$J$-equation on holomorphic vector bundles

TL;DR

This paper introduces the $J$-equation for holomorphic vector bundles on compact Kähler manifolds, exploring its properties, stability conditions, and applications to related geometric equations.

Contribution

It defines the $J$-equation for vector bundles, establishes algebraic stability criteria, and connects to the deformed Hermitian-Yang-Mills equation in specific regimes.

Findings

Established algebraic (asymptotic) $J$-stability on Kähler surfaces.

Derived a numerical criterion for vortex bundles via dimensional reduction.

Applied results to the vector bundle version of the deformed Hermitian-Yang-Mills equation.

Abstract

We introduce the $J$-equation on holomorphic vector bundles over compact K\"ahler manifolds and investigate some fundamental properties as well as examples of solutions. In particular, we provide an algebraic condition called (asymptotic) $J$-stability in terms of subbundles on compact K\"ahler surfaces, and a numerical criterion on vortex bundles via dimensional reduction. Also, we discuss an application for the vector bundle version of the deformed Hermitian-Yang-Mills equation in the small volume regime.