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On semi-continuity problems for minimal log discrepancies

Yusuke Nakamura

Abstract

We show the semi-continuity property of minimal log discrepancies for varieties which have a crepant resolution in the category of Deligne-Mumford stacks. Using this property, we also prove the ideal-adic semi-continuity problem for toric pairs.

On semi-continuity problems for minimal log discrepancies

Abstract

We show the semi-continuity property of minimal log discrepancies for varieties which have a crepant resolution in the category of Deligne-Mumford stacks. Using this property, we also prove the ideal-adic semi-continuity problem for toric pairs.

Paper Structure

This paper contains 14 sections, 16 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minimal log discrepancies
  4. Notation and remarks on Deligne-Mumford stacks
  5. Yasuda's twisted jet stacks
  6. Motivic integration
  7. The transformation rule
  8. Minimal log discrepancies and jet schemes
  9. Lower semi-continuity problems on varieties with quotient singularities
  10. Minimal log discrepancies and twisted jet stacks
  11. Proof of Theorem \ref{['thm:main1']}
  12. The ideal-adic semi-continuity problem on varieties with a $\mathbb{C} ^{*}$-action
  13. Proof of Proposition \ref{['prop:main2']}
  14. The toric case

Key Result

Theorem 1.2

Let $X$ be a $\mathbb{Q}$-Gorenstein normal variety. Assume $X$ has a crepant resolution in the category of Deligne-Mumford stacks. Then, the followings hold. Especially, Conjecture conj:LSC holds for the variety $X$.

Theorems & Definitions (38)

  • Conjecture 1.1: LSC conjecture
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Conjecture 1.5: Mustaţă
  • Proposition 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 2.1
  • ...and 28 more