Workshop on Fano spherical varieties - 2025

Invited Speakers 

• Lorenzo Barban (Institute for Basic Science-CCG) 

• Yoshinori Hashimoto (Osaka Metropolitan University)

• In-Kyun Kim (KIAS-HCMC)

• Jeong-Seop Kim (KIAS) 

• Yoosik Kim (Pusan National University)

• Minseong Kwon (KAIST/IBS-CCG) 

• Eunjeong Lee (Ewha Womans University) 

• Yan Li (Beijing Institute of Technology) 

• Satoshi Nakamura (Institute of Science Tokyo)

• Feng Shao (Ewha Womans University) 

• Naoto Yotsutani (Kagawa University)

Titles and Abstracts 

• Lorenzo Barban (Institute for Basic Science-CCG) 

Title: Geometric realization of birational maps 

Abstract: In this talk we introduce a projective algebraic version of the notion of cobordism coming from Morse theory, which we call geometric realization 

 Given a birational map $\phi$ among normal projective varieties, a geometric realization of $\phi$ is a normal projective $\mathbb{C}^*$-variety $X$ such that the birational map among geometric quotients of $X$ by $\mathbb{C}^*$ parametrizing general orbits coincides with $\phi$.  We survey which class of birational maps admits a geometric realization, providing specific examples for maps among Mori dream spaces. This talk is based on joint works with E. Romano, G. Occhetta, L. Solá Conde and S. Urbinati. 

• Yoshinori Hashimoto (Osaka Metropolitan University) 

Title: Calabi’s extremal metrics - review of basics and relative K-stability 

Abstract: The extremal metrics were introduced by E. Calabi as a "least curved metric" on a compact Kähler manifold, and they are widely regarded as a class of canonical Kähler metrics partly because they generalise constant scalar curvature Kähler (cscK) and Kähler-Einstein metrics. In the first talk, we review the basics of the extremal metrics, starting from the definition. We put a particular emphasis on the modified Mabuchi energy, and explain that the existence of the extremal metrics can be characterised by the asymptotic slope of the modified Mabuchi energy along geodesic rays, as proved by W. He. The topic of the second talk is the conjectured correspondence between the existence of extremal metrics and an algebro-geometric notion called relative K-stability, proposed by G. Székelyhidi. We start from the definition of the relative K-stability and explain the well-established relationship to the asymptotic slope of the modified Mabuchi energy. Finally, we explain how some results for cscK metrics which involve non-Archimedean metrics can be generalised to the extremal metrics.

• In-Kyun Kim (KIAS-HCMC)

Title: K-stability of blow-ups of the weighted projective planes 

Abstract: The K-stability of smooth del Pezzo surfaces is well understood. For instance, the projective plane is K-polystable, but it becomes K-unstable after one or two blow-ups, regains K-stability after three blow-ups, and retains this stability until the Fano condition is violated after eight blow-ups. 

 In this talk, we investigate the K-stability of blow-ups of weighted projective planes with weights (1, 1, n) and demonstrate that singular del Pezzo surfaces arising from n+4 blow-ups are K-stable. This result highlights a striking parallel between the stability patterns of the projective plane and the weighted projective plane, offering new insights into the broader relationship between blow-ups and K-stability in algebraic geometry.

• Jeong-Seop Kim (KIAS) 

Title: Bigness of tangent bundles of manifolds of Picard number ≥ 2 

Abstract: After Mori's solution to Hartshorne's conjecture regarding ample tangent bundles, a series of questions has arisen concerning various positivity of tangent bundles. In the first part of the talk, I will introduce some results focused on the question of big tangent bundles, with recent progress initiated by A. Höring, J. Liu, and F. Shao. I will then present various examples and counterexamples of smooth projective varieties with big tangent bundles, including Fano threefolds of Picard number ≥ 2, weak del Pezzo surfaces, and projective bundles. 

• Yoosik Kim (Pusan National University)

Title: Newton–Okounkov bodies of flag varieties and their applications to symplectic topology

Abstract: In this talk, I will discuss the construction of Newton–Okounkov bodies for flag varieties using the cluster structure of a unipotent cell and talk about their applications in symplectic topology. This talk is based on ongoing joint work with Yunhyung Cho, Myungho Kim, and Euiyong Park. 

• Minseong Kwon (KAIST/IBS-CCG) 

Title: Homogeneous Legendrian subvarieties of nilpotent orbits 

Abstract: A nilpotent orbit is an adjoint orbit of a nilpotent element in the projectivized semi-simple Lie algebra. It is well-known that each nilpotent orbit is equipped with a natural contact structure, that is, a maximally non-integrable hyperplane distribution. Indeed, every known example of Fano contact manifolds can be realized as a nilpotent orbit. In this talk, I will discuss Legendrian submanifolds of nilpotent orbits, i.e. integral submanifolds of maximal dimension. First, I will recall contact geometry of nilpotent orbits. Next, I will present a classification of homogeneous Legendrian subvarieties of nilpotent orbits, and a description of their moduli spaces. 

