Workshop on K-stability

Monday, January 5, 2026

9:00 - 9:10 Registration and Openning

9:10 - 10:10 Ruadhai Dervan| Arcs and stability of pairs 

10:40-11:40 Jingjun Han| On boundedness in general type MMP

13:30-14:30 Minghao Miao| The volume of K-semistable Fano manifolds

15:00-16:00 Chuyu Zhou| On finiteness of K-moduli compactifications 

Tuesday, January 6, 2026

9:10 - 10:10 Kento Fujita| Another view on smooth prime Fano threefolds of degree 22 with infinite automorphism groups 

10:10-10:20 Group Photo

10:40-11:40 Chenyang Xu| Properness of K-moduli 

13:30-14:30 Junyao Peng| Asymptotics of stability thresholds 

15:00-16:00 Linsheng Wang| K-polystability of Fano cone singularities  

16:20-17:20 Lightning talks

18:00-20:30 Banquet

Wednesday, January 7, 2026

9:10 - 10:10 Mattias Jonsson (online)| On the Yau--Tian--Donaldson conjecture for cscK metrics

10:40-11:40 Kewei Zhang| Canonical metrics and K-beta stability

13:30-14:30 Theodoros Papazachariou| Some explicit examples of K-moduli spaces

Ruadhai Dervan

Title: Arcs and stability of pairs

Abstract: Paul introduced the theory of stability of pairs about twenty years ago, as a generalisation of geometric invariant theory to the case when one does not have an ample line bundle. I will begin by discussing some algebro-geometric results around the theory of stability of pairs, such as a numerical criterion (through Donaldson's theory of arcs), and a result stating that stability of pairs is a very general condition. These results have applications to K-stability, where they give an answer to a question of Tian around the behaviour of the Mabuchi functional, and where they show that (uniform) arc K-stability is a very general condition in families. This is joint work with Rémi Reboulet.

Kento Fujita 藤田健人

Title: Another view on smooth prime Fano threefolds of degree 22 with infinite automorphism groups

Abstract: All smooth Fano threefolds with infinite automorphism groups are understood due to Prokhorov, Kuznetsov and Shramov by use of deep studies of their Hilbert scheme of lines. I will present as our joint work with Adrien Dubouloz and Takashi Kishimoto an alternative and self-contained proof of it, allowing us to use several properties on the smooth quintic del Pezzo threefold. Moreover, I would like to explain an interesting elementary link joining prime Fano threefolds of degree 22 with Fano threefolds of No. 2.21 in Mori-Mukai’s list. 

Jingjun Han 韩京俊

Title: On boundedness in general type MMP

Abstract: One of the main open problems in the Minimal Model Program (MMP) is the termination. Motivated by local volumes introduced by Chi Li, we introduce log canonical volume which is non-decreasing in any sequence of MMP for general type varieties. As a result, in such kind of MMP, we show that (1) the Cartier index of any Weil Q-Cartier is uniformly bounded; (2) every fiber of the extremal contractions or the flips is bounded (3) the set of minimal log discrepancies belongs to a finite set. This is a joint work with Lu Qi and Ziquan Zhuang. 

Mattias Jonsson (online)

Title: On the Yau--Tian--Donaldson conjecture for cscK metrics

Abstract: I will report on joint work with S. Boucksom, in which we prove that the existence of a (weighted) cscK metric on a polarized manifold (X,L) is equivalent to a condition of algebro-geometric nature, formulated in terms of non-Archimedean pluripotential theory.

Minghao Miao 缪铭昊

Title: The Volume of K-Semistable Fano Manifolds

Abstract: In 2015, K. Fujita showed that for any n-dimensional K-semistable Fano manifold, the anti-canonical volume is always less than or equal to that of complex projective space (CP^n). In this talk, I will discuss my recent joint work with Chi Li on characterizing the second-largest volume. We prove that for any n-dimensional K-semistable Fano manifold X that is not isomorphic to CPⁿ, the volume is at most 2n^n, with the equality holds if and only if X is a smooth quadric hypersurface or CP^1 × CP^{n-1}. This result applies, in particular, to all Fano manifolds admitting Kähler–Einstein metrics. Our proof is based on a new connection between K-stability and minimal rational curves.

