ON THE STABILITY OF FLAT COMPLEX VECTOR BUNDLES OVER PARALLELIZABLE MANIFOLDS

Sur la stabilité des fibrés plats au-dessus des variétés parallélisables

Indranil Biswas (1) , Sorin Dumitrescu (2) , Manfred Lehn (3)

1 TIFR - Tata Institute for Fundamental Research
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 MI - Institut für Mathematik

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.

Mots clés: Parallelizable manifold; stable bundle; cocompact lattice; deformations

Domaines: Mathématiques [math]

Indranil Biswas, Sorin Dumitrescu, Manfred Lehn. ON THE STABILITY OF FLAT COMPLEX VECTOR BUNDLES OVER PARALLELIZABLE MANIFOLDS. Comptes Rendus. Mathématique, 2018, 356 (10), pp.1030-1035. ⟨10.1016/j.crma.2018.08.001⟩. ⟨hal-01588411v2⟩