Comptes Rendus
no. 3
Algebraic geometry/Analytic geometry
The direct image of the relative dualizing sheaf needs not be semiample
[L'image directe du faisceau dualisant relatif n'est pas nécessairement semi-ample]

Présenté par : 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.

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.

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.

Reçu le :
Accepté le :
Publié le :
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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