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

Abstract

We explore connections between existence of $\mathbb{k}$-rational points for Fano varieties defined over $\mathbb{k}$, 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 $\mathbb{k}\subset \mathbb{C}$ admit (orbifold) Kähler--Einstein metrics if they do not have $\mathbb{k}$-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 $\mathbb{k}\subset \mathbb{C}$ whose geometric model is strictly K-semistable.

44 pages