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Comptes Rendus Mathématique
Volume 352, n° 3
pages 241-244 (mars 2014)
Doi : 10.1016/j.crma.2013.12.015
Received : 19 November 2013 ;  accepted : 17 December 2013
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
 

Fabrizio Catanese , Michael Dettweiler
 Mathematisches Institut, Universität Bayreuth, 95447 Bayreuth, Germany 

Abstract

We provide details for the proof of Fujita's second theorem and prove that for a Kähler fibre space   over a smooth projective curve B , the direct image of the relative dualizing sheaf   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.

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Résumé

Nous donnons des détails sur la démonstration du second théorème de Fujita et nous montrons que l'image directe du fibré canonique relatif   d'une fibration   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.

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1  We remark that, while unitary flatness of a bundle implies numerical semipositivity, flatness alone does not, as shown by the following result ([[1]], Thm. 4): Let   be a Kodaira fibration, i.e., X is a surface and all the fibres of f are smooth curves not all isomorphic to each other. Then the direct image sheaf   has strictly positive degree hence   is a flat bundle which is not numerically semipositive .


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