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

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 Kähler metric, to be constant. In this paper, we introduce and study dHYM equations with variable Kähler 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 Kähler–Yang–Mills system of Álvarez–Cónsul, Garcia–Fernandez and García–Prada, and the coupled Kähler–Einstein equations of Hultgren–Witt Nyström. 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 Székelyhidi.