Workshop on Fano spherical varieties - 2026

Invited Speakers

• Lorenzo Barban (IBS Center for Complex Geometry) 

• Yonghwa Cho (Gyeongsang National University) 

• Sung Rak Choi (Yonsei University) 

• Jaehyun Hong (IBS Center for Complex Geometry) 

• DongSeon Hwang (IBS Center for Complex Geometry) 

• Taekgyu Hwang (IBS Center for Complex Geometry) 

• Minyoung Jeon (IBS Center for Complex Geometry) 

• In-Kyun Kim (KIAS) 

• Shin-young Kim (Kangwon National University) 

• Minseong Kwon (Morningside Center of Mathematics in AMSS, CAS) 

• Donggun Lee (IBS Center for Complex Geometry) 

• Eunjeong Lee (Chungbuk National University)

• Yingqi Liu (IBS Center for Complex Geometry) 

• Zhijun Luo (IBS Center for Complex Geometry) 

• Haesong Seo (IBS Center for Complex Geometry) 

• Guolei Zhong (IBS Center for Complex Geometry) 

Program (tentative) 

• February 2 (Mon) 15:00~15:30 Registration & Discussion, 15:30~16:20 In-Kyun Kim, 16:40~17:30 Taekgyu Hwang, 18:00~ Dinner 

• February 3 (Tue) 10:00~10:50 DongSeon Hwang, 11:10~12:00 Sung Rak Choi, 12:00~14:00 Lunch, 14:00~14:50 Lorenzo Barban,  15:10~16:00 Eunjeong Lee, 16:20~17:10 Guolei Zhong, 18:00~ Dinner 

• February 4 (Wed) 10:00~10:50 Minseong Kwon, 11:10~12:00 Shin-young Kim, 12:00~14:00 Lunch, 14:00~18:00 Discussion for joint works, 18:00~ Dinner 

• February 5 (Thur) 10:00~10:50 Minyoung Jeon, 11:10~12:00 Zhijun Luo, 12:00~14:00 Lunch, 14:00~14:50 Jaehyun Hong, 15:10~16:00 Donggun Lee, 16:20~17:10 Yingqi Liu, 18:00~ Dinner 

• February 6 (Fri) 10:00~10:50 Haesong Seo, 11:10~12:00 Yonghwa Cho, 12:00~14:00 Lunch, 14:00~18:00 Discussion 

Titles and Abstracts 

• Lorenzo Barban (IBS Center for Complex Geometry) 

Title: An algebro-geometric cobordism for birational maps among Mori dream spaces 

Abstract: In this talk we construct geometric realizations, that is an algebro-geometric version of cobordism, for birational maps among Mori Dream Spaces. We show that these geometric realizations are Mori Dream Spaces as well, and that they can be constructed so that they induce factorizations of the original birational map as compositions of wall-crossings. We present toric examples and discuss when a geometric realization is Fano. This seminar is based on a joint work with G. Occhetta and L. Solá Conde. 

• Yonghwa Cho (Gyeongsang National University) 

Title: Maximally nodal sextic surfaces and linear determinantal representations 

Abstract: A significant result by Casnati and Catanese tells that an (half-)even set of nodes is related to a symmetric map between vector bundles on the ambient projective 3-space. In some cases, it happens that the vector bundles split into a direct sum of line bundles, realizing the symmetric map as a symmetric matrix of homogeneous forms. For instance, in a joint work with F. Catanese and M. Kiermaier, we showed that every maximally nodal sextic surface (with 65 nodes) is defined by the determinant of the 4x4 matrix of homogeneous forms of diagonal degrees (1,1,1,3). It is proved by showing that the vector bundles associated to a half-even set of cardinality 31 split into the direct sum of line bundles. In this talk, we present our recent result proving that every maximally nodal sextic surface contains a half-even set of cardinality 35 which realizes the sextic as a determinant of 6x6 matrix of linear forms. This in particular proves that the cokernel of this matrix defines an Ulrich sheaf of rank 1. We also present an explicit 6x6 matrix of linear forms, whose determinant is the equation of Barth's sextic surface.

• Sung Rak Choi (Yonsei University) 

Title: A valuative approach to -K-MMP 

Abstract: We first define and study "triples" that are more general than the pairs and generalized pairs. In this framework, we prove that the log canonical thresholds of klt triples can be computed by quasimonomial valuations which extends the previous results by Xu. As an application of the results on triples, we study how to run -K-MMP. The results are based on the joint work with S. Jang, D. Kim, and D. Lee. 

