On coupled constant scalar curvature K\"ahler metrics

We provide a moment map interpretation for the coupled K"ahler-Einstein equations introduced by Hultgren and Witt Nystr"om, and in the process introduce a more general system of equations, which we call coupled cscK equations. A differentio-geometric formulation of the corresponding Futaki invariant is obtained and a notion of K-polystability is defined for this new system. Finally, motivated by a result of Sz'ekelyhidi, we prove that if there is a solution to our equations, then small K-polystable perturbations of the underlying complex structure and polarizations also admit coupled cscK metrics.