A remark on "Connections and Higgs fields on a principal bundle"

TL;DR

This paper investigates the existence of flat holomorphic and unitary flat connections on certain vector bundles over non-Kaehler and Gauduchon manifolds, providing counterexamples to common assumptions.

Contribution

It demonstrates that unipotent and topologically trivial stable vector bundles can lack flat connections on non-Kaehler and Gauduchon manifolds, respectively.

Findings

Unipotent vector bundles on non-Kaehler manifolds may not admit flat holomorphic connections.

Stable vector bundles on Gauduchon manifolds can lack unitary flat connections.

Counterexamples challenge the generality of flat connection existence in complex geometry.

Abstract

We show that a unipotent vector bundle on a non-Kaehler compact complex manifold does not admit a flat holomorphic connection in general. We also construct examples of topologically trivial stable vector bundle on compact Gauduchon manifold that does not admit any unitary flat connection.