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Skoda-Zeriahi type integrability and entropy compactness for some measure with $L^1$-density

Takahiro Aoi

TL;DR

The paper advances Skoda–Zeriahi type integrability to measures with $L^1$-density $d\mu_\gamma$ on bounded pseudoconvex domains by introducing a log-log threshold $\nu_D$ to quantify potential singularities. It proves a positivity lower bound for the integrability threshold of $\mathcal{U}\subset PSH(X,\omega)$ relative to $d\mu_\gamma$, and develops a local $L^p(d\mu_\gamma)$-convergence theory that transfers $L^1$-convergence to stronger norms under uniform exponential integrability. A central achievement is the entropy-compactness theorem for families of Poincaré-type Kähler potentials in the finite energy space $\mathcal{E}^1(X,\omega)$, with entropy measured against the singular measure $d\mu_\gamma$, enabling variational approaches to canonical metrics of Poincaré type. The results connect integrability, entropy, and geometric analysis in a way that extends classical $L^p$-density theories to highly singular measures and informs Mabuchi-type variational problems in settings with cusp-like asymptotics.

Abstract

In this paper, we prove the Skoda-Zeriahi type integrability theorem with respect to some measure with $L^1$-density. In addition, we introduce the log-log threshold in order to detect singularities of Kähler potentials. We prove the positivity of the integrability threshold for such a measure and Kähler potentials with uniform log-log threshold. As an application, we prove the entropy compactness theorem for a family of potential functions of Poincaré type Kähler metrics with uniform log-log threshold. The Ohsawa-Takegoshi $L^2$-extension theorem and Skoda-Zeriahi's integrability theorem play a very important role in this paper.

Skoda-Zeriahi type integrability and entropy compactness for some measure with $L^1$-density

TL;DR

The paper advances Skoda–Zeriahi type integrability to measures with -density on bounded pseudoconvex domains by introducing a log-log threshold to quantify potential singularities. It proves a positivity lower bound for the integrability threshold of relative to , and develops a local -convergence theory that transfers -convergence to stronger norms under uniform exponential integrability. A central achievement is the entropy-compactness theorem for families of Poincaré-type Kähler potentials in the finite energy space , with entropy measured against the singular measure , enabling variational approaches to canonical metrics of Poincaré type. The results connect integrability, entropy, and geometric analysis in a way that extends classical -density theories to highly singular measures and informs Mabuchi-type variational problems in settings with cusp-like asymptotics.

Abstract

In this paper, we prove the Skoda-Zeriahi type integrability theorem with respect to some measure with -density. In addition, we introduce the log-log threshold in order to detect singularities of Kähler potentials. We prove the positivity of the integrability threshold for such a measure and Kähler potentials with uniform log-log threshold. As an application, we prove the entropy compactness theorem for a family of potential functions of Poincaré type Kähler metrics with uniform log-log threshold. The Ohsawa-Takegoshi -extension theorem and Skoda-Zeriahi's integrability theorem play a very important role in this paper.
Paper Structure (9 sections, 20 theorems, 68 equations)

This paper contains 9 sections, 20 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Skoda-Zeriahi type integrability
  3. Local $L^p(d \mu_\gamma)$-convergence
  4. Integrability threshold
  5. Nonpluripolar product and full mass
  6. Positivity of integrability threshold
  7. Entropy compactness theorem
  8. Finite energy space and $d_1$-metric
  9. Proof of Theorem \ref{['ent cpt']}

Key Result

Theorem 1.1

(Ze, Sk, see GZ) Let $\Omega \subset \mathbb{C}^n$ be an open subset and $\mathcal{U}$ be an $L^1$-compact family in ${\rm PSH}(\Omega)$. For a compact subset $K \Subset \Omega$, we set $\mathcal{V}(\mathcal{U},K ):= \sup_{x \in K} \{ \nu_\varphi (x)\,\, | \,\, \varphi \in \mathcal{U} \}$, where $

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3: log-log threshold
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Remark 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 46 more