• Jaehyun Hong (IBS Center for Complex Geometry)

Title: Geometry of regular semisimple Lusztig varieties 

Abstract: In a series of papers, Lusztig developed a theory of characters of a reductive algebraic group G by using perverse sheaves on G. To get appropriate perverse sheaves on G (called character sheaves), he considered a family of subvarieties of the flag variety G/B parameterized by elements in G; now, we call Lusztig varieties. In this talk, we will explain how they are related to two interesting families of subvarieties of the flag variety, Schubert varieties and Hessenberg varieties. 

Regular semisimple Lusztig varieties share many nice properties with Schubert varieties. They are normal Cohen-Macaulay, have rational singularities, and are of Fano type. We construct a flat degeneration of regular semisimple Lusztig varieties to regular semisimple Hessenberg varieties and compare their cohomology spaces. This is joint work with P. Brosnan and D. Lee. 

• DongSeon Hwang (IBS Center for Complex Geometry) 

Title: Automorphism groups of toroidal horospherical varieties 

Abstract: We present our recent work on the structure of the identity component of the automorphism group of a smooth, complete, toroidal horospherical variety by generalizing the notion of Demazure roots using the toric bundle structure. In particular, we provide a criterion for the reductivity of $\mathrm{Aut}^0(X)$ in terms of an analogous notion of Demazure roots for such toric bundles, i.e., projective toric bundles over rational homogeneous spaces. As an application, we prove the K-unstability of certain P^1-bundles over rational homogeneous spaces. This is joint work in progress with Lorenzo Barban and Minseong Kwon. 

• Taekgyu Hwang (IBS Center for Complex Geometry)

Title: Strong Calabi dream manifolds and volume polynomials 

Abstract: A compact Kähler manifold is called a strong Calabi dream manifold if it admits a constant scalar curvature metric in each Kähler class. Such a condition is very restrictive and we expect that the only toric examples are the products of complex projective spaces. I am going to present a strategy to use volume polynomials for this problem. 

• Minyoung Jeon (IBS Center for Complex Geometry) 

Title: Irreducible characteristic cycles for K-orbit closures  

Abstract: The symmetric subgroup K=GL(p)×GL(q) acts on a flag variety X of GL(n) with finitely many orbits, and we study the closures of these orbits. In this talk, we construct a small resolution of a certain family of these orbit closures with smooth, strongly reduced fibers. Using this resolution, we show that their characteristic cycles are irreducible. As a consequence, Jones’ result applies and allows us to compute the Chern–Mather classes. This is joint work with William Graham and Scott Larson. 

• In-Kyun Kim (KIAS) 

Title: K-stability and Cylindricity for Blow-ups of Weighted Projective Spaces 

Abstract: The K-stability and cylindricity of smooth del Pezzo surfaces have been well studied. Naturally, one may ask whether similar phenomena occur for blow-ups of singular del Pezzo surfaces. In this talk, I discuss the K-stability and cylindricity of singular del Pezzo surfaces obtained as blow-ups of weighted projective planes. I describe the conditions under which these blow-ups remain Fano and explore the relationship between the existence of cylindrical structures and their K-stability. 

• Shin-young Kim (Kangwon National University) 

Title: Deformation rigidity of some quasi-homogeneous varieties with Picard number one

Abstract: We investigate the global deformation rigidity problem of rational homogeneous manifolds of Picard number one which were developed by Hwang, Mok and others. In particular, we focus on the role of varieties of minimal rational tangents. Starting with similar ideas, we introduce some recent global deformation rigidity results of some quasi-homogeneous varieties, symmetric varieties and horospherical varieties, with Picard number one.

• Minseong Kwon (Morningside Center of Mathematics in AMSS, CAS) 

Title: Sphericality of spaces of conics on rational homogeneous spaces 

Abstract: After Mori's celebrated works, rational curves have played a prominent role in the study of Fano manifolds. Among them, the geometry of lines on rational homogeneous spaces is now well understood, and can be described in terms of Tits geometry. Namely, for a rational homogeneous space X of Picard number 1, the space of lines on X is homogeneous if and only if X is associated to a long simple root. In this talk, I will discuss conics on X, and prove the following: the space of conics on X is spherical if and only if X is associated to a long simple root. This talk is based on a joint work in progress with Nicolas Perrin. 

