相关论文: Semi-continuity of complex singularity exponents a…
Given a Fano manifold $(X,\omega)$ we develop a variational approach to characterize analytically the existence of K\"ahler-Einstein metrics with prescribed singularities, assuming that these singularities can be approximated algebraically.…
We run the continuity method for Mabuchi's generalization of K\"{a}hler-Einstein metrics, assuming the existence of an extremal K\"{a}hler metric. It gives an analytic proof (without minimal model program) of the recent existence result…
Given a compact polarized manifold $(X,L)$, we introduce two new stability thresholds in terms of singularity types of global quasi-plurisubharmonic functions on $X$. We prove that in the Fano setting, the new invariants can effectively…
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 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.…
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…
In this paper we prove the existence of coupled K\"ahler-Einstein metrics on complex manifolds whose canonical bundle is ample. These metrics were introduced and their existence in the said case was proven by Hultgren and Nystr\"om using…
We study degenerate complex Monge-Amp\`ere equations of the form $(\omega+dd^c \varphi)^n = e^{t \varphi} \mu$ where $\omega$ is a big semi-positive form on a compact K\"ahler manifold $X$ of dimension $n$, $t \in \R^+$, and $\mu=f\omega^n$…
In this paper we provide new necessary and sufficient conditions for the existence of K\"ahler-Einstein metrics on small deformations of a Fano K\"ahler-Einstein manifold. We also show that the Weil-Petersson metric can be approximated by…
We explore connections between existence of $\Bbbk$-rational points for Fano varieties defined over $\Bbbk$, a subfield of $\mathbb{C}$, and existence of K\"ahler-Einstein metrics on their geometric models. First, we show that geometric…
On a compact K\"ahler manifold $(X,\omega)$, we study the strong continuity of solutions with prescribed singularities of complex Monge-Amp\`ere equations with integrable Lebesgue densities. Moreover, we give sufficient conditions for the…
Given a one-parameter family of $\mathbb{Q}$-Fano varieties such that the central fibre admits a unique K\"ahler-Einstein metric, we provide an analytic method to show that the neighboring fibre admits a unique K\"ahler-Einstein metric. Our…
The notion of coupled K\"ahler-Einstein metrics was introduced recently by Hultgren-WittNystr\"om. In this paper we discuss deformation of a coupled K\"ahler-Einstein metrics on a Fano manifold. In particular we obtain a necessary and…
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 propose new types of canonical metrics on K\"ahler manifolds, called coupled K\"ahler-Einstein metrics, generalizing K\"ahler-Einstein metrics. We prove existence and uniqueness results in the cases when the canonical bundle is ample and…
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.
Motivated by the notion of multiplier Hermitian-Einstein metric of type $\sigma$ introduced by Mabuchi, we introduce the notion of $\sigma$-extremal K\"{a}hler metrics on compact K\"{a}hler manifolds, which generalizes Calabi's extremal…
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 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…
The global holomorphic \alpha-invariant introduced by Tian is closely related with the study in the existence of Kahler-Einstein metric. We apply the result of Tian, Lu and Zelditch on polarized Kahler metrics to approximate…