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

Présenté par : Jean-Pierre Demailly

Indranil Biswas1 ; Jaya N. Iyer2
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, Volume 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
Indranil Biswas 1 ; Jaya N. Iyer 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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Indranil Biswas; Jaya N. Iyer. Holomorphic connections on some complex manifolds. Comptes Rendus. Mathématique, Volume 344 (2007) no. 9, pp. 577-580. doi : 10.1016/j.crma.2007.03.030. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.03.030/

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

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

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

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