Workshop on Fano spherical varieties - 2024 Spring

Invited Speakers

• Yunhyung Cho (Sungkyunkwan University) 

• Jaehyun Hong (Institute for Basic Science) 

• Shin-young Kim (Ewha Womans University)

• Minseong Kwon (KAIST/IBS-CCG) 

• Eunjeong Lee (Chungbuk National University) 

• Yan Li (Beijing Institute of Technology) - online talk 

• Sungmin Yoo (Incheon National University)

• Naoto Yotsutani (Kagawa University)

Program (Booklet)    

• February 14 (Wednesday) 15:00~16:00 Eunjeong Lee, 16:30~17:30 Sungmin Yoo, 18:30~ Dinner & Discussion 

• February 15 (Thursday) 10:00~11:00 Naoto Yotsutani (Lecture 1), 11:20~12:20 Yan Li (Lecture 1), Lunch, 14:30~15:30 Shin-young Kim, 16:00~17:00 Minseong Kwon, 17:00~18:00 Discussion, 18:00~ Dinner & Discussion 

• February 16 (Friday) 10:00~11:00 Naoto Yotsutani  (Lecture 2), 11:20~12:20 Yan Li (Lecture 2), Lunch, 14:00~18:00 Discussion, 18:00~ Dinner & Discussion 

• February 17 (Saturday) 09:45~10:45 Jaehyun Hong, 11:00~12:00 Yunhyung Cho, Lunch & Discussion 

Titles and Abstracts 

• Yunhyung Cho (Sungkyunkwan University) 

Title: Maximal tori in the Hamiltonian diffeomorphism groups and their mutations 

Abstract: Let M be a compact symplectic toric manifold. It is known by McDuff that the number of maximal tori is finite up to conjugation by elements of symplectomorphism groups. She also posed an interesting question whether the number is one when a symplectic form is monotone. Recently, this conjecture was solved in some special classes of manifolds such as Bott manifolds or small / large Picard numbers. In this talk, I will introduce two ideas to attack the conjecture: (1) c_1-cohomological rigidity of Fano manifolds (by C.-Lee-Masuda-Park) and (2) mutations of polytopes (Pabiniak-Tolman).

• Jaehyun Hong (Institute for Basic Science) 

Title: Rigidity of smooth horospherical varieties of Picard number one 

Abstract: A homogeneous space G/H is called horospherical if G is reductive and H contains the unipotent radical of a Borel subgroup of G. In this case, the normalizer P of H in G is parabolic, and the morphism from G/H to G/P is a torus bundle over a rational homogeneous variety. A horospherical variety is a normal G-variety having an open dense G-orbit that is horospherical. Examples include rational homogeneous varieties, toric varieties, and odd symplectic Grassmannians. Pasquier classified smooth projective horospherical varieties of Picard number one, and interesting questions are whether they can be characterized by the variety of minimal rational tangents at a general point and whether they are rigid under Kahler deformation, as in the case of rational homogeneous varieties of Picard number one. Hwang and Li confirm this characterization and rigidity hold for odd symplectic Grassmannians and smooth horospherical variety of type G_2. In this talk, we will focus on the remaining cases. This is joint work with Shin-young Kim.

• Shin-young Kim (Ewha Womans University)

Title: Minimal rational curves on complete symmetric varieties 

Abstract: We describe the families of minimal rational curves on any complete symmetric variety, and the corresponding varieties of minimal rational tangents. In particular, we prove that these varieties are homogeneous and that for non-exceptional irreducible wonderful varieties, there is a unique family of minimal rational curves, and hence a unique variety of minimal rational tangents. This talk is based on a work with M. Brion and N. Perrin.

