# On coupled constant scalar curvature K\"ahler metrics

## TL;DR

This paper introduces a geometric framework for coupled constant scalar curvature K"ahler (cscK) metrics, generalizing existing equations, defining stability notions, and proving existence results under perturbations.

## Contribution

It provides a moment map interpretation for coupled cscK equations, introduces a new system of equations, and establishes a stability criterion and existence results for solutions.

## Key findings

- Moment map interpretation for coupled cscK equations
- Introduction of a more general coupled cscK system
- Existence of solutions under small K-polystable perturbations

## Abstract

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.