Geometry/Analytic geometry
On the stability of flat complex vector bundles over parallelizable manifolds
Comptes Rendus. Mathématique, Volume 356 (2018) no. 10, pp. 1030-1035.

We investigate the flat holomorphic vector bundles over compact complex parallelizable manifolds G/Γ, where G is a complex connected Lie group and Γ is a cocompact lattice in it. The main result proved here is a structure theorem for flat holomorphic vector bundles Eρ associated with any irreducible representation ρ:ΓGL(r,C). More precisely, we prove that Eρ is holomorphically isomorphic to a vector bundle of the form En, where E is a stable vector bundle. All the rational Chern classes of E vanish, in particular, its degree is zero.

We deduce a stability result for flat holomorphic vector bundles Eρ of rank 2 over G/Γ. If an irreducible representation ρ:ΓGL(2,C) satisfies the condition that the induced homomorphism ΓPGL(2,C) does not extend to a homomorphism from G, then Eρ is proved to be stable.

Nous étudions les fibrés holomorphes plats sur les variétés parallélisables compactes G/Γ (avec G un groupe de Lie connexe complexe et Γ un réseau cocompact). Notre résultat principal décrit les fibrés holomorphes plats Eρ associés à des représentations irréductibles ρ:ΓGL(r,C). Nous démontrons que ces fibrés Eρ sont isomorphes à une somme directe En, avec E un fibré vectoriel stable de degré zéro.

Nous en déduisons un résultat de stabilité concernant les fibrés holomorphes plats Eρ de rang 2 sur les quotients G/Γ. Si ρ:ΓGL(2,C) est une représentation irréductible telle que le morphisme induit ρ:ΓPGL(2,C) ne s'étend pas à G, alors Eρ est stable.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.08.001
Biswas, Indranil 1; Dumitrescu, Sorin 2; Lehn, Manfred 3

1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
2 Université Côte d'Azur, CNRS, LJAD, France
3 Institut für Mathematik, Johannes Gutenberg Universität Mainz, 55099 Mainz, Germany
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Biswas, Indranil; Dumitrescu, Sorin; Lehn, Manfred. On the stability of flat complex vector bundles over parallelizable manifolds. Comptes Rendus. Mathématique, Volume 356 (2018) no. 10, pp. 1030-1035. doi : 10.1016/j.crma.2018.08.001. http://www.numdam.org/articles/10.1016/j.crma.2018.08.001/

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