• Minseong Kwon (KAIST/IBS-CCG) 

Title: Spherical geometry of Hilbert schemes of conics in adjoint varieties

Abstract: We study spaces of rational curves of degree 2, i.e. conics, in adjoint varieties. Adjoint varieties are rational homogeneous spaces defined as the projectivizations of the minimal nilpotent orbits in the simple Lie algebras. Each adjoint variety is equipped with a complex-contact structure, which imposes restrictive conditions on behavior of rational curves. In this talk, first we show that for each adjoint variety not of type A nor C, the space of nonsingular conics is a spherical variety, using contact geometry. Next, we present how the colored data of the Hilbert scheme of conics can be computed based on geometry of linear subspaces and the Hilbert-Chow morphism.

• Eunjeong Lee (Chungbuk National University) 

Title: Toric Schubert varieties in Grassmannians

Abstract: Let G be a semisimple Lie group over $\mathbb{C}$, T a maximal torus of G, and B a Borel subgroup of G containing T. Let P be a parabolic subgroup which is a closed subgroup containing the Borel subgroup B. The homogeneous space G/P is a smooth projective algebraic variety, called a partial flag variety or a rational homogeneous manifold. When P is maximal, then we call G/P a Grassmannian variety. The left multiplication of T on G induces that on G/P. Schubert varieties are T-invariant subvarieties of the partial flag varieties. In this talk, we study toric Schubert varieties in flag varieties, especially in Grassmannians, with respect to the action of the torus T. Indeed, we present an explicit description of the fan of a Gorenstein toric Schubert variety in a Grassmannian, and we prove that any Gorenstein toric Schubert variety in a Grassmannian is Fano. This talk is based on joint work with Shin-young Kim. 

• Yan Li (Beijing Institute of Technology) - online LINK:   https://gnu-ac-kr.zoom.us/j/9166851017     Meeting ID:   916 685 1017

Title: Weighted K-stability of Q-Fano spherical varieties 

Abstract: Let G be a connected, complex reductive Lie group and X a Q-Fano G-spherical variety. In this talk, based on a recent work of Han Jiyuan and Li Chi, we compute the weighed non-Archimedean functionals of a G-equivariant normal test configurations of X in terms of its combinatorial data. Also we define a modified Futaki invariant with respect to certain weight function g, and give an expression in terms of intersection numbers. Finally we show the equivalence of different notations of stability and gives a stability criterion on Q-Fano spherical varieties, which is also a criterion of existence of Kahler-Ricci g-solitons.

• Sungmin Yoo (Incheon National University)

Title: Bergman metrics and Kähler-Einstein metrics on projective manifolds 

Abstract: A basic problem in differential geometry is to find canonical, or best, metrics on a given manifold. By the famous uniformization theorem, every Riemann surface admits a Riemannian metric of constant curvature surfaces. As a higher dimensional generalization of this result, we will introduce two canonical Kähler metrics for Kähler manifolds, especially for projective manifolds: Bergman metrics and Kähler-Einstein metrics. Both metrics can be considered as higher dimensional generalizations of the Poincaré metirc. In this talk, we will study several results related to the Yau-Tian-Donaldson conjecture and Kazhdan's Theorem, which tell us that these two metrics are deeply related to each other. 

• Naoto Yotsutani (Kagawa University)

Title: Numerical semistability of projective toric varieties

Abstract: Numerical semistability is one notion of GIT stability, which is defined by the inclusion of the weight polytopes (Chow/Hurwitz polytopes). It was proved by Paul that the K-energy of a smooth linearly normal projective variety $X$ restricted to the Bergman metrics is bounded from below if and only if it is numerically semistable. In this talk, we provide a necessary and sufficient condition for a given smooth toric variety $X_P$ to be numerically semistable, building upon the works of Gelfand-Kapranov-Zelevinsky (A-Resultants/A-Discriminants). Applying this result to a smooth polarized toric variety $(X_P, L_P)$, we prove that $(X_P, L_P)$ is asymptotically numerically semistable if and only if it is K-semistable for toric degenerations.

Accommodation 

• Main Hotel: New Raon Stay Hotel (166 Yeongcheongang-ro, Jinju-si, Gyeongsangnam-do, Korea)  +82-55-751-1111