On the stability of flat complex vector bundles over parallelizable manifolds

Abstract

We investigate the flat holomorphic vector bundles over compact complex parallelizable manifolds $G/\Gamma$, where $G$ is a complex connected Lie group and $\Gamma$ is a cocompact lattice in it. The main result proved here is a structure theorem for flat holomorphic vector bundles $E_{\rho }$ associated to any irreducible representation $\rho :\Gamma \to \text{GL}(r,\mathbb{C})$. More precisely, we prove that $E_{\rho }$ is holomorphically isomorphic to a vector bundle of the form $E^{\oplus n}$, 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 $E_{\rho }$ of rank 2 over $G/\Gamma$. If an irreducible representation $\rho :\Gamma \to \text{GL}(2,\mathbb{C})$ satisfies the conditionmthat the induced homomorphism $\Gamma \to \mathrm{PGL}(2,\mathbb{C})$ does not extend to a homomorphism from $G$, then $E_{\rho }$ is proved to be stable.

Comptes Rendus Mathématique (to appear)