Deformed Hermitian Yang-Mills connections, extended gauge group and scalar curvature

TL;DR

This paper introduces and studies coupled deformed Hermitian Yang-Mills equations with variable K"ahler metrics, connecting complex geometry, stability conditions, and scalar curvature through an extended gauge group framework.

Contribution

It develops a new coupled system of equations involving dHYM and scalar curvature, extending previous models and linking stability notions in complex geometry.

Findings

Derived general properties of the coupled dHYM equations.

Established limits and special cases recovering known systems.

Analyzed solutions on abelian varieties with source terms.

Abstract

The deformed Hermitian Yang-Mills (dHYM) equation is a special Lagrangian type condition in complex geometry. It requires the complex analogue of the Lagrangian phase, defined for Chern connections on holomorphic line bundles using a background K\"ahler metric, to be constant. In this paper we introduce and study dHYM equations with variable K\"ahler metric. These are coupled equations involving both the Lagrangian phase and the radius function, at the same time. They are obtained by using the extended gauge group to couple the moment map interpretation of dHYM connections, due to Collins-Yau and mirror to Thomas' moment map for special Lagrangians, to the Donaldson-Fujiki picture of scalar curvature as a moment map. As a consequence one expects that solutions should satisfy a mixture of K-stability and Bridgeland-type stability. In special limits, or in special cases, we recover the K\"ahler-Yang-Mills system of \'Alvarez-C\'onsul, Garcia-Fernandez and Garc\'ia-Prada, and the coupled K\"ahler-Einstein equations of Hultgren-Witt Nystr\"om. After establishing several general results we focus on the equations and their large/small radius limits on abelian varieties, with a source term, following ideas of Feng and Sz\'ekelyhidi.