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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. 

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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. 

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3. 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 . 

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4. Special Lagrangian submanifolds of log Calabi-Yau manifolds

   Collins, Tristan C. ; Jacob, A. ; Lin, Y.-S. ( January 2021 , Duke mathematical journal)

   null (Ed.)

   We study the existence of special Lagrangian submanifolds of log Calabi–Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian and Yau. We prove that if X is a Tian–Yau manifold and if the compact Calabi–Yau manifold at infinity admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface or a rational elliptic surface and D is a smooth divisor in the linear system of K_Y with D^2=d, then X=Y/D admits a special Lagrangian torus fibration, as conjectured by Strominger–Yau–Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung and Yau. In the special case that Y is a rational elliptic surface or Y=P^2, we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kähler rotation, X can be compactified to the complement of a Kodaira type I_d fiber appearing as a singular fiber in a rational elliptic surface. 

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5. K-MODULI OF CURVES ON A QUADRIC SURFACE AND K3 SURFACES

   https://doi.org/10.1017/S1474748021000384  

   ASCHER, KENNETH ; DEVLEMING, KRISTIN ; LIU, YUCHEN ( September 2021 , Journal of the Institute of Mathematics of Jussieu)

   Abstract We show that the K-moduli spaces of log Fano pairs $\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$ , where C is a $(4,4)$ curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ , complete intersection curves in $\mathbb {P}^3$ . This, together with recent results by Laza and O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$ curves on $\mathbb {P}^1\times \mathbb {P}^1$ and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces. 

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