Related papers: K-stability of birationally superrigid Fano 3-fold…

We prove birational superrigidity of Fano double hypersurfaces of index one with quadratic and multi-quadratic singularities, satisfying certain regularity conditions, and give an effective explicit lower bound for the codimension of the…

We prove birational superrigidity of Fano cyclic covers of index 1 over hypersurfaces in the projective space.

We prove K-stability of smooth Fano 3-folds of Picard rank 3 and degree 20 that satisfy very explicit generality condition.

We determine the rationality of very general quasismooth Fano 3-fold weighted hypersurfaces completely and determine the stable rationality of them except for cubic 3-folds. More precisely we prove that (i) very general Fano 3-fold weighted…

We establish birational superrigidity for a large class of singular projective Fano hypersurfaces of index one. In the special case of isolated singularities, our result applies for instance to: (1) hypersurfaces with semi-homogeneous…

We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of K\"ahler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, (b) to compute the stability thresholds…

We give conditions for a uniruled variety of dimension at least 2 to be non-solid. This study provides further evidence to a conjecture by Abban and Okada on the solidity of Fano 3-folds. To complement our results we write explicit…

We give a purely algebro-geometric proof that if the alpha-invariant of a Q-Fano variety X is greater than dim X/(dim X+1), then (X,O(-K_X)) is K-stable. The key of our proof is a relation among the Seshadri constants, the alpha-invariant…

We prove that every smooth Fano threefold from the family No 2.8 is K-stable. Such a Fano threefold is a double cover of the blow-up of $\mathbb{P}^3$ at one point branched along an anti-canonical divisor.

We prove birational superrigidity of every hypersurface of degree N in P^N with singular locus of dimension s, under the assumption that N is at least 2s+8 and it has only quadratic singularities of rank at least N-s. Combined with the…

We show that certain Galois covers of K-semistable Fano varieties are K-stable. We use this to give some new examples of Fano manifolds admitting K\"ahler-Einstein metrics, including hypersurfaces, double solids and threefolds.

We introduce a notion of K-stability for adjoint foliated structures via test configurations and the foliated Donaldson-Futaki invariant. We prove reduction to special test configurations for adjoint Fano foliated structures by showing that…

We prove that for every $\epsilon>0$, there is a birationally super-rigid Fano variety $X$ such that $\frac{1}{2}\leqslant\alpha(X)\leqslant \frac{1}{2}+\epsilon$. Also we show that for every $\epsilon>0$, there is a Fano variety $X$ and a…

Let X be a Fano manifold. G.Tian proves that if X admits a Kaehler-Einstein metric, then it satisfies two different stability conditions: one involving the Futaki invariant of a special degeneration of X, the other Hilbert-Mumford-stability…

We show a relation between the birational superrigidity of Fano manifold and its slope stability in the sense of Ross-Thomas.

We apply a recent theorem of Li and the first author to give some criteria for the K-stability of Fano varieties in terms of anticanonical Q-divisors. First, we propose a condition in terms of certain anticanonical Q-divisors of given Fano…

Tian's criterion for K-stability states that a Fano variety of dimension $n$ whose alpha invariant is greater than $\frac{n}{n+1}$ is K-stable. We show that this criterion is sharp by constructing singular Fano varieties with alpha…

We prove that all smooth Fano threefolds in the families 2.1, 2.2, 2.3, 2.4, 2.6 and 2.7 are K-stable, and we also prove that smooth Fano threefolds in the family 2.5 that satisfy one very explicit generality condition are K-stable.

We study the K-stability of singular Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system is base-point-free but not very ample.

We prove that smooth Fano 3-folds in the families 2.18 and 3.4 are K-stable.