Entropies in $\mu$-framework of canonical metrics and K-stability, II -- Non-archimedean aspect: non-archimedean $\mu$-entropy and $\mu$K-semistability

Abstract

This is the second in a series of two papers studying $\mu$-cscK metrics and $\mu$K-stability from a new perspective, inspired by observations on $\mu$-character in arXiv:2004.06393 and on Perelman's $W$-entropy in the first paper arXiv:2101.11197. This second paper is devoted to studying a non-archimedean counterpart of Perelman's $\mu$-entropy. The concept originally appeared as $\mu$-character of polarized family in the previous research arXiv:2004.06393, where we used it to introduce an analogue of CM line bundle adapted to $\mu$K-stability. We firstly show some differential of the characteristic $\mu$-entropy $\mathbf{\check{\mu}}^\lambda$ is the minus of $\mu^\lambda$-Futaki invariant, which connects $\mu^\lambda$K-semistability to the maximization of characteristic $\mu^\lambda$-entropy. It in particular provides us a criterion for $\mu^\lambda$K-semistability working without detecting the vector $\xi$ involved in the $\mu^\lambda_\xi$-Futaki invariant. In the latter part, we propose a non-archimedean pluripotential approach to the maximization problem. In order to adjust the characteristic $\mu$-entropy $\mathbf{\check{\mu}}^\lambda$ to Boucksom--Jonsson's non-archimedean framework, we introduce a natural modification $\mathbf{\check{\mu}}^\lambda_{\mathrm{NA}}$ which we call non-archimedean $\mu$-entropy. We extend the non-archimedean $\mu$-entropy from the set of test configurations to a space $\mathcal{E}^{\exp} (X, L)$ of non-archimedean psh metrics on the Berkovich space $X^{\mathrm{NA}}$, which is endowed with a complete metric structure. We introduce a measure $\int \chi \mathcal{D}_\varphi$ on Berkovich space called moment measure for this sake, which can be considered as a hybrid of Monge--Amp\`ere measure and Duistermaat--Heckman measure.