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

Abstract

This is the second in a series of two papers studying µ-cscK metrics and µK-stability from a new perspective, inspired by observations on µcharacter in [Ino3] and on Perelman's W -entropy in the first paper [Ino4].This second paper is devoted to studying a non-archimedean counterpart of Perelman's µ-entropy. The concept originally appeared as µ-character of polarized family in the previous research [Ino3], where we used it to introduce an analogue of CM line bundle adapted to µK-stability.We firstly show some differential of the characteristic µ-entropy μλ ch is the minus of µ λ -Futaki invariant, which connects µ λ K-semistability to the maximization of characteristic µ λ -entropy. It in particular provides us a criterion for µ λ K-semistability working without detecting the vector ξ involved in the µ λ ξ -Futaki invariant. We observe a family of filtrations {F ξ+τ (X ,L) } τ ∈[0,∞) associated to a test configuration (X , L) and a vector field ξ acting on (X, L) to consider the differential. We conceptualize such family of filtrations as polyhedral configuration and study its generalities. The concept implicitly appeared in many literatures involved in R-test configuration.In the latter part, we propose a non-archimedean pluripotential approach to the maximization problem. In order to adjust the characteristic µ-entropy μλ ch to Boucksom-Jonsson's non-archimedean framework, we introduce a natural modification μλ NA which we call non-archimedean µ-entropy. We extend the non-archimedean µ-entropy from the set of test configurations to a space E exp NA (X, L) of non-archimedean psh metrics on the Berkovich space X NA , which is endowed with a complete metric structure. We introduce moment measure χDϕ on Berkovich space for this sake, which can be considered as a hybrid of Monge-Ampère measure and Duistermaat-Heckman measure.We also compare our µ-framework with other frameworks: H-entropy framework in the context of Kähler-Ricci flow and Calabi energy framework in the context of Calabi flow. Some illustrations by toric examples are attached in Appendix.