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On minimal log discrepancies on varieties with fixed Gorenstein index

Yusuke Nakamura

Abstract

We generalize the rationality theorem of the accumulation points of log canonical thresholds which was proved by Hacon, M\textsuperscript{c}Kernan, and Xu. Further, we apply the rationality to the ACC problem on the minimal log discrepancies. We study the set of log discrepancies on varieties with fixed Gorenstein index. As a corollary, we prove that the minimal log discrepancies of three-dimensional canonical pairs with fixed coefficients satisfy the ACC.

On minimal log discrepancies on varieties with fixed Gorenstein index

Abstract

We generalize the rationality theorem of the accumulation points of log canonical thresholds which was proved by Hacon, M\textsuperscript{c}Kernan, and Xu. Further, we apply the rationality to the ACC problem on the minimal log discrepancies. We study the set of log discrepancies on varieties with fixed Gorenstein index. As a corollary, we prove that the minimal log discrepancies of three-dimensional canonical pairs with fixed coefficients satisfy the ACC.

Paper Structure

This paper contains 8 sections, 18 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minimal log discrepancies
  4. Extraction of divisors
  5. ACC for log canonical thresholds
  6. Accumulation points of log canonical thresholds
  7. Perturbation of irrational coefficients of lc pairs
  8. Proof of main theorem and corollaries

Key Result

Theorem 1.2

Fix $d \in \mathbb{Z} _{>0}$, $r \in \mathbb{Z} _{>0}$ and a finite subset $I \subset [0, + \infty)$. Then the following set is discrete in $\mathbb{R}$. Here we denote by $P(d,r)$ the set of all $d$-dimensional lc pairs $(X, \mathfrak{a})$ such that $r K_X$ is a Cartier divisor.

Theorems & Definitions (47)

  • Conjecture 1.1: ACC conjecture Shokurov:models
  • Theorem 1.2
  • Corollary 1.3
  • Conjecture 1.4: BDD conjecture
  • Corollary 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • ...and 37 more