Complex structures on products of circle bundles over complex manifolds

Sankaran, Parameswaran; Thakur, Ajay Singh

doi:10.1016/j.crma.2011.02.016

Algebraic Geometry/Analytic Geometry

Complex structures on products of circle bundles over complex manifolds
[Structures complexes sur les produits de fibrés en cercles au dessus des variétés complexes]

Présenté par : Jean-Pierre Demailly

Sankaran, Parameswaran ¹ ; Thakur, Ajay Singh ¹

¹ The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India

Comptes Rendus. Mathématique, Tome 349 (2011) no. 7-8, pp. 437-439.

Résumé
Abstract

Dans ce Note, on propose une construction de structures complexes sur le produit de deux fibré en cercles associés aux fibrés en droites, amples, négatifs sur des variétés drapeaux $X_{i} = G_{i} / P_{i}$ , $i = 1, 2$ , où les $G_{i}$ sont des groupes de Lie linéaires connexes, complexes, semi-simples et les $P_{i} \subset G_{i}$ sont des sous-groupes paraboliques. La variété construite S nʼest pas symplectique et donc nʼest pas kählérienne. On démontre que le groupe ${Pic}^{0} (S)$ des fibrés en droites holomorphes topologiquement triviaux est isomorphe aux nombres complexes $C$ .

We propose, in this Note, a construction of complex structures on the product of two circle bundles associated to negative ample line bundles over flag varieties $X_{i} : = G_{i} / P_{i}$ , $i = 1, 2$ , where the $G_{i}$ are complex semisimple linear Lie groups and the $P_{i} \subset G_{i}$ are parabolic subgroups. The resulting manifold S is non-symplectic and hence non-Kählerian. We show that the group ${Pic}^{0} (S)$ of topologically trivial holomorphic line bundles on S is isomorphic to $C$ .

Reçu le : 2010-12-01
Accepté le : 2011-02-17
Publié le : 2011-03-02

DOI : 10.1016/j.crma.2011.02.016

Affiliations des auteurs :

Sankaran, Parameswaran ¹ ; Thakur, Ajay Singh ¹

¹ The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India

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     title = {Complex structures on products of circle bundles over complex manifolds},
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Sankaran, Parameswaran; Thakur, Ajay Singh. Complex structures on products of circle bundles over complex manifolds. Comptes Rendus. Mathématique, Tome 349 (2011) no. 7-8, pp. 437-439. doi : 10.1016/j.crma.2011.02.016. http://www.numdam.org/articles/10.1016/j.crma.2011.02.016/

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