• Eunjeong Lee (Ewha Womans University) 

Title: Quaternions and polygons in the 5-dimensional Euclidean space 

Abstract: In this talk, we present a perspective on polygon spaces and 2-Grassmannians or real and complex numbers based on the work of Hausmann and Knutson in 1997. Using spin representation, our approach provides a pathway for generalizing their results to higher dimensions to quaternions and octonions. 

 In this talk, we focus on polygon space over 5-dimensional Euclidean spaces and quaternions. We construct a connection between the representation of the quaternionic unitary group and the standard representation of SO(5), which ultimately leads to the Spin(5)-representation of the quaternionic 2-Stiefel manifold. This is a joint work with Prof. Jae-Hyouk Lee. 

• Yan Li (Beijing Institute of Technology) 

Title: K-stability of Fano G-varieties of complexity 1 

Abstract: Let G be a connected, complex reductive Lie group that acts on a normal variety X. Fix any Borel subgroup B of G. The codimension of B-orbits at general positions is called the complexity of the G-action. Spherical varieties are precisely G-varieties of complexity 0. The general structure theory of G-varieties (referred as Luna-Vust theory now) have been developed by Luna-Vust, Brion, Knop, Timashev, etc. during the past decades. Especially, in cases of complexity 0 and 1, the variety admits nice combinatorial structure. 

 The famous Yau-Tian-Donaldson conjecture confirms that on a Fano variety, the existence of canonical metrics is equivalent to certain algebraic stability (K-stability for Einstein metric and its variants for others). On Fano spherical varieties, there is nice combinatorial criterion of various kinds of stability. 

 In this lecture, I will introduce a combinatorial criterion of K-stability for Fano G-varieties of complexity 1. The first part of the lecture devotes to a review of Luna-Vust theory and in particular its application to G-varieties of complexity 1. In the second part, I will show the combinatorial criterion. 

• Satoshi Nakamura (Institute of Science Tokyo)

Title: Recent progress on the existence of Calabi’s extremal metrics and Mabuchi’s soliton on Fano manifolds 

Abstract: I will discuss several generalizations of the Kähler-Einstein metric, which may appear even when a Fano manifold is obstructed by Futaki’s invariant. One such generalization is the extremal metric introduced by Calabi. It is defined in terms of the scalar curvature so that makes sense for general polarized manifolds. Mabuchi introduced an analogous notion, called the Mabuchi soliton, in terms of the Ricci potential function. When the Futaki invariant vanishes, these two notions coincide with the Kähler-Einstein metric. However these are different metrics in general, one may ask whether their existence conditions are equivalent. This is the central topic of my talk. 

 In the first lecture, I will review foundational concepts for these metrics, focusing on existence results in terms of the coercivity of corresponding energy functionals. In the second lecture, I will discuss a recent result which states that the existence of the extremal metric in the first Chern class is equivalent to the existence of the Mabuchi soliton, assuming a necessary condition on the Mabuchi constant. This result was first proved by Apostolov-Lahdili-Nitta. They used the minimal model program in algebraic geometry. Hisamoto and the speaker gave a direct analytic proof using a continuity method for the Mabuchi soliton. 

• Feng Shao (Ewha Womans University) 

Title: Bigness of tangent bundles and dynamical rigidity of Fano manifolds of Picard number 1 

Abstract: Lazarsfeld raised the problem that any surjective morphism f from a rational homogeneous space S of Picard number 1 to a projective manifold X must satisfy that either X is a projective space or f is an isomorphism. This problem was completely resolved in the affirmative by Hwang and Mok. In this talk, we will provide a partial extension of Lazarsfeld's problem to Fano manifolds of Picard number 1 that have big tangent bundles. As an application, we study the bigness of the tangent bundles of Fano manifolds with Picard number 1. This talk is based on my joint work with Guolei Zhong. 

• Naoto Yotsutani (Kagawa University)

Title: Extremal Kähler metrics and destabilizers for relative K-polystability of toric varieties

Abstract: It was conjectured by Székelyhidi that a polarized manifold admits an extremal Kähler metric in the class of polarization if and only if it is relatively K-polystable. Furthermore, the folklore conjecture states that every toric Fano manifold admits an extremal Kähler metric in its first Chern class. For a given toric Fano manifold X, we provide a destabilizing convex function on the corresponding moment polytope P to clarify the relative K-unstability of X. Applying this criteria into a certain toric Fano manifold, we prove that there exists a toric Fano manifold of dimension 10 that does not admit an extremal Kähler metric. 

Accommodation for invited speakers 

• Building 414(Sangnam Institute of Management), Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 03722, Korea.
  | Tel. +82-02-2123-4263~7
  | Email. sangnam@yonsei.ac.kr
  In Korean:  상남경영원, 연세대학교

• Map and Direction to the Accomodation

Webpages of previous workshops

• 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.