Entropies in $μ$-framework of canonical metrics and K-stability, I -- Archimedean aspect: Perelman's W-entropy and $μ$-cscK metrics

Abstract

This is the first in a series of two papers (cf. [Ino4]) studying µ-cscK metrics and µK-stability, from a new perspective evoked from observations in [Ino3] and in this first paper.The first paper is about a characterization of µ-cscK metrics in terms of Perelman's W -entropy W λ . We regard Perelman's W -entropy as a functional on the tangent bundle T H(X, L) of the space H(X, L) of Kähler metrics in a given Kähler class L. The critical points of W λ turn out to be µ λ -cscK metrics. When λ ≤ 0, the supremum along the fibres gives a smooth functional on H(X, L), which we call µ-entropy. Then µ λ -cscK metrics are also characterized as critical points of this functional, similarly as extremal metric is characterized as the critical points of Calabi functional.We also prove the W -entropy is monotonic along geodesics, following Berman-Berndtsson's subharmonicity argument. Studying the limit of the W -entropy, we obtain a lower bound of the µ-entropy. This bound is not just analogous, but indeed related to Donaldson's lower bound on Calabi functional by the extremal limit λ → −∞.

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

“…This result is an algebraic counterpart of Theorem 1.4 in [Ino4] on Perelman's µ-entropy. We will reinterpret this theorem as Theorem 1.10 in terms of nonarchimedean formalism.…”

“…In this second paper of the series, we explore an invariant for test configuration called µ-entropy in the first paper [Ino4] and its connection to µK-semistability introduced in [Ino3] (see also [Lah1,Ino2]). This paper consists of three parts.…”

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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.

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“…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].…”

Extremal metrics on fibrations

“…In the projective case, R-test configuration can be encoded using R-filtrations of the coordinate ring of (X, L); the reason we use the above geometric interpretation is the lack of a good analogue of such filtrations in the Kähler setting. We also note that R-test configuration are essentially the same as what Inoue calls polyhedral test configurations [22] and what were originally called R-degenerations in [13]; we use the language of Boucksom-Jonsson [5]. These degenerations seem to have first appeared in the work of Chen-Sun-Wang [8].…”