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Today's seminar
Seminar information archive ～05/21｜Today's seminar 05/22 | Future seminars 05/23～
2026/05/22
Algebraic Geometry Seminar
13:15-14:45 Room #117 (Graduate School of Math. Sci. Bldg.)
Justin Sawon (University of North Carolina Chapel Hill)
Classification results for Lagrangian fibrations
Justin Sawon (University of North Carolina Chapel Hill)
Classification results for Lagrangian fibrations
[ Abstract ]
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).
Colloquium
15:30-16:30 Room #NISSAY Lecture Hall (Graduate School of Math. Sci. Bldg.)
Evgeny Shinder (University of Sheffield / University of Tokyo)
Gromov's cancellation question in birational algebraic geometry
Evgeny Shinder (University of Sheffield / University of Tokyo)
Gromov's cancellation question in birational algebraic geometry
[ Abstract ]
Gromov's 1999 cancellation question is: given two open embeddings of a variety U into a variety X, do they always have isomorphic closed complements? In my joint work with Hsueh-Yung Lin we reformulate this question in terms of the structure of the Grothendieck ring of varieties and answer it in various situations. The answer will be positive or negative depending on the dimension of varieties and the ground field. Finally, I will present an application to the structure of the Cremona group of birational self-maps of the projective space.
Gromov's 1999 cancellation question is: given two open embeddings of a variety U into a variety X, do they always have isomorphic closed complements? In my joint work with Hsueh-Yung Lin we reformulate this question in terms of the structure of the Grothendieck ring of varieties and answer it in various situations. The answer will be positive or negative depending on the dimension of varieties and the ground field. Finally, I will present an application to the structure of the Cremona group of birational self-maps of the projective space.
