Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stable degeneration of families of klt singularities with constant local volume

Zhiyuan Chen

TL;DR

The paper proves that in a locally stable family of klt singularities with constant local volume, the minimizing valuations yield a flat, bicontinuousIdeal sequence whose associated graded algebra degenerates coherently to a locally stable family of K-semistable log Fano cone singularities. It develops and extends Xu–Zhuang’s framework to families via Kollár models, ensuring finite generation in families and providing a canonical degenerating family compatible across fibers. The result culminates in a stratified representability statement for when degenerate, and is illustrated by an explicit unibranch plane-curve example, where Puiseux characteristics govern the local-volume behavior and K-semistable degenerations. Overall, the work advances a family-level K-stability theory for singularities by establishing flatness and coherence of minimizing valuations under constant local volume, enabling controlled degenerations to K-semistable cones with torus actions.

Abstract

We prove that for a locally stable family of klt singularities with constant local volume, the ideal sequences of the minimizing valuations for the normalized volume function form a family of ideals with flat cosupport, which induces a degeneration to a locally stable family of K-semistable log Fano cone singularities. Our proof is a family version of the method of C. Xu and Z. Zhuang proving finite generation by Kollár models and multiple degenerations.

Stable degeneration of families of klt singularities with constant local volume

TL;DR

The paper proves that in a locally stable family of klt singularities with constant local volume, the minimizing valuations yield a flat, bicontinuousIdeal sequence whose associated graded algebra degenerates coherently to a locally stable family of K-semistable log Fano cone singularities. It develops and extends Xu–Zhuang’s framework to families via Kollár models, ensuring finite generation in families and providing a canonical degenerating family compatible across fibers. The result culminates in a stratified representability statement for when degenerate, and is illustrated by an explicit unibranch plane-curve example, where Puiseux characteristics govern the local-volume behavior and K-semistable degenerations. Overall, the work advances a family-level K-stability theory for singularities by establishing flatness and coherence of minimizing valuations under constant local volume, enabling controlled degenerations to K-semistable cones with torus actions.

Abstract

We prove that for a locally stable family of klt singularities with constant local volume, the ideal sequences of the minimizing valuations for the normalized volume function form a family of ideals with flat cosupport, which induces a degeneration to a locally stable family of K-semistable log Fano cone singularities. Our proof is a family version of the method of C. Xu and Z. Zhuang proving finite generation by Kollár models and multiple degenerations.
Paper Structure (26 sections, 32 theorems, 224 equations)

This paper contains 26 sections, 32 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Birational geometry and singularities
  4. Qdlt pairs
  5. Models
  6. Locally stable families
  7. Valuations
  8. Real valuations
  9. Quasi-monomial valuations
  10. Log discrepancies
  11. Local volumes
  12. Ideal sequences
  13. Normalized volumes and the minimizers
  14. Families of log Fano cone singularities
  15. Special valuations
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $S$ be a semi-normal scheme essentially of finite type over a field of characteristic zero. Let $\pi \colon (X, \Delta) \to S$ be a locally stable family of klt pairs, with a section $x \colon S \to X$ of $\pi$, such that is a locally constant function on $S$, where $(X_s, \Delta_s)$ is the fiber over a point $s \in S$, and $x_s = x(s) \in X_s$.We will say $\pi \colon (X, \Delta) \to S$ with

Theorems & Definitions (88)

  • Theorem 1.1: Theorem \ref{['K-semistable_degen_over_semi-normal']}
  • Theorem 1.2: Theorem \ref{['construct_Kol_model_of_minimizer_over_DVR']}
  • Theorem 1.3: Corollary \ref{['representability']}
  • Example 1.4
  • Definition 2.1: cf. dFKX, XZ_stable_deg
  • Lemma 2.2: dFKX
  • Definition 2.3
  • Lemma 2.4: cf. BCHM
  • proof
  • Definition 2.5
  • ...and 78 more