Coupled free energy and a diagonal approach to coupled K\"{a}hler-Einstein metrics

arXiv:2611.06968v1 Announce Type: new Abstract: In this paper, we introduce the coupled free energy $F_{-\gamma}^{\rm cp}$ which is a natural generalization of the free energy functional defined by Berman. In the Fano case, our coupled free energy provides a diagonal approach to the coercivity of the coupled Ding functional. We also define the non-Archimedean coupled free energy $F_{-\gamma}^{\rm cp,NA}$ on the space of finite energy measures on the Berkovich analytification. As a consequence of the regularization process of the non-Archimedean entropy established by Boucksom-Jonsson, Chi Li and Trusiani, we prove that the existence of a unique coupled K\"{a}hler-Einstein metric is equivalent to the stability condition that there exists $\epsilon >0$ such that $F_{-1-\epsilon}^{\rm cp,NA} \geq 0$ on the space of measures associated with a test configuration for the anticanonical line bundle. As a coupled version of the work by Berman, we show that the law of the empirical measure satisfies a large deviation principle whose rate function is the coupled free energy under the assumption that a conjecture by Hultgren is true. We also introduce a coupled version of the microscopic stability thresholds and prove that a uniformly coupled Gibbs stable Fano manifold has a unique coupled K\"{a}hler-Einstein metric. This is a generalization of the results by Fujita-Odaka and Berman.

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