相关论文: Existence and deformations of Kahler-Einstein metr…

We survey recent results on the existence of K\"ahler-Einstein metrics on certain smoothable Fano varieties, focusing on the importance of such metrics in the construction of compact algebraic moduli spaces of K-polystable Fano varieties.…

In this article we prove the existence of Kahler-Ricci solitons on smoothable, K-stable Q-Fano varieties. We also investigate the behavior of twisted Kahler-Ricci solitons in the Gromov-Hausdorff topology under this smoothing family.

We annnounce a proof of the fact that a K-stable Fano manifold admits a Kahler-Einstein metric and give a brief outline of the proof.

We show that there exist K\"ahler-Einstein metrics on two exceptional Pasquier's two-orbits varieties. As an application, we will provide a new example of K-unstable Fano manifold with Picard number one.

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 show that if a Fano manifold $M$ is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then $M$ admits a K\"ahler-Einstein metric. This is a strengthening of the solution of the…

It is shown that any, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their…

We consider coupled K\"ahler-Einstein metrics and weighted solitons on Fano manifolds. These are natural generalizations of K\"ahler-Einstein metrics. As in the case of K\"ahler-Einstein metrics, the existence is known to be equivalent to…

In this paper, we prove an existence result for K\"ahler-Einstein metrics on $\mathbb Q$-Fano compactifications of Lie groups. As an application, we classify $\mathbb Q$-Fano compactifications of $SO_4(\mathbb C)$ which admit a…

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 relate the global log canonical threshold of a variety with torus action to the global log canonical threshold of its quotient. We apply this to certain Fano varieties and use Tian's criterion to prove the existence of Kahler-Einstein…

We prove the $K$-polystability of all smooth complex Fano threefolds admitting an effective action of $\text{SL}_2$ but not of a 2-torus or 3-torus. In particular, the existence of K\"{a}hler-Einstein metrics on varieties in the families…

We prove a criterion for K-stability of a $\mathbb{Q}$-Fano spherical variety with respect to equivariant special test configurations, in terms of its moment polytope and some combinatorial data associated to the open orbit. Combined with…

We give a criterion for the existence of a K\"ahler-Einstein metric on a Fano manifold $M$ in terms of the higher algebraic alpha-invariants $\alpha_{m,k}(M)$.

We consider Fano manifolds M that admit a collection of finite automorphism groups G_1, ..., G_k, such that the quotients M/G_i are smooth Fano manifolds possessing a Kaehler-Einstein metric. Under some numerical and smoothness assumptions…

Let $Y$ be a compact K\"ahler normal space and $\alpha \in H^{1,1}(Y,\mathbb{R})$ a K\"ahler class. We study metric properties of the space $\mathcal{H}_\alpha$ of K\"ahler metrics in $\alpha$ using Mabuchi geodesics. We extend several…

Let $X$ be a $\mathbb Q$-Fano variety admitting a K\"ahler-Einstein metric. We prove that up to a finite quasi-\'etale cover, $X$ splits isometrically as a product of K\"ahler-Einstein $\mathbb Q$-Fano varieties whose tangent sheaf is…

Symmetric varieties are normal equivarient open embeddings of symmetric homogeneous spaces, and they are interesting examples of spherical varieties. We prove that all smooth Fano symmetric varieties with Picard number one admit…

This is the third and final paper in a series which establish results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle approaches…

We prove an existence result for twisted K\"ahler-Einstein metrics, assuming an appropriate twisted K-stability condition. An improvement over earlier results is that certain non-negative twisting forms are allowed.