Comptes Rendus
no. 3
Algebraic geometry/Analytic geometry
The direct image of the relative dualizing sheaf needs not be semiample

Presented by: Claire Voisin

Fabrizio Catanese1; Michael Dettweiler1
1 Mathematisches Institut, Universität Bayreuth, 95447 Bayreuth, Germany
Comptes Rendus. Mathématique, Volume 352 (2014) no. 3, pp. 241-244.

We provide details for the proof of Fujita's second theorem and prove that for a Kähler fibre space f:X→B over a smooth projective curve B, the direct image of the relative dualizing sheaf V:=f⁎ωX/B is the direct sum of an ample and a unitary flat bundle. We also show that V needs not be semiample, which is our main result.

Nous donnons des détails sur la démonstration du second théorème de Fujita et nous montrons que l'image directe du fibré canonique relatif V:=f⁎ωX/B d'une fibration f:X→B sur une courbe B est la somme directe d'un fibré vectoriel ample et d'un fibré vectoriel unitairement plat si l'espace total X est une variété kählérienne compacte. Nous montrons en outre que V n'est en général pas semi-ample, ce qui constitue notre résultat principal.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.12.015

Fabrizio Catanese 1; Michael Dettweiler 1

1 Mathematisches Institut, Universität Bayreuth, 95447 Bayreuth, Germany 
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Fabrizio Catanese; Michael Dettweiler. The direct image of the relative dualizing sheaf needs not be semiample. Comptes Rendus. Mathématique, Volume 352 (2014) no. 3, pp. 241-244. doi : 10.1016/j.crma.2013.12.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.12.015/

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