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代数幾何学セミナー
過去の記録 ～04/28｜次回の予定｜今後の予定 04/29～
| 開催情報 | 金曜日 13:30～15:00 数理科学研究科棟(駒場) 118号室 |
|---|---|
| 担当者 | 權業 善範、河上 龍郎 、榎園 誠 |
今後の予定
2026年05月12日(火)
13:30-15:00 数理科学研究科棟(駒場) 128号室
斎藤 秀司 氏 (東京大学)
Birational lattices in the cohomology of the structure sheaf over non-archimedean fields
斎藤 秀司 氏 (東京大学)
Birational lattices in the cohomology of the structure sheaf over non-archimedean fields
[ 講演概要 ]
We show that the cohomology of the structure sheaf of smooth and proper schemes over a complete non-archimedean field K with the ring R of integers of characteristic zero, can be refined to a birational cohomology theory of smooth (not necessarily proper) schemes with values in R-lattices, and the same holds for K of positive characteristic in dimensions at most 3.
As one application, we obtain that the automorphism group of the function field of a proper smooth variety X of dimension at most 3 over any field of positive characteristic acts quasi-unipotently on the cohomology of the canonical sheaf of X.
The proof relies on some results from rigid analytic geometry on the cohomology of twisted integral rigid structure sheaves due to Bartenwerfer and van der Put.
We show that the cohomology of the structure sheaf of smooth and proper schemes over a complete non-archimedean field K with the ring R of integers of characteristic zero, can be refined to a birational cohomology theory of smooth (not necessarily proper) schemes with values in R-lattices, and the same holds for K of positive characteristic in dimensions at most 3.
As one application, we obtain that the automorphism group of the function field of a proper smooth variety X of dimension at most 3 over any field of positive characteristic acts quasi-unipotently on the cohomology of the canonical sheaf of X.
The proof relies on some results from rigid analytic geometry on the cohomology of twisted integral rigid structure sheaves due to Bartenwerfer and van der Put.
2026年05月22日(金)
13:15-14:45 数理科学研究科棟(駒場) 117号室
Justin Sawon 氏 (University of North Carolina Chapel Hill)
Classification results for Lagrangian fibrations
Justin Sawon 氏 (University of North Carolina Chapel Hill)
Classification results for Lagrangian fibrations
[ 講演概要 ]
A Lagrangian fibration on a holomorphic symplectic manifold or variety is one whose general fibre is an abelian variety that is Lagrangian with respect to the symplectic form. Examples were constructed by Beauville/Mukai whose fibres are Jacobians of curves, and by Markushevich-Tikhomirov, Arbarella-Sacca-Ferretti, Matteini, S-Shen, and Brakkee-Camere-Grossi-Pertusi-Sacca-Viktorova whose fibres are Prym varieties of curves with involutions. In all of these examples the family of curves is a linear system on a K3 surface, suggesting the question: is this always the case? Markushevich answered this affirmatively in the genus two case: if the relative compactified Jacobian of a family of genus two curves is a Lagrangian fibration then the curves all lie on a K3 surface, and the Lagrangian fibration is a Beauville-Mukai system. In this talk I will describe our generalization of this result to higher genus, and also to relative Prym varieties of genus three covers with involutions (joint work with Xuqiang Qin).
A Lagrangian fibration on a holomorphic symplectic manifold or variety is one whose general fibre is an abelian variety that is Lagrangian with respect to the symplectic form. Examples were constructed by Beauville/Mukai whose fibres are Jacobians of curves, and by Markushevich-Tikhomirov, Arbarella-Sacca-Ferretti, Matteini, S-Shen, and Brakkee-Camere-Grossi-Pertusi-Sacca-Viktorova whose fibres are Prym varieties of curves with involutions. In all of these examples the family of curves is a linear system on a K3 surface, suggesting the question: is this always the case? Markushevich answered this affirmatively in the genus two case: if the relative compactified Jacobian of a family of genus two curves is a Lagrangian fibration then the curves all lie on a K3 surface, and the Lagrangian fibration is a Beauville-Mukai system. In this talk I will describe our generalization of this result to higher genus, and also to relative Prym varieties of genus three covers with involutions (joint work with Xuqiang Qin).
2026年06月05日(金)
13:15-14:45 数理科学研究科棟(駒場) 117号室
Young-Hoon Kiem 氏 (Korea Institute for Advanced Study)
TBA
Young-Hoon Kiem 氏 (Korea Institute for Advanced Study)
TBA
[ 講演概要 ]
TBA
TBA
