On the stability of flat complex vector bundles over parallelizable manifolds

TL;DR

This paper studies the structure and stability of flat holomorphic vector bundles over compact complex parallelizable manifolds, revealing their decomposition into stable bundles with vanishing Chern classes and establishing stability criteria for rank 2 bundles.

Contribution

It provides a structure theorem for flat holomorphic vector bundles over such manifolds, showing they decompose into stable bundles with zero Chern classes, and establishes stability conditions for rank 2 bundles.

Findings

Flat holomorphic vector bundles decompose into stable bundles with zero Chern classes.

All rational Chern classes of the stable bundle vanish.

Rank 2 bundles are stable under certain irreducibility conditions.

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 $\rho : \Gamma \rightarrow \text{GL}(r,{\mathbb C})$. 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 $\rho : \Gamma\rightarrow \text{GL}(2, \mathbb {C})$ satisfies the conditionmthat the induced homomorphism $\Gamma\rightarrow {\rm PGL}(2, {\mathbb C})$ does not extend to a homomorphism from $G$, then $E_{\rho}$ is proved to be stable.