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Register a new userReductivity of the automorphism group of K-polystable Fano varieties
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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. 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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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 hypersurfaces at generalized Eckardt points and for cubic surfaces at all points, and (c) to provide a new algebraic proof of Tians criterion for K-stability, amongst other applications.
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 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 of every K-unstable Fano variety is computed by a divisorial valuation, then such K-moduli spaces are proper. The argument relies on studying certain optimal destabilizing test configurations and constructing a Theta-stratification on the moduli stack of Fano varieties.
We study the probability that an $(n - m)$-dimensional linear subspace in $mathbb{P}^n$ or a collection of points spanning such a linear subspace is contained in an $m$-dimensional variety $Y subset mathbb{P}^n$. This involves a strategy used by Galkin--Shinder to connect properties of a cubic hypersurface to its Fano variety of lines via cut and paste relations in the Grothendieck ring of varieties. Generalizing this idea to varieties of higher codimension and degree, we can measure growth rates of weighted probabilities of $k$-planes contained in a sequence of varieties with varying initial parameters over a finite field. In the course of doing this, we move an identity motivated by rationality problems involving cubic hypersurfaces to a motivic statistics setting associated with cohomological stability.
We give a survey of the recent progress on the study of K-stability of Fano varieties by an algebro-geometric approach.
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