Yang-Mills bar connections over compact K\"ahler manifolds

TL;DR

This paper introduces a new Yang-Mills bar equation on complex vector bundles over compact Hermitian manifolds, proves existence of solutions on K"ahler manifolds, and explores the flow and rigidity properties of these connections.

Contribution

It defines the Yang-Mills bar equation as an Euler-Lagrange equation, establishes existence results, and investigates the rigidity of holomorphic connections on K"ahler manifolds.

Findings

Existence of non-trivial solutions on compact K"ahler manifolds

Short time existence of the negative Yang-Mills bar gradient flow

Rigidity of holomorphic connections under positive Ricci curvature

Abstract

In this note we introduce a Yang-Mills bar equation on complex vector bundles over compact Hermitian manifolds as the Euler-Lagrange equation for a Yang-Mills bar functional. We show the existence of a non-trivial solution of this equation over compact K\"ahler manifolds as well as a short time existence of the negative Yang-Mills bar gradient flow. We also show a rigidity of holomorphic connections among a class of Yang-Mills bar connections over compact K\"ahler manifolds of positive Ricci curvature.