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K-stability of Fano 3-folds in the World of Null-A

Hamid Abban, Ivan Cheltsov, Takashi Kishimoto, Frederic Mangolte

TL;DR

The authors establish a geometric analogue of arithmetic phenomena for smooth Fano 3-folds by introducing Condition (A): every finite abelian subgroup of Aut(X) fixes a point. They prove that any non-K-polystable smooth Fano 3-fold satisfies (A) with exactly eight exceptional deformation families where (A) fails, and they provide explicit abelian-group actions lacking fixed points to realize these exceptions. The approach combines the Mori–Mukai classification, equivariant birational geometry, and δ-invariant estimates to control fixed points and stability across 105 deformation families, yielding a precise list of exceptional cases and a broad set of families for which (A) holds. This leads to a clearer understanding of the interaction between K-stability and automorphism-group actions, informing the moduli theory of Fano 3-folds and revealing a geometric-arithmetic parallel in stability phenomena.

Abstract

A variety is said to satisfy Condition (A) if every finite abelian subgroup of its automorphism group has a fixed point. We show that a smooth Fano 3-fold not satisfying Condition (A) is K-polystable unless it is contained in eight exceptional deformation families (seven of them consists of one smooth member, and one of them has one-parameter moduli).

K-stability of Fano 3-folds in the World of Null-A

TL;DR

The authors establish a geometric analogue of arithmetic phenomena for smooth Fano 3-folds by introducing Condition (A): every finite abelian subgroup of Aut(X) fixes a point. They prove that any non-K-polystable smooth Fano 3-fold satisfies (A) with exactly eight exceptional deformation families where (A) fails, and they provide explicit abelian-group actions lacking fixed points to realize these exceptions. The approach combines the Mori–Mukai classification, equivariant birational geometry, and δ-invariant estimates to control fixed points and stability across 105 deformation families, yielding a precise list of exceptional cases and a broad set of families for which (A) holds. This leads to a clearer understanding of the interaction between K-stability and automorphism-group actions, informing the moduli theory of Fano 3-folds and revealing a geometric-arithmetic parallel in stability phenomena.

Abstract

A variety is said to satisfy Condition (A) if every finite abelian subgroup of its automorphism group has a fixed point. We show that a smooth Fano 3-fold not satisfying Condition (A) is K-polystable unless it is contained in eight exceptional deformation families (seven of them consists of one smooth member, and one of them has one-parameter moduli).
Paper Structure (11 sections, 48 theorems, 76 equations)

This paper contains 11 sections, 48 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Plan of the proof
  4. Preliminary results
  5. Smooth Fano 3-folds satisfying Condition $\mathbf{(A)}$
  6. Families containing K-polystable members
  7. Families not containing K-polystable members
  8. K-polystable Fano 3-folds not satisfying Condition $\mathbf{(A)}$
  9. Examples of smooth Fano 3-folds not satisfying Condition $\mathbf{(A)}$
  10. K-unstable smooth Fano 3-folds not satisfying Condition $\mathbf{(A)}$.
  11. K-polystable smooth Fano 3-folds not satisfying Condition $\mathbf{(A)}$.

Key Result

Theorem 2.1

Let $X$ and $X^\prime$ be smooth varieties acted on by a finite abelian group $A$. Suppose that there exists an $A$-equivariant birational map $\psi\colon X\dasharrow X^\prime$. Then $X$ contains a point fixed by $A$ if and only if $X^\prime$ contains a point fixed by $A$.

Theorems & Definitions (113)

  • Theorem 2.1: ReYou00
  • Corollary 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 103 more