Year: 2026
Type: article
Abstract: We explore connections between existence of k $\Bbbk $ double struck k -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.
Source: Forum of Mathematics Sigma
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $2,650
OpenAlex ID: W4404345919