ON THE STABILITY OF FLAT COMPLEX VECTOR BUNDLES OVER PARALLELIZABLE MANIFOLDS
Sur la stabilité des fibrés plats au-dessus des variétés parallélisables
Résumé
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 to any irreducible representation ρ : Γ −→ GL(d, C). More precisely, we prove that Eρ is holomorphically isomorphic to a vector bundle of the form E⊕n , 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 E ρ of rank 2 over compact quotients SL(2, C)/Γ. If an irreducible homomorphism ρ from Γ to SL(2, C) satisfies the condition that the projection Γ → PGL(2, C), obtained by composing of ρ with the projection of SL(2, C) to PGL(2, C), does not extend to SL(2, C), then E ρ is proved to be stable.
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)
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