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1. K-stability for varieties with a big anticanonical class

   Xu, Chenyang (August 2023, Épijournal Géom. Algébrique)

   We extend the algebraic K-stability theory to projective klt pairs with a big anticanonical class. While in general such a pair could behave pathologically, it is observed in this note that the K-semistability condition will force them to have a klt anticanonical model, whose stability property is the same as that of the original pair. 

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2. On boundedness of singularities and minimal log discrepancies of Kollár components

   https://doi.org/10.1090/jag/822  

   Zhuang, Ziquan (January 2024, Journal of Algebraic Geometry)

   Recent study in K-stability suggests that Kawamata log terminal (klt) singularities whose local volumes are bounded away from zero should be bounded up to special degeneration. We show that this is true in dimension three, or when the minimal log discrepancies of Kollár components are bounded from above. We conjecture that the minimal log discrepancies of Kollár components are always bounded from above, and verify it in dimension three when the local volumes are bounded away from zero. We also answer a question from Han, Liu, and Qi on the relation between log canonical thresholds and local volumes. 

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3. Stable degenerations of singularities

   https://doi.org/10.1090/jams/1055  

   Xu, Chenyang; Zhuang, Ziquan (July 2025, Journal of the American Mathematical Society)

   For any Kawamata log terminal (klt) singularity and any minimizer of its normalized volume function, we prove that the associated graded ring is always finitely generated, as conjectured by Chi Li. As a consequence, we complete the last step of establishing the Stable Degeneration Conjecture proposed by Chi Li and the first named author for an arbitrary klt singularity. 

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4. Klt varieties of general type with small volume

   https://doi.org/10.2422/2036-2145.202109_010  

   Totaro, Burt; Wang, Chengxi (January 2023, ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE)

   By Hacon-McKernan-Xu, there is a positive lower bound in each dimension for the volumes of all klt varieties with ample canonical class. We show that these bounds must go to zero extremely fast as the dimension increases, by constructing a klt n-fold with ample canonical class whose volume is less than 1/2^{2^n}. These examples should be close to optimal. We also construct, for every n, a klt Fano variety of dimension n such that the space of sections of the mth power of the anticanonical bundle is zero for all m from 1 to about 2^{2^n}. Here again there is some bound in each dimension, by Birkar’s theorem on boundedness of complements, and we are showing that the bound must increase extremely fast with the dimension. 

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5. A non-Archimedean approach to K-stability, II: Divisorial stability and openness

   https://doi.org/10.1515/crelle-2023-0062  

   Boucksom, Sébastien; Jonsson, Mattias (October 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   To any projective pair (X,B) equipped with an ample Q-line bundle L (or even any ample numerical class), we attach a new invariant $$\beta(\mu)$$, defined on convex combinations $$\mu$$ of divisorial valuations on X , viewed as point masses on the Berkovich analytification of X . The construction is based on non-Archimedean pluripotential theory, and extends the Dervan–Legendre invariant for a single valuation – itself specializing to Li and Fujita’s valuative invariant in the Fano case, which detects K-stability. Using our $$\beta$$-invariant, we define divisorial (semi)stability, and show that divisorial semistability implies (X,B) is sublc (i.e. its log discrepancy function is non-negative), and that divisorial stability is an open condition with respect to the polarization L. We also show that divisorial stability implies uniform K-stability in the usual sense of (ample) test configurations, and that it is equivalent to uniform K-stability with respect to all norms/filtrations on the section ring of (X,L), as considered by Chi Li. 

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