf ¯ / K pf ) where K pf / K is a certain p -adic local field with residue field the smallest perfect field k pf containing k .

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