This is a reading seminar on K-moduli theory. Our main reference is the K-stability book by Chenyang Xu. Other references (original papers) are also listed below. Basics on K-stability and stack theory are assumed.
The typical schedule is on Friday 4 pm (EST) via Zoom. The following is a tentative schedule.
Sep.22 2023: Families of stable log Fano families. (Junyan Zhao; notes, video)
Sep.29 2023: Boundedness of K-semistable log Fano pairs (Junyan Zhao; notes, video).
Oct.08 2023: Openness of K-(semi)stability (Lu Qi; notes, video)
Oct.12 2023: Construction of the K-moduli stack (Junyao Peng; notes, video)
Oct.20 2023: Review of Alper's good moduli spaces (Junyao Peng; notes, video)
Oct.27 2023: Existence of K-moduli spaces I (Junyao Peng; notes, video)
Nov.01 2023: Existence of K-moduli spaces II (Junyan Zhao; notes, video)
Properness of K-moduli spaces I (Theta-stratification)
Properness of K-moduli spaces II (Optimal destabilization theorem)
Harder-Narasimhan filtration for a family
Semi-positivity of CM line bundles
Reduced stability I
Reduced stability II
Twisted K-stability
Twisted families
Positivity of CM line bundles I
Positivity of CM line bundles II
Other References:
1) Blum, Harold, et al. "On properness of K-moduli spaces and optimal degenerations of Fano varieties." Selecta Mathematica 27.4 (2021): 73.
2) Xu, Chenyang, and Ziquan Zhuang. "On positivity of the CM line bundle on K-moduli spaces." Annals of mathematics 192.3 (2020): 1005-1068.
3) Codogni, Giulio, and Zsolt Patakfalvi. "Positivity of the CM line bundle for families of K-stable klt Fano varieties." Inventiones mathematicae 223 (2021): 811-894.
4) Jiang, Chen. "Boundedness of Q-Fano varieties with degrees and alpha-invariants bounded from below", Ann. Sci. Éc. Norm. Supér. (4) 53 (2020), no. 4, 1235--1248.
5) Li, Chi, Xiaowei Wang, and Chenyang Xu. "Algebraicity of the metric tangent cones and equivariant K-stability." Journal of the American Mathematical Society 34.4 (2021): 1175-1214.
6) Blum, Harold, Yuchen Liu, and Chenyang Xu. "Openness of K-semistability for Fano varieties." Duke Mathematical Journal 171.13 (2022): 2753-2797.
7) Liu, Yuchen, Chenyang Xu, and Ziquan Zhuang. "Finite generation for valuations computing stability thresholds and applications to K-stability." Annals of Mathematics 196.2 (2022): 507-566.
8) Alper, Jarod, et al. "Reductivity of the automorphism group of K-polystable Fano varieties." Inventiones mathematicae 222.3 (2020): 995-1032.
9) Blum, Harold, and Chenyang Xu. "Uniqueness of K-polystable degenerations of Fano varieties." Annals of Mathematics 190.2 (2019): 609-656.
10) Xu, Chenyang. "A minimizing valuation is quasi-monomial." Annals of Mathematics 191.3 (2020): 1003-1030.
11) Alper, Jarod. "Notes on stacks and moduli." (pdf)