dHYM connections coupled to a variable Kähler metric

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 Kahler metric, to be constant. In this paper we introduce and study dHYM equations with variable Kahler 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 Kahler-Yang-Mills system of Alvarez-Consul, Garcia-Fernandez and Garcıa-Prada, and ́the coupled Kahler-Einstein equations of Hultgren-Witt Nystrom. After establishing several general results we focus on the equations and their large/small radius limits on abelian varieties, with a source term, and on ruled surfaces, allowing solutions to develop conical singularities.