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We show that for a K-unstable Fano variety, any divisorial valuation computing its stability threshold induces a non-trivial special test configuration preserving the stability threshold. When such a divisorial valuation exists, we show that the Fano variety degenerates to a uniquely determined twisted K-polystable Fano variety. We also show that the stability threshold can be approximated by divisorial valuations induced by special test configurations. As an application of the above results and the analytic work of Datar, Szekelyhidi, and Ross, we deduce that greatest Ricci lower bounds of Fano manifolds of fixed dimension form a finite set of rational numbers. As a key step in the proofs, we adapt the process of Li and Xu producing special test configurations to twisted K-stability in the sense of Dervan.
We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of Kahler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, (b) to compute the stability thresholds for hypersu
We establish an algebraic approach to prove the properness of moduli spaces of K-polystable Fano varieties and reduce the problem to a conjecture on destabilizations of K-unstable Fano varieties. Specifically, we prove that if the stability threshold
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 $Theta$-reductivity of the moduli of K-semistable log Fano pairs. A
We give a survey of the recent progress on the study of K-stability of Fano varieties by an algebro-geometric approach.
We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3-fold with obstructed deformations. In one case, the K-moduli space