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

   https://doi.org/10.46298/epiga.2023.10231  

   Xu, Chenyang (September 2023, Épijournal de Géométrie Algébrique)

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

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2. 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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3. 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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4. Wall crossing for K‐moduli spaces of plane curves

   https://doi.org/10.1112/plms.12615  

   Ascher, Kenneth; DeVleming, Kristin; Liu, Yuchen (June 2024, Proceedings of the London Mathematical Society)

   Abstract We construct proper good moduli spaces parametrizing K‐polystable ‐Gorenstein smoothable log Fano pairs , where is a Fano variety and is a rational multiple of the anticanonical divisor. We then establish a wall‐crossing framework of these K‐moduli spaces as varies. The main application in this paper is the case of plane curves of degree as boundary divisors of . In this case, we show that when the coefficient is small, the K‐moduli space of these pairs is isomorphic to the GIT moduli space. We then show that the first wall crossing of these K‐moduli spaces are weighted blow‐ups of Kirwan type. We also describe all wall crossings for degree 4,5,6 and relate the final K‐moduli spaces to Hacking's compactification and the moduli of K3 surfaces. 

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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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