K-stability of pointless del Pezzo surfaces and Fano 3-folds
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929572195336192 |
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| author | Abban, Hamid Cheltsov, Ivan Kishimoto, Takashi Mangolte, Frederic |
| author_facet | Abban, Hamid Cheltsov, Ivan Kishimoto, Takashi Mangolte, Frederic |
| contents | We explore connections between existence of $\Bbbk$-rational points for Fano varieties defined over $\Bbbk$, 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 $\Bbbk\subset\mathbb{C}$ admit (orbifold) Kähler--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 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 $\Bbbk\subset\mathbb{C}$ whose geometric model is strictly K-semistable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_00767 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | K-stability of pointless del Pezzo surfaces and Fano 3-folds Abban, Hamid Cheltsov, Ivan Kishimoto, Takashi Mangolte, Frederic Algebraic Geometry 14J45, 32Q20 We explore connections between existence of $\Bbbk$-rational points for Fano varieties defined over $\Bbbk$, 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 $\Bbbk\subset\mathbb{C}$ admit (orbifold) Kähler--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 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 $\Bbbk\subset\mathbb{C}$ whose geometric model is strictly K-semistable. |
| title | K-stability of pointless del Pezzo surfaces and Fano 3-folds |
| topic | Algebraic Geometry 14J45, 32Q20 |
| url | https://arxiv.org/abs/2411.00767 |