Special Kähler geometry and holomorphic Lagrangian fibrations
Given a holomorphic Lagrangian fibration of a compact hyperkähler manifold, we use the differential geometry of the special Kähler metric that exists on the base away from the discriminant locus, and show that the pullback of the tangent bundle of the base to the total space of a family of minimal r...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2024-06-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.629/ |
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| author | Li, Yang Tosatti, Valentino |
| author_facet | Li, Yang Tosatti, Valentino |
| author_sort | Li, Yang |
| collection | DOAJ |
| description | Given a holomorphic Lagrangian fibration of a compact hyperkähler manifold, we use the differential geometry of the special Kähler metric that exists on the base away from the discriminant locus, and show that the pullback of the tangent bundle of the base to the total space of a family of minimal rational curves admits a parallel splitting. The splitting is nontrivial when the base is not half-dimensional projective space. Combining this with results of Voisin, Hwang and Bakker–Schnell, we deduce that the base must be projective space, a result first proved by Hwang. |
| format | Article |
| id | doaj-art-20577a3acceb47b0b93bd82e4e14f999 |
| 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-20577a3acceb47b0b93bd82e4e14f9992025-02-07T11:13:31ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-06-01362S117119610.5802/crmath.62910.5802/crmath.629Special Kähler geometry and holomorphic Lagrangian fibrationsLi, Yang0Tosatti, Valentino1Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USACourant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York, NY 10012, USAGiven a holomorphic Lagrangian fibration of a compact hyperkähler manifold, we use the differential geometry of the special Kähler metric that exists on the base away from the discriminant locus, and show that the pullback of the tangent bundle of the base to the total space of a family of minimal rational curves admits a parallel splitting. The splitting is nontrivial when the base is not half-dimensional projective space. Combining this with results of Voisin, Hwang and Bakker–Schnell, we deduce that the base must be projective space, a result first proved by Hwang.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.629/ |
| spellingShingle | Li, Yang Tosatti, Valentino Special Kähler geometry and holomorphic Lagrangian fibrations Comptes Rendus. Mathématique |
| title | Special Kähler geometry and holomorphic Lagrangian fibrations |
| title_full | Special Kähler geometry and holomorphic Lagrangian fibrations |
| title_fullStr | Special Kähler geometry and holomorphic Lagrangian fibrations |
| title_full_unstemmed | Special Kähler geometry and holomorphic Lagrangian fibrations |
| title_short | Special Kähler geometry and holomorphic Lagrangian fibrations |
| title_sort | special kahler geometry and holomorphic lagrangian fibrations |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.629/ |
| work_keys_str_mv | AT liyang specialkahlergeometryandholomorphiclagrangianfibrations AT tosattivalentino specialkahlergeometryandholomorphiclagrangianfibrations |