We prove that on any log Fano pair of dimension n whose stability threshold is less than
n+1n, any valuation computing the stability threshold has a finitely generated associated …

In this paper, we prove the openness of K-semistability in families of log Fano pairs by
showing that the stability threshold is a constructible function on the fibers. We also prove …

We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we
show that the moduli stack of K-stable Q-Fano varieties is separated. Together with recently …

In this paper, we consider the CM line bundle on the K-moduli space, ie, the moduli space
parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace …

Algebraic varieties are shapes defined by polynomial equations. Smooth Fano threefolds
are a fundamental subclass that can be thought of as higher-dimensional generalizations of …

We prove two new results on the $ K $-polystability of $\mathbb {Q} $-Fano varieties based
on purely algebro-geometric arguments. The first one says that any $ K $-semistable log …

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 …

K-stability of Fano varieties: an algebro-geometric approach Page 1 EMS Surv. Math. Sci. 8 (2021),
265–354 DOI 10.4171/EMSS/51 © 2021 European Mathematical Society Published by EMS …

We prove that the K-moduli space of cubic threefolds is identical to their geometric invariant
theory (GIT) moduli. More precisely, the K-semistability, K-polystability, and K-stability of …

Abstract For any Q Q-Gorenstein klt singularity (X, o), we introduce a normalized volume
function vol vol^ that is defined on the space of real valuations centered at o and consider …