More Like this

1. Reductivity of the automorphism group of K-polystable Fano varieties.

   Alper, Jarod ; Blum, Harold ; Halpern-Leistner, Daniel ; Xu, Chenyang ( July 2020 , Inventiones mathematicae)

   null (Ed.)

   We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and Θ-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a separated good moduli space. 

   more » « less
2. Algebraicity of the metric tangent cones and equivariant K-stability

   https://doi.org/10.1090/jams/974  

   Li, Chi ; Wang, Xiaowei ; Xu, Chenyang ( October 2021 , Journal of the American Mathematical Society)

   We prove two new results on the K K -polystability of Q \mathbb {Q} -Fano varieties based on purely algebro-geometric arguments. The first one says that any K K -semistable log Fano cone has a special degeneration to a uniquely determined K K -polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to complete the proof of Donaldson-Sun’s conjecture which says that the metric tangent cone of any point appearing on a Gromov-Hausdorff limit of Kähler-Einstein Fano manifolds depends only on the algebraic structure of the singularity. The second result says that for any log Fano variety with the torus action, K K -polystability is equivalent to equivariant K K -polystability, that is, to check K K -polystability, it is sufficient to check special test configurations which are equivariant under the torus action. 

   more » « less
3. Klt varieties with conjecturally minimal volume

   https://doi.org/10.1093/imrn/rnad047  

   Totaro, Burt ( March 2023 , International Mathematics Research Notices)

   We construct klt projective varieties with ample canonical class and the smallest known volume. We also find exceptional klt Fano varieties with the smallest known anti-canonical volume. We conjecture that our examples have the smallest volume in every dimension, and we give low-dimensional evidence for that. In order to improve on earlier examples, we are forced to consider weighted hypersurfaces that are not quasi-smooth. We show that our Fano varieties are exceptional by computing their global log canonical threshold (or alpha-invariant) exactly; it is extremely large, roughly 2^{2^n} in dimension n. These examples give improved lower bounds in Birkar’s theorem on boundedness of complements for Fano varieties. 

   more » « less
4. K-stability and birational models of moduli of quartic K3 surfaces

   https://doi.org/10.1007/s00222-022-01170-5  

   Ascher, Kenneth ; DeVleming, Kristin ; Liu, Yuchen ( May 2023 , Inventiones mathematicae)

   Abstract We show that the K-moduli spaces of log Fano pairs $$({\mathbb {P}}^3, cS)$$ ( P 3 , c S ) where S is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily–Borel compactification of moduli of quartic K3 surfaces as c varies in the interval (0, 1). We completely describe the wall crossings of these K-moduli spaces. As the main application, we verify Laza–O’Grady’s prediction on the Hassett–Keel–Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of $${\mathbb {P}}^3$$ P 3 . 

   more » « less
5. 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)

   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.

    

   more » « less