A decomposition theorem for $\mathbb{Q}$-Fano Kähler–Einstein varieties
Let $X$ be a $\mathbb{Q}$-Fano variety admitting a Kähler–Einstein metric. We prove that up to a finite quasi-étale cover, $X$ splits isometrically as a product of Kähler–Einstein $\mathbb{Q}$-Fano varieties whose tangent sheaf is stable with respect to the anticanonical polarization. This relies am...
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Académie des sciences
2024-06-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.612/ |
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| author | Druel, Stéphane Guenancia, Henri Păun, Mihai |
| author_facet | Druel, Stéphane Guenancia, Henri Păun, Mihai |
| author_sort | Druel, Stéphane |
| collection | DOAJ |
| description | Let $X$ be a $\mathbb{Q}$-Fano variety admitting a Kähler–Einstein metric. We prove that up to a finite quasi-étale cover, $X$ splits isometrically as a product of Kähler–Einstein $\mathbb{Q}$-Fano varieties whose tangent sheaf is stable with respect to the anticanonical polarization. This relies among other things on a very general splitting theorem for algebraically integrable foliations. We also prove that the canonical extension of $T_X$ by $\mathscr{O}_X$ is polystable with respect to the anticanonical polarization. |
| format | Article |
| id | doaj-art-cf67fa6fa64e4c06b85ca5cdfc9041eb |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-06-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-cf67fa6fa64e4c06b85ca5cdfc9041eb2025-02-07T11:13:30ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-06-01362S19311810.5802/crmath.61210.5802/crmath.612A decomposition theorem for $\mathbb{Q}$-Fano Kähler–Einstein varietiesDruel, Stéphane0Guenancia, Henri1Păun, Mihai2Univ Lyon, CNRS, Université Claude Bernard Lyon 1, UMR 5208, Institut Camille Jordan, F-69622 Villeurbanne, FranceInstitut de Mathématiques de Toulouse, Université Paul Sabatier, 31062 Toulouse Cedex 9, FranceLehrstuhl für Mathematik VIII, Universität Bayreuth, 95440 Bayreuth, GermanyLet $X$ be a $\mathbb{Q}$-Fano variety admitting a Kähler–Einstein metric. We prove that up to a finite quasi-étale cover, $X$ splits isometrically as a product of Kähler–Einstein $\mathbb{Q}$-Fano varieties whose tangent sheaf is stable with respect to the anticanonical polarization. This relies among other things on a very general splitting theorem for algebraically integrable foliations. We also prove that the canonical extension of $T_X$ by $\mathscr{O}_X$ is polystable with respect to the anticanonical polarization.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.612/$\mathbb{Q}$-Fano varietiessingular Kähler–Einstein metricsstable reflexive sheavesalgebraically integrable foliations |
| spellingShingle | Druel, Stéphane Guenancia, Henri Păun, Mihai A decomposition theorem for $\mathbb{Q}$-Fano Kähler–Einstein varieties Comptes Rendus. Mathématique $\mathbb{Q}$-Fano varieties singular Kähler–Einstein metrics stable reflexive sheaves algebraically integrable foliations |
| title | A decomposition theorem for $\mathbb{Q}$-Fano Kähler–Einstein varieties |
| title_full | A decomposition theorem for $\mathbb{Q}$-Fano Kähler–Einstein varieties |
| title_fullStr | A decomposition theorem for $\mathbb{Q}$-Fano Kähler–Einstein varieties |
| title_full_unstemmed | A decomposition theorem for $\mathbb{Q}$-Fano Kähler–Einstein varieties |
| title_short | A decomposition theorem for $\mathbb{Q}$-Fano Kähler–Einstein varieties |
| title_sort | decomposition theorem for mathbb q fano kahler einstein varieties |
| topic | $\mathbb{Q}$-Fano varieties singular Kähler–Einstein metrics stable reflexive sheaves algebraically integrable foliations |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.612/ |
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