• Donggun Lee (IBS Center for Complex Geometry)

Title: Lusztig characters and Hessenberg characters 

Abstract: This talk follows Prof. Jaehyun Hong’s talk.  Smooth regular semisimple Lusztig varieties degenerate to regular semisimple Hessenberg varieties, and for such pairs the singular cohomology groups are isomorphic as representations of the Weyl group. In this talk, we explain how this relationship can be lifted to the level of Hecke characters. Based on joint work in progress with Patrick Brosnan and Jaehyun Hong. 

• Eunjeong Lee (Chungbuk National University)

Title: Toric varieties in partial flag varieties 

Abstract: Let G be a semisimple Lie group, T a maximal torus of G, and B a Borel subgroup of G containing T. For a parabolic subgroup P containing B, the homogeneous space G/P is a smooth projective algebraic variety, called a (partial) flag variety. If P is minimal, that is, P=B, then we call G/B a full flag variety. When P is maximal, we call G/P a Grassmannian. The left multiplication of T on G induces an action on G/P. By taking closures of T-orbits, we obtain lots of toric varieties including toric Schubert varieties. In this talk, we study the properties of these toric varieties. In particular, we consider toric Schubert varieties in G/P. This talk is partially based on joint work with Inseo Kim. 

• Yingqi Liu (IBS Center for Complex Geometry) 

Title: On linear sections of spinor tenfold 

Abstract: Vinberg’s θ-representations provide a classification of orbits in certain coregular representations, generalizing Jordan’s theory for simple Lie algebras. In recent years, increasing attention has been paid to their geometric applications. In this talk, I will discuss our study of linear sections of the spinor tenfold. Building on the work of Kuznetsov, we show how Vinberg’s θ-representations can be used to study the moduli space and the explicit geometry of these linear sections. This is joint work with L. Manivel. 

• Zhijun Luo (IBS Center for Complex Geometry) 

Title: Equivariant compactifications of unipotent algebraic groups 

Abstract: The study of equivariant compactifications (ECs) of algebraic groups has a long and rich history. For reductive groups, the geometry of their EC is often understood via combinatorial methods. In contrast, for unipotent groups, there is currently no uniform framework to describe the geometry of their equivariant compactifications. In this talk, we focus on two basic examples of unipotent groups: the vector group $\mathbb{G}_a^n$, which is the simplest unipotent group, and the Heisenberg group $\mathbb{H}_{2n+1}$, which exhibits slightly more intricate.The study of equivariant compactifications of these groups naturally leads to two fundamental problems: the existence of equivariant compactifications under suitable conditions, and the classification of equivariant structures on a given equivariant compactification when the number of such structures is finite. In this talk, I will address these two problems under certain assumptions. This work is based on joint work with Cong Ding. 

• Haesong Seo (IBS Center for Complex Geometry) 

Title: Hyperbolicity problems in general Fano threefolds and in spherical varieties 

Abstract: A smooth projective variety is (Brody) hyperbolic if it does not admit a nonconstant holomorphic map from the complex plane. Demailly proved that hyperbolic manifolds are algebraically hyperbolic, i.e., there is a bound for the degree of curves lying on the manifold in terms of their genera. In this talk, we investigate algebraic hyperbolicity of (very general) hypersurfaces in general Fano threefolds with Picard number one and in spherical varieties. This is based on a joint work with Minseong Kwon. 

• Guolei Zhong (IBS Center for Complex Geometry) 

Title: Dynamical toric conjecture in dimension three 

Abstract: A surjective holomorphic self-map of a projective manifold is said to have dominant topological degree if its last dynamical degree is strictly larger than the other dynamical degrees. Several previous works by Meng-Zhang and Yoshikawa show that, equipped with such a self-map, the underlying manifold, up to a finite cover, admits a Fano type fibration over an abelian variety. Moreover, it is conjectured that a general periodic fiber must be toric. In this talk, I will survey our recent progress toward this conjecture from the aspects of the equivariant minimal model program and the positivity of tangent bundle. 

Webpages of previous workshops

• Workshop on Fano spherical varieties - 2025 (February 3-7, 2025), Yonsei University, Seoul, Korea. 

• Workshop on Fano spherical varieties - 2024 Spring (February 14-17, 2024), Gyeongsang National University, Jinju, Korea. 

• Workshop on Fano spherical varieties - 2022 Spring (February 3-5, 2022), Online Workshop.