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Some inequalities for the weighted log canonical thresholds

Nguyen Xuan Hong

TL;DR

This work analyzes weighted log canonical thresholds $c_t(\varphi)$ for plurisubharmonic functions and their relation to the Lelong number $\nu_\varphi(0)$. It establishes sharp criteria for finiteness and precise inequalities linking $c_t(\varphi)$ to $\nu_\varphi(0)$, including an exact extremal formula $c_t(\varphi) = (t+n)/\nu_\varphi(0)$ under suitable Cegrell-class conditions. The authors develop a hierarchy of slope inequalities across index parameters and provide a quantitative lower bound for increments $c_{t+s}(\varphi)-c_t(\varphi)$, incorporating the higher-order concentrations $e_j(\varphi)$ and their role in complex singularity theory. The results extend Skoda-type bounds, connect to the strong openness conjecture, and include sharpness remarks with explicit function examples.

Abstract

In this paper, we will study the relations on the weighted log canonical thresholds of plurisubharmonic functions. We prove some inequalities for the weighted log canonical thresholds.

Some inequalities for the weighted log canonical thresholds

TL;DR

This work analyzes weighted log canonical thresholds for plurisubharmonic functions and their relation to the Lelong number . It establishes sharp criteria for finiteness and precise inequalities linking to , including an exact extremal formula under suitable Cegrell-class conditions. The authors develop a hierarchy of slope inequalities across index parameters and provide a quantitative lower bound for increments , incorporating the higher-order concentrations and their role in complex singularity theory. The results extend Skoda-type bounds, connect to the strong openness conjecture, and include sharpness remarks with explicit function examples.

Abstract

In this paper, we will study the relations on the weighted log canonical thresholds of plurisubharmonic functions. We prove some inequalities for the weighted log canonical thresholds.

Paper Structure

This paper contains 5 sections, 12 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['le3..']}
  3. Proof of Theorem \ref{['th2']}
  4. Proof of Theorem \ref{['th.']}
  5. Proof of Theorem \ref{['th']}

Key Result

Theorem 1.1

Let $\Omega$ be a domain in $\mathbb C^n$ containing the origin $0$. Assume that $t>-n$ is a real number and $\varphi$ is a plurisubharmonic function in $\Omega$. Then, $\nu_\varphi(0) \in (0,+\infty)$ if and only if $c_t(\varphi)\in (0,+\infty)$. Moreover, provided that $\nu_\varphi(0) >0$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['le3..']}
  • Lemma 3.1
  • ...and 14 more