On coupled constant scalar curvature Kähler metrics

Abstract

We provide a moment map interpretation for the coupled Kähler-Einstein equations introduced by Hultgren and Witt Nyström, 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 Székelyhidi, 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.

21 pages. Corrected minor typos, included additional citations, clarified the definition of the coupled Futaki invariant