论文标题
$μ$ $ $ - 规范指标和k稳定性的熵，ii-非架构的方面：非架构
Entropies in $μ$-framework of canonical metrics and K-stability, II -- Non-archimedean aspect: non-archimedean $μ$-entropy and $μ$K-semistability
论文作者
论文摘要
这是两篇研究$ $ -CSCK指标的第二篇论文中的第二篇，从新的角度来看，启发了Arxiv $μ$ - character：2004.06393以及Perelman的$ W $ -Entropy在第一篇论文Arxiv：2101.111197中的观察。 第二篇论文致力于研究Perelman的$μ$ entropy的非Archimedean对应物。该概念最初以$μ$ $ $ - 偏光族的character在先前的研究ARXIV：2004.06393中，在那里我们使用它引入了CM线束的类似物，适用于$ $ $ k稳定性。 首先，我们显示了特征性$μ$ -entropy $ \ mathbf {\checkμ}^λ$的差异，是$μ^λ$ -futaki不变性的负，它连接$μ^λ$ k semistability conlence uncipitiants $μ^λ$ k- semistability。它特别为我们提供了一个$μ^λ$ k的标准，而无需检测到$μ^λ_ξ$ -futaki不变的矢量$ξ$。 在后一部分中，我们提出了针对最大化问题的非架构的多能元方法。为了调整特征$ $ $ -Entropy $ \ Mathbf {\checkμ}^λ$，向boucksom---琼斯森的非架构框架，我们引入了自然修改$ \ mathbf {\ mathbf {\checkμ}^λ_{^λ_{\ mathrm {na na rm {na} $ non-non-non-non-non-non-non-ark $ arark。我们将非Archimedean $ $ $ entropy从一组测试配置扩展到空间$ \ MATHCAL {e}^{\ exp}（X，x，x，x，x，x，l）$ berkimedean PSH指标上的Berkovich Space $ x^{\ Mathrm {na}} $的berkovich space $ x^{\ x^{\ x，na}} $。为此，我们在伯科维奇空间上引入了一个尺寸$ \ intχ\ Mathcal {d}_φ$，以此为artment Measues，可以将其视为Monge的混合体 - AmpèreMeasureMesuare and Duistermaat-Heckman-Heckman Measure。
This is the second in a series of two papers studying $μ$-cscK metrics and $μ$K-stability from a new perspective, inspired by observations on $μ$-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 $μ$-entropy. The concept originally appeared as $μ$-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 $μ$K-stability. We firstly show some differential of the characteristic $μ$-entropy $\mathbf{\checkμ}^λ$ 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. In the latter part, we propose a non-archimedean pluripotential approach to the maximization problem. In order to adjust the characteristic $μ$-entropy $\mathbf{\checkμ}^λ$ to Boucksom--Jonsson's non-archimedean framework, we introduce a natural modification $\mathbf{\checkμ}^λ_{\mathrm{NA}}$ which we call non-archimedean $μ$-entropy. We extend the non-archimedean $μ$-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 $\int χ\mathcal{D}_φ$ on Berkovich space called moment measure for this sake, which can be considered as a hybrid of Monge--Ampère measure and Duistermaat--Heckman measure.
