We give a variational proof of a version of the Yau–Tian–Donaldson conjecture for twisted
Kähler–Einstein currents, and use this to express the greatest (twisted) Ricci lower bound in …

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 …

Consider a polarized complex manifold (X, L) and a ray of positive metrics on L defined by a
positive metric on a test configuration for (X, L). For many common functionals in Kähler …

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 …

Let (X, D) be a polarized log variety with an effective holomorphic torus action, and Θ be a
closed positive torus invariant (1, 1)‐current. For any smooth positive function g defined on …

We prove an algebraic version of the Hamilton–Tian conjecture for all log Fano pairs. More
precisely, we show that any log Fano pair admits a canonical two-step degeneration to a …

Using quantization techniques, we show that the ı-invariant of Fujita–Odaka coincides with
the optimal exponent in a certain Moser–Trudinger type inequality. Consequently, we obtain …

We study K-stability properties of a smooth Fano variety X using non-Archimedean
geometry, specifically the Berkovich analytification of X with respect to the trivial absolute …

We prove existence of twisted Kähler–Einstein metrics in big cohomology classes, using a
divisorial stability condition. In particular, when− KX -K_X is big, we obtain a uniform Yau …

We show that uniform K-stability is a Zariski open condition in Q-Gorenstein families of Q-
Fano varieties. To prove this result, we consider the behavior of the stability threshold in …