$Z$-critical equations for holomorphic vector bundles on K\"ahler surfaces

TL;DR

This paper establishes a link between the existence of certain special Hermitian metrics on rank 2 holomorphic bundles over K"ahler surfaces and their stability, with implications for equations like deformed Hermitian Yang-Mills.

Contribution

It proves that $Z$-positive and $Z$-critical Hermitian metrics imply $Z$-stability for rank 2 bundles on K"ahler surfaces, extending stability results to new equations.

Findings

Existence of $Z$-critical metrics implies $Z$-stability.

Results apply to deformed Hermitian Yang-Mills and almost Hermite-Einstein equations.

Examples of $Z$-unstable bundles and metrics outside large volume limit.

Abstract

We prove that the existence of a $Z$-positive and $Z$-critical Hermitian metric on a rank 2 holomorphic vector bundle over a compact K\"ahler surface implies that the bundle is $Z$-stable. As particular cases, we obtain stability results for the deformed Hermitian Yang-Mills equation and the almost Hermite-Einstein equation for rank 2 bundles over surfaces. We show examples of $Z$-unstable bundles and $Z$-critical metrics away from the large volume limit.