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

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 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 ⊂ 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 ⊂ ℂ double struck k subset of double struck upper C whose geometric model is strictly K-semistable.

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BibTeX RIS

BibTeX

@article{abban2026kstability,
  title = {{K-stability of pointless del Pezzo surfaces and Fano 3-folds}},
  author = {Hamid Abban and Ivan Cheltsov and Takashi Kishimoto and Frederic Mangolte},
  journal = {Forum of Mathematics, Sigma},
  year = {2026},
  month = oct,
  doi = {10.1017/fms.2026.10266},
  url = {https://www.cambridge.org/core/product/identifier/S2050509426102667/type/journal_article},
  note = {CC BY 4.0},
}

RIS

TY  - JOUR
TI  - K-stability of pointless del Pezzo surfaces and Fano 3-folds
AU  - Hamid Abban
AU  - Ivan Cheltsov
AU  - Takashi Kishimoto
AU  - Frederic Mangolte
PY  - 2026
DA  - 2026/10/22
JO  - Forum of Mathematics, Sigma
DO  - 10.1017/fms.2026.10266
UR  - https://www.cambridge.org/core/product/identifier/S2050509426102667/type/journal_article
AB  - 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 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 ⊂ 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 ⊂ ℂ double struck k subset of double struck upper C whose geometric model is strictly K-semistable.
SN  - 2050-5094
ER  -

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