On Gauduchon K\"ahler-like manifolds

TL;DR

This paper investigates the conditions under which Gauduchon connections on compact Hermitian manifolds imply the metric is K"ahler, proving a conjecture and exploring duality phenomena among these connections.

Contribution

The paper proves the conjecture that two K"ahler-like Gauduchon connections imply a K"ahler metric and discusses partial results related to the conjecture involving a single such connection.

Findings

Proof of the conjecture that two K"ahler-like Gauduchon connections imply a K"ahler metric

Discovery of a duality phenomenon among Gauduchon connections

Partial answers to the conjecture involving a single K"ahler-like Gauduchon connection

Abstract

In a paper by Angella, Otal, Ugarte, and Villacampa, the authors conjectured that on a compact Hermitian manifold, if a Gauduchon connection other than Chern or Strominger is K\"ahler-like, then the Hermitian metric must be K\"ahler. They also conjectured that if two Gauduchon connections are both K\"ahler-like, then the metric must be K\"ahler. In this paper, we discuss some partial answers to the first conjecture, and give a proof to the second conjecture. In the process, we discovered an interesting `duality' phenomenon amongst Gauduchon connections, which seems to be intimately tied to the question, though we do not know if there is any underlying reason for that from physics.