Geodesic
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

Catanese, Fabrizio1 ; Dettweiler, Michael1
1 Mathematisches Institut, Universität Bayreuth, 95447 Bayreuth, Germany
Comptes Rendus. Mathématique, Tome 352 (2014) no. 3, pp. 241-244.

Voir la notice de l'article provenant de la source Numdam

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

Catanese, Fabrizio 1 ; Dettweiler, Michael 1

1 Mathematisches Institut, Universität Bayreuth, 95447 Bayreuth, Germany 
@article{CRMATH_2014__352_3_241_0,
     author = {Catanese, Fabrizio and Dettweiler, Michael},
     title = {The direct image of the relative dualizing sheaf needs not be semiample},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {241--244},
     publisher = {Elsevier},
     volume = {352},
     number = {3},
     year = {2014},
     doi = {10.1016/j.crma.2013.12.015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2013.12.015/}
}
TY  - JOUR
AU  - Catanese, Fabrizio
AU  - Dettweiler, Michael
TI  - The direct image of the relative dualizing sheaf needs not be semiample
JO  - Comptes Rendus. Mathématique
PY  - 2014
SP  - 241
EP  - 244
VL  - 352
IS  - 3
PB  - Elsevier
UR  - http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2013.12.015/
DO  - 10.1016/j.crma.2013.12.015
LA  - en
ID  - CRMATH_2014__352_3_241_0
ER  - 
%0 Journal Article
%A Catanese, Fabrizio
%A Dettweiler, Michael
%T The direct image of the relative dualizing sheaf needs not be semiample
%J Comptes Rendus. Mathématique
%D 2014
%P 241-244
%V 352
%N 3
%I Elsevier
%U http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2013.12.015/
%R 10.1016/j.crma.2013.12.015
%G en
%F CRMATH_2014__352_3_241_0
Catanese, Fabrizio; Dettweiler, Michael. The direct image of the relative dualizing sheaf needs not be semiample. Comptes Rendus. Mathématique, Tome 352 (2014) no. 3, pp. 241-244. doi : 10.1016/j.crma.2013.12.015. http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2013.12.015/

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

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

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

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

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

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

[9] Hartshorne, R. 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. (Ueno, Kenji, ed.) (Progress in Mathematics), Volume vol. 39, Birkhäuser, Boston, Mass. (1983), pp. 591-630 (Open problems: Classification of algebraic and analytic manifolds, 1982)

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

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

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

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

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

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

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

Cité par Sources :