Hermitian manifolds with flat Gauduchon connections

TL;DR

This paper classifies compact Hermitian manifolds with flat Gauduchon connections, proving they are Kähler unless they have flat Chern or Bismut connections, and extends results to Kähler-like conditions and non-compact cases.

Contribution

It completes the classification of such manifolds, confirming conjectures and extending results to broader conditions and non-compact cases.

Findings

Manifolds with flat Gauduchon connections are mostly Kähler.

Confirmed Yang and Zheng's conjecture for compact cases.

Extended classification to Kähler-like conditions and non-compact manifolds.

Abstract

We complete the classification of compact Hermitian manifolds admitting a flat Gauduchon connection. In particular, we establish a conjecture of Yang and Zheng, showing that apart from the cases of a flat Chern or Bismut connection, such manifolds are K\"ahler. More generally, we prove the same result holds when the flatness assumption is replaced by the so-called K\"ahler-like condition, proving a conjecture of Angella, Otal, Ugarte and Villacampa. We also treat the non-compact case.