Theodoros Papazachariou

Title: Some explicit examples of K-moduli spaces

Abstract: In the past decade, K-stability has made extraordinary progress in constructing moduli spaces of Fano varieties and log Fano pairs. This construction, however, is not explicit, and needs to be studied on a case-by-case basis to explicitly describe specific examples of moduli spaces for Fano varieties. Recently, the focus has shifted to explicit descriptions of such K-moduli of Fano threefolds, which has resulted in the full moduli description of a number of families. The methods used to obtain these descriptions rely either on explicit computations of the invariants defining K-stability, or on the moduli continuity method, which relates K-moduli spaces to other natural compactifications such as GIT, which are more easily approachable. In this talk, I will explain how the moduli continuity method allows us to obtain new explicit K-moduli descriptions for four more families (2.11, 3.3, 3.5, 3.8) via relating the K-moduli to explicit GIT quotients.

Junyao Peng 彭俊尧

Title: Asymptotics of stability thresholds

Abstract: We study asymptotic behavior of the stability thresholds of a big line bundle, and prove explicit bounds on the error terms. This answers Jin--Rubinstein--Tian's questions affirmatively. A key step in our proof is to show that the stability thresholds of a big line bundle can always be computed by quasi-monomial valuations. This generalizes Blum--Jonsson's result on the stability thresholds of an ample line bundle.

Linsheng Wang 王淋生

Title: K-polystability of Fano cone singularities  

Abstract: In this talk, I would like to introduce a recent work about K-polystability of log Fano cone singularities. Precisely, we prove that a K-semistable log Fano cone singularity is K-polystable for normal test configurations if and only if it is K-polystable for special test configurations.

Chenyang Xu 许晨阳

Title: Properness of K-moduli
Abstract: (Joint with Harold Blum, Yuchen Liu, and Ziquan Zhuang) 

We present a new proof of the properness of K-moduli spaces. While our approach still depends on the higher-rank finite generation theorem, it avoids the use of Halpern-Leistner’s Θ-stratification theory. Instead, we develop a purely birational method, rooted in a relative framework for K-stability, which provides a more direct geometric proof of properness.

Kewei Zhang 张科伟

Title: Canonical metrics and K-beta stability

Abstract: The Yau-Tian-Donaldson conjecture predicts that existence of constant scalar curvature Kahler (cscK) metrics is equivalent to (uniform) K-stability. In this talk I will introduce a new stability notion called K-beta stability, which we show to be equivalent with the existence of cscK metrics. This talk is based on my joint work with T. Darvas.

Chuyu Zhou 周楚宇

Title: On finiteness of K-moduli compactifications

Abstract: Given a family of K-semistable log Fano manifolds with changeable coefficient, we hope that there are only finitely many K-moduli compactifications as we change the coefficient, which clearly implies the boundedness of K-semistable degenerations.  In this talk, we will see that this boundedness is enough to establish the whole non-proportional wall crossing theory for K-stability, i.e. there is a finite semi-algebraic chamber decomposition to control the variation of K-(semi/poly)stability.

Zhiming Guo 郭志明

TBA

Yijue Hu 胡忆珏

TBA

Pietro Mesquita-Piccione 

Title: K-stability for models on Kähler manifolds

Abstract: Based on joint work with David Witt Nyström, this talk will address the extension of K-stability for models to non-algebraic Kähler manifolds. I will present an overview of the theory in this more general setting. If time permits, I will also explain how the non-Archimedean Monge–Ampère equation yields an explicit valuative criterion for K-stability for models.

Yuxin Zhang 张宇鑫

TBA