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.

中文翻译：

---

关于耦合常数标量曲率 Kähler 度量

我们为 Hultgren 和 Witt Nystr\"om 引入的耦合 K\"ahler-Einstein 方程提供了矩图解释，并在此过程中引入了一个更通用的方程组，我们称之为耦合 cscK 方程。获得了相应 Futaki 不变量的微分几何公式，并为这个新系统定义了 K 多稳定性的概念。最后，在 Sz\'ekelyhidi 的结果的推动下，我们证明如果我们的方程有解，那么底层复杂结构和极化的小 K 多稳态扰动也允许耦合 cscK 度量。