On the stability of flat complex vector bundles over parallelizable manifolds

Abstract

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:Γ⟶GL(r,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 Eρ of rank 2 over G/Γ. If an irreducible representation ρ:Γ⟶GL(2,C) satisfies the condition that the induced homomorphism Γ⟶PGL(2,C) does not extend to a homomorphism from G, then Eρ is proved to be stable.