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Stable degenerations of singularities

Chenyang Xu, Ziquan Zhuang

Abstract

For any Kawamata log terminal (klt) singularity and any minimizer of its normalized volume function, we prove that the associated graded ring is always finitely generated, as conjectured by Chi Li. As a consequence, we complete the last step of establishing the Stable Degeneration Conjecture proposed by Chi Li and the first named author for an arbitrary klt singularity.

Stable degenerations of singularities

Abstract

For any Kawamata log terminal (klt) singularity and any minimizer of its normalized volume function, we prove that the associated graded ring is always finitely generated, as conjectured by Chi Li. As a consequence, we complete the last step of establishing the Stable Degeneration Conjecture proposed by Chi Li and the first named author for an arbitrary klt singularity.
Paper Structure (18 sections, 39 theorems, 173 equations)

This paper contains 18 sections, 39 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Main theorem
  3. Finite generation via multiple degeneration
  4. Preliminaries
  5. Qdlt pairs
  6. Valuations and models
  7. Valuations
  8. Models
  9. Normalized volume
  10. Kollár components
  11. Lc places of special $\mathbb{Q}$-complements
  12. Special $\mathbb{Q}$-complements
  13. Kollár models
  14. Construction of Kollár models
  15. Kollár models and finite generation
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $x\in (X=\mathrm{Spec}(R),\Delta)$ be a klt singularity. Let $v$ be a minimizer of $\widehat{\rm vol}$ on $\mathrm{Val}_{X,x}$. Then the associated graded algebra ${\rm gr}_v R:=\bigoplus_{\lambda\in \mathbb{R}_{\ge 0}}\mathfrak{a}_\lambda/\mathfrak{a}_{>\lambda}$ is finitely generated, where

Theorems & Definitions (87)

  • Theorem 1.1: Local higher rank finite generation
  • Theorem 1.2: Stable Degeneration Conjecture
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5: LXZ-HRFG
  • Theorem 1.6
  • Conjecture 1.7
  • Definition 2.1
  • Definition 2.2: dFKX-dualcomplex*Definition 35
  • Lemma 2.3
  • ...and 77 more