Coupled equations for Kähler metrics and Yang-Mills connections

We introduce a new system of partial differential equations coupling a Kähler metric on a compact complex manifold X and a connection on a principal bundle over X. These equations intertwine two well studied quantities, the first being the curvature of a Hermite--Yang--Mills connection (HYM) and the second being the scalar curvature of a Kähler metric. They depend on a positive real parameter $\alpha$ and have an interpretation in terms of a moment map, where the group of symmetries is an extension of the gauge group of the bundle that moves the base X. The problem considered merges the well-studied theories of Hermitian-Yang-Mills connections (obtained for $\alpha>0$) and constant scalar curvature Kähler metrics (which correspond to $\alpha=0$) into a unique theory. We use the moment map interpretation of the coupled equations to give necessary and sufficient conditions for the existence of solutions. Building on the work of A. Futaki, we provide an obstruction using an adapted version of the Futaki invariant for the coupled equations. We give a sufficient condition, obtained via a deformation argument, that is satisfied in a large family of examples. Relying on previous work of S. K. Donaldson, we define an algebraic (poly)stability condition for a pair consisting of a polarized variety and a holomorphic vector bundle, and conjecture that the existence of solutions implies the polystability of the pair. This is joint work with Luis Alvarez-Consul and Oscar Garcia-Prada (Madrid).