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

k $\Bbbk $ double struck k

, a subfield of

C $\mathbb {C}$ double struck upper 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

k ⊂ C $\Bbbk \subset \mathbb {C}$ double struck k subset of double struck upper C

admit (orbifold) Kähler–Einstein metrics if they do not have

k $\Bbbk $ double struck 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 $12$ 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

k ⊂ C $\Bbbk \subset \mathbb {C}$ double struck k subset of double struck upper C

whose geometric model is strictly K-semistable.