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.

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.

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/

[1] F. Catanese; M. Dettweiler Answer to a question by Fujita on variation of Hodge structures, 2013 (preprint, 26 pages) | arXiv

[2] P. Deligne; G.D. Mostow Monodromy of hypergeometric functions and non-lattice integral monodromy, Publ. Math. IHÉS, Volume 63 (1986), pp. 5-89

[3] Takao Fujita On Kähler fiber spaces over curves, J. Math. Soc. Jpn., Volume 30 (1978) no. 4, pp. 779-794

[4] Takao Fujita The sheaf of relative canonical forms of a Kähler fiber space over a curve, Proc. Jpn. Acad., Ser. A, Math. Sci., Volume 54 (1978) no. 7, pp. 183-184

[5] P. Griffiths Periods of integrals on algebraic manifolds. III. Some global differential-geometric properties of the period mapping, Publ. Math. IHÉS, Volume 38 (1970), pp. 125-180

[6] P. Griffiths Topics in Transcendental Algebraic Geometry, Annals of Mathematics Studies, vol. 106, Princeton University Press, 1984

[7] P. Griffiths; J. Harris Principles of Algebraic Geometry, Pure and Applied Mathematics, Wiley-Interscience, New York, 1978

[8] P. Griffiths; W. Schmid Recent developments in Hodge theory: A discussion of techniques and results, 1973 (1975), pp. 31-127

[9] R. Hartshorne Ample vector bundles on curves, Nagoya Math. J., Volume 43 (1971), pp. 73-89

[10] Classification of algebraic and analytic manifolds, Proc. Symp. Katata/Jap. (Kenji Ueno, ed.) (Progress in Mathematics), Volume vol. 39, Birkhäuser, Boston, Mass. (1983), pp. 591-630 (Open problems: Classification of algebraic and analytic manifolds, 1982)

[11] Y. Kawamata Kodaira dimension of algebraic fiber spaces over curves, Invent. Math., Volume 66 (1982) no. 1, pp. 57-71

[12] G. Kempf; F.F. Knudsen; D. Mumford; B. Saint Donat Toroidal Embeddings, I, Lecture Notes in Mathematics, vol. 739, Springer, 1973 (viii+209 p)

[13] M. Kohno Global Analysis in Linear Differential Equations, Kluwer Academic Publishers, 1999

[14] J. Kollár Higher direct images of dualizing sheaves. I, II, Ann. Math. (2), Volume 123 (1986), pp. 11-42

[15] R. Lazarsfeld Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3, vol. 49, Springer-Verlag, Berlin, 2004 (xviii+385 p)

[16] C.A.M. Peters A criterion for flatness of Hodge bundles over curves and geometric applications, Math. Ann., Volume 268 (1984) no. 1, pp. 1-19

[17] W. Schmid Variation of Hodge structure: The singularities of the period mapping, Invent. Math., Volume 22 (1973), pp. 211-319

[18] H.A. Schwarz Über diejenigen Fälle in welchen die Gaussische hypergeometrische Reihe eine algebraische Funktion ihres vierten Elements darstellt, J. Reine Angew. Math., Volume 75 (1873), pp. 292-335

[19] S. Zucker Hodge theory with degenerating coefficients: L2-cohomology in the Poincaré metric, Ann. Math. (2), Volume 109 (1979), pp. 415-476

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