Hermitian-Yang-Mills connections on some complete non-compact K\"ahler manifolds

TL;DR

This paper establishes an algebraic criterion linking a new stability condition to the existence of Hermitian-Yang-Mills metrics on vector bundles over certain complete non-compact K"ahler manifolds, extending classical results.

Contribution

It introduces a generalized stability notion for vector bundles on non-compact K"ahler manifolds and proves its equivalence to the existence of Hermitian-Yang-Mills metrics.

Findings

New stability condition is necessary and sufficient for existence of Hermitian-Yang-Mills metrics.

Provides algebraic criteria for metrics on bundles over non-compact K"ahler manifolds.

Extends classical Hermitian-Yang-Mills theory to non-compact settings.

Abstract

We give an algebraic criterion for the existence of projectively Hermitian-Yang-Mills metrics on a holomorphic vector bundle $E$ over some complete non-compact K\"ahler manifolds $(X,\omega)$, where $X$ is the complement of a divisor in a compact K\"ahler manifold and we impose some conditions on the cohomology class and the asymptotic behaviour of the K\"ahler form $\omega$. We introduce the notion of stability with respect to a pair of $(1,1)$-classes which generalizes the standard slope stability. We prove that this new stability condition is both sufficient and necessary for the existence of projectively Hermitian-Yang-Mills metrics in our setting.