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 ${\check{\mu }}^{\lambda }$ 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 ${\mu }_{\xi }^{\lambda }$-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 ${\check{\mu }}^{\lambda }$ to Boucksom--Jonsson's non-archimedean framework, we introduce a natural modification ${\check{\mu }}_{\mathrm{NA}}^{\lambda }$ 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 $\int \chi {\mathcal{D}}_{\phi }$ on Berkovich space called moment measure for this sake, which can be considered as a hybrid of Monge--Ampère measure and Duistermaat--Heckman measure.

136 pages, Comments are very welcome!