Hermitian-Einstein connections on principal bundles over flat affine manifolds

TL;DR

This paper establishes a correspondence between Hermitian-Einstein structures and polystability for flat principal bundles over special flat affine manifolds, extending complex geometric stability results to affine settings.

Contribution

It proves that flat principal G-bundles over such manifolds admit Hermitian-Einstein structures if and only if they are polystable, and establishes the uniqueness of these structures and filtrations.

Findings

Hermitian-Einstein structures exist iff the bundle is polystable

Unique Hermitian-Einstein connections for polystable bundles

Existence and uniqueness of Harder-Narasimhan filtration for flat vector bundles

Abstract

Let $M$ be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric $g$ and a covariant constant volume form. Let $G$ be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal $G$-bundle $E_G$ over $M$ admits a Hermitian-Einstein structure if and only if $E_G$ is polystable. A polystable flat principal $G$--bundle over $M$ admits a unique Hermitian-Einstein connection. We also prove the existence and uniqueness of a Harder-Narasimhan filtration for flat vector bundles over $M$.