no. 9
Algebraic Geometry
Holomorphic connections on some complex manifolds
[Connexions holomorphes sur quelques variétés complexes]

Présenté par : Jean-Pierre Demailly

Biswas, Indranil1 ; Iyer, Jaya N.2
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
2 The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
Comptes Rendus. Mathématique, Tome 344 (2007) no. 9, pp. 577-580.

Soit M une variété complexe compacte connexe, munie d'une submersion holomorphe M→T, où T est un tore complexe, telle que les fibres soient rationnellement connexes. Soit E un fibré vectoriel holomorphe sur M admettant une connexion. Alors E admet une connexion holomorphe plate. Un énoncé similaire vaut pour tout quotient fini de M.

Let M be a compact connected complex manifold equipped with a holomorphic submersion to a complex torus such that the fibers are all rationally connected. Then any holomorphic vector bundle over M admitting a holomorphic connection actually admits a flat holomorphic connection. A similar statement is valid for any finite quotient of M.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.03.030
Biswas, Indranil 1 ; Iyer, Jaya N. 2

1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
2 The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India 
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Biswas, Indranil; Iyer, Jaya N. Holomorphic connections on some complex manifolds. Comptes Rendus. Mathématique, Tome 344 (2007) no. 9, pp. 577-580. doi : 10.1016/j.crma.2007.03.030. http://www.numdam.org/articles/10.1016/j.crma.2007.03.030/

[1] Araujo, C.; Kollár, J. Rational curves on varieties, Budapest, 2001 (Bolyai Soc. Math. Stud.), Volume vol. 12, Springer, Berlin (2003), pp. 13-68

[2] Atiyah, M.F. Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc., Volume 85 (1957), pp. 181-207

[3] I. Biswas, T.L. Gómez, Connections and Higgs fields on a principal bundle, Ann. Global Anal. Geom., in press

[4] Campana, F. On twistor spaces of the class C, J. Differential Geom., Volume 33 (1991), pp. 541-549

[5] Demailly, J.-P.; Peternell, T.; Schneider, M. Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom., Volume 3 (1994), pp. 295-345

[6] Grothendieck, A. Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math., Volume 79 (1957), pp. 121-138

[7] Kollár, J. Fundamental groups of rationally connected varieties, Michigan Math. J., Volume 48 (2000), pp. 359-368

[8] Kollár, J.; Miyaoka, Y.; Mori, S. Rationally connected varieties, J. Algebraic Geom., Volume 1 (1992), pp. 429-448

[9] Kollár, J.; Miyaoka, Y.; Mori, S. Rational connectedness and boundedness for Fano manifolds, J. Differential Geom., Volume 36 (1992), pp. 765-779

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