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K-stable smooth Fano threefolds of Picard rank two

Ivan Cheltsov, Elena Denisova, Kento Fujita

TL;DR

The paper advances the classification of K-stable smooth Fano threefolds by proving that all smooth members in six Picard rank-two deformation families, № 2.1, 2.2, 2.3, 2.4, 2.6, and 2.7, are K-stable, with family № 2.5 K-stable under a generality condition. The authors apply Abban–Zhuang theory to the valuative criterion, expressing stability through δ-invariants and S-quantities, and perform extensive, case-by-case δ-analysis across diverse fibration structures (del Pezzo fibrations, conic bundles, and blow-ups). Their approach yields explicit δ>1 bounds for centers over X, ruling out K-unstability in these families. This work strengthens the broader program toward a complete K-stability classification of smooth Fano threefolds and provides a blueprint for handling remaining uncertain families via detailed intersection-theoretic and Zariski-decomposition computations.

Abstract

We prove that all smooth Fano threefolds in the families 2.1, 2.2, 2.3, 2.4, 2.6 and 2.7 are K-stable, and we also prove that smooth Fano threefolds in the family 2.5 that satisfy one very explicit generality condition are K-stable.

K-stable smooth Fano threefolds of Picard rank two

TL;DR

The paper advances the classification of K-stable smooth Fano threefolds by proving that all smooth members in six Picard rank-two deformation families, № 2.1, 2.2, 2.3, 2.4, 2.6, and 2.7, are K-stable, with family № 2.5 K-stable under a generality condition. The authors apply Abban–Zhuang theory to the valuative criterion, expressing stability through δ-invariants and S-quantities, and perform extensive, case-by-case δ-analysis across diverse fibration structures (del Pezzo fibrations, conic bundles, and blow-ups). Their approach yields explicit δ>1 bounds for centers over X, ruling out K-unstability in these families. This work strengthens the broader program toward a complete K-stability classification of smooth Fano threefolds and provides a blueprint for handling remaining uncertain families via detailed intersection-theoretic and Zariski-decomposition computations.

Abstract

We prove that all smooth Fano threefolds in the families 2.1, 2.2, 2.3, 2.4, 2.6 and 2.7 are K-stable, and we also prove that smooth Fano threefolds in the family 2.5 that satisfy one very explicit generality condition are K-stable.
Paper Structure (6 sections, 40 theorems, 335 equations)

This paper contains 6 sections, 40 theorems, 335 equations.

Table of Contents

  1. Introduction
  2. Families № 2.1, № 2.3, № 2.5
  3. Family № 2.2.
  4. Family № 2.4.
  5. Family № 2.6 (Verra threefolds)
  6. Family № 2.7

Key Result

Lemma 2.1

Let $O$ be a point in $X$, let $A$ be the fiber of the morphism $\phi$ such that $O\in A$. Suppose that $A$ has at most Du Val singularities at the point $O$. Then

Theorems & Definitions (75)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 65 more