Hodge metrics and the curvature of higher direct images

Mourougane, Christophe; Takayama, Shigeharu

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 6, pp. 905-924.

Nous utilisons la théorie de représentation par formes harmoniques des classes de cohomologie relative développée par Takegoshi et la structure des calculs de courbure de fibrés images directes développée par Berndtsson, pour étudier les images directes supérieures par un morphisme lisse du fibré canonique relatif tensorisé par un fibré vectoriel holomorphe hermitien semi-positif. Nous montrons qu'elles sont localement libres et que, munies de métriques convenables de type Hodge, elles sont à courbure semi-positive.

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.

Classification : 32L10, 14F10, 32J25, 14D07
Keywords: higher direct images, Hodge metrics, harmonic theory for relative cohomology, Nakano positivity
Mot clés : images directes supérieures, métriques de Hodge, théorie harmonique pour la cohomologie relative, positivité de Nakano

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