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Algebraic Geometry Seminar
Seminar information archive ～01/13｜Next seminar｜Future seminars 01/14～
| Date, time & place | Friday 13:30 - 15:00 118Room #118 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | GONGYO Yoshinori, KAWAKAMI Tatsuro, ENOKIZONO Makoto |
Future seminars
2026/01/14
13:30-15:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Radu Laza (Stony Brook University)
Remarks on Lagrangian Fibrations on Hyperkähler Manifolds
Radu Laza (Stony Brook University)
Remarks on Lagrangian Fibrations on Hyperkähler Manifolds
[ Abstract ]
Hyperkähler manifolds are fundamental building blocks in the classification theory of algebraic varieties. A central problem is the finiteness of their deformation types, and, more ambitiously, the classification of these types. A natural approach to these questions is through the study of Lagrangian fibrations. In particular, the SYZ conjecture predicts that every deformation class of hyperkähler manifolds contains a member admitting a Lagrangian fibration.
In this talk, I will discuss several recent results on Lagrangian fibrations on hyperkähler manifolds. I will focus in particular on the special case of isotrivial Lagrangian fibrations, and on the striking fact that no such fibration exists in the exceptional OG10 deformation type. I will also briefly mention general boundedness results for Lagrangian fibrations, as well as results concerning the structure of their singular fibers. This latter part is largely based on the work of other authors, with some additional perspective and commentary of my own.
Hyperkähler manifolds are fundamental building blocks in the classification theory of algebraic varieties. A central problem is the finiteness of their deformation types, and, more ambitiously, the classification of these types. A natural approach to these questions is through the study of Lagrangian fibrations. In particular, the SYZ conjecture predicts that every deformation class of hyperkähler manifolds contains a member admitting a Lagrangian fibration.
In this talk, I will discuss several recent results on Lagrangian fibrations on hyperkähler manifolds. I will focus in particular on the special case of isotrivial Lagrangian fibrations, and on the striking fact that no such fibration exists in the exceptional OG10 deformation type. I will also briefly mention general boundedness results for Lagrangian fibrations, as well as results concerning the structure of their singular fibers. This latter part is largely based on the work of other authors, with some additional perspective and commentary of my own.
2026/01/16
13:30-15:00 Room #118 (Graduate School of Math. Sci. Bldg.)
Ryu Tomonaga (University of Tokyo)
On d-tilting bundles on d-folds
Ryu Tomonaga (University of Tokyo)
On d-tilting bundles on d-folds
[ Abstract ]
A d-tilting bundle is a tilting bundle whose endomorphism algebra has global dimension at most d. On d-dimensional smooth projective varieties, d-tilting bundles generalize geometric helices and play an important role in connections with tilting bundles on the total space of the canonical bundle (Calabi-Yau completion), non-commutative crepant resolutions and higher Auslander-Reiten theory.
In this talk, we prove the following results. First, if a d-dimensional smooth projective variety has a d-tilting bundle, then it is weak Fano. Second, every weak del Pezzo surface has a 2-tilting bundle. As an application, we show that every singular del Pezzo cone admits a non-commutative crepant resolution.
If time permits, we will also present a classification of d-tilting bundles consisting of line bundles on d-dimensional smooth toric Fano stacks of Picard number one or two.
A d-tilting bundle is a tilting bundle whose endomorphism algebra has global dimension at most d. On d-dimensional smooth projective varieties, d-tilting bundles generalize geometric helices and play an important role in connections with tilting bundles on the total space of the canonical bundle (Calabi-Yau completion), non-commutative crepant resolutions and higher Auslander-Reiten theory.
In this talk, we prove the following results. First, if a d-dimensional smooth projective variety has a d-tilting bundle, then it is weak Fano. Second, every weak del Pezzo surface has a 2-tilting bundle. As an application, we show that every singular del Pezzo cone admits a non-commutative crepant resolution.
If time permits, we will also present a classification of d-tilting bundles consisting of line bundles on d-dimensional smooth toric Fano stacks of Picard number one or two.
