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

TL;DR

This paper investigates the relationship between the existence of rational points on Fano varieties over subfields of complex numbers and the presence of Kähler-Einstein metrics on their models, providing new examples and criteria for such metrics.

Contribution

It establishes conditions under which Fano varieties without rational points admit Kähler-Einstein metrics, and explicitly constructs new examples of such Fano 3-folds.

Findings

Geometric models of certain del Pezzo surfaces admit Kähler-Einstein metrics if they lack rational points.

Most smooth Fano 3-folds also admit Kähler-Einstein metrics, with 8 exceptions.

New examples of prime Fano 3-folds of genus 12 with Kähler-Einstein metrics are identified.

Abstract

We explore connections between existence of $\Bbbk$-rational points for Fano varieties defined over $\Bbbk$, a subfield of $\mathbb{C}$, and existence of K\"ahler-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) K\"ahler--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 K\"ahler-Einstein metrics. In particular, we obtain new examples of prime Fano 3-folds of genus $12$ that admit K\"ahler--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.