On coupled constant scalar curvature Kähler metrics

We provide a moment map interpretation for the coupled Kähler–Einstein equations introduced in [16], 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.

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Received 20 March 2019

Accepted 2 September 2019

Published 28 October 2020