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K-stability of pointless del Pezzo surfaces and Fano 3-folds

Hamid Abban, Ivan Cheltsov, Takashi Kishimoto, Frederic Mangolte

Abstract

We explore connections between existence of $\Bbbk$-rational points for Fano varieties defined over $\Bbbk$, a subfield of $\mathbb{C}$, and existence of Kähler-Einstein metrics on their geometric models. First, we show that geometric models of del Pezzo surfaces with at worst quotient singularities defined over $\Bbbk\subset\mathbb{C}$ admit (orbifold) Kähler--Einstein metrics if they do not have $\Bbbk$-rational points. Then we prove the same result for smooth Fano 3-folds with 8 exceptions. Consequently, we explicitly describe several families of pointless Fano 3-folds whose geometric models admit Kähler-Einstein metrics. In particular, we obtain new examples of prime Fano 3-folds of genus $12$ that admit Kähler--Einstein metrics. Our result can also be used to prove existence of rational points for certain Fano varieties, for example for any smooth Fano 3-fold over $\Bbbk\subset\mathbb{C}$ whose geometric model is strictly K-semistable.

K-stability of pointless del Pezzo surfaces and Fano 3-folds

Abstract

We explore connections between existence of -rational points for Fano varieties defined over , a subfield of , and existence of Kähler-Einstein metrics on their geometric models. First, we show that geometric models of del Pezzo surfaces with at worst quotient singularities defined over admit (orbifold) Kähler--Einstein metrics if they do not have -rational points. Then we prove the same result for smooth Fano 3-folds with 8 exceptions. Consequently, we explicitly describe several families of pointless Fano 3-folds whose geometric models admit Kähler-Einstein metrics. In particular, we obtain new examples of prime Fano 3-folds of genus that admit Kähler--Einstein metrics. Our result can also be used to prove existence of rational points for certain Fano varieties, for example for any smooth Fano 3-fold over whose geometric model is strictly K-semistable.

Paper Structure

This paper contains 13 sections, 68 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Preliminaries
  4. Singular del Pezzo surfaces
  5. On $\alpha$-invariants of del Pezzo surfaces
  6. Real del Pezzo surfaces; a warm up
  7. Del Pezzo surfaces of Picard rank one
  8. The proof of Theorem \ref{['theorem:1']}
  9. Smooth Fano 3-folds with $\Bbbk$-points
  10. Families containing K-polystable members
  11. K-unstable families
  12. Pointless K-polystable Fano 3-folds
  13. Examples of pointless smooth Fano 3-folds

Key Result

Theorem A

Let $S$ be a del Pezzo surface with quotient singularities defined over a subfield $\Bbbk$ of $\mathbb{C}$. Assume the geometric model of $S$ does not admit an orbifold Kähler--Einstein metric. Then $S$ has a $\Bbbk$-rational point.

Theorems & Definitions (149)

  • Theorem A
  • Theorem B
  • Corollary 1.1
  • Example 1.2
  • Remark 1.3
  • proof : Proof of Theorem \ref{['theorem:2']}
  • Lemma 2.1: Lang--Nishimura Lemma
  • proof
  • Lemma 2.2
  • proof
  • ...and 139 more