Related papers: Twisted K\"ahler-Einstein metrics

We prove that an extremal metric on a polarised smooth complex projective variety exists if it is $\mathbb{G}$-uniformly $K$-stable relative to the extremal torus over models, extending a result due to Chi Li for constant scalar curvature…

K\"ahler-Einstein currents, also known as singular K\"ahler-Einstein metrics, have been introduced and constructed a little over a decade ago. These currents live on mildly singular compact K\"ahler spaces $X$ and their two defining…

This paper extends our earlier results to higher dimensions using a different approach, based on the rigidity of complex structures on certain domains.

This paper is a survey of some recent progress on the study of Calabi's extremal K\"ahler metrics. We first discuss the Yau-Tian-Donaldson conjecture relating the existence of extremal metrics to an algebro-geometric stability notion and we…

In joint work with Chen and Weber, the author has elsewhere shown that CP2#2(-CP2) admits an Einstein metric. The present paper gives a new and rather different proof of this fact. Our results include new existence theorems for extremal…

The existence of \emph{weak conical K\"ahler-Einstein} metrics along smooth hypersurfaces with angle between $0$ and $2\pi$ is obtained by studying a smooth continuity method and a \emph{local Moser's iteration} technique. In the case of…

Let $X$ be a canonically polarized variety, i.e. a complex projective variety such that its canonical class $K_{X}$ defines an ample $\Q-$line bundle, and satisfying the conditions $G_1$ and $S_2$. Our main result says that $X$ admits a…

This is an invitation to the probabilistic approach for constructing K\"ahler-Einstein metrics on complex projective algebraic manifolds X. The metrics in question emerge in the large N-limit from a canonical way of sampling N points on X,…

We show that a polarized affine variety admits a Ricci flat K\"ahler cone metric, if and only if it is K-stable. This generalizes Chen-Donaldson-Sun's solution of the Yau-Tian-Donaldson conjecture to K\"ahler cones, or equivalently,…

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 call a metric quasi-Einstein if the $m$-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the…

We prove the finite step termination of bubble trees for singularity formation of polarized K\"ahler-Einstein metrics in the non-collapsing situation. We also raise several questions and conjectures in connection with algebraic geometry and…

In this expository paper we review on the existence problem of Einstein-Maxwell K\"ahler metrics, and make several remarks. Firstly, we consider a slightly more general set-up than Einstein-Maxwell K\"ahler metrics, and give extensions of…

We introduce an analogue of Bridgeland's stability conditions for polarised varieties. Much as Bridgeland stability is modelled on slope stability of coherent sheaves, our notion of Z-stability is modelled on the notion of K-stability of…

In this paper we study a generalization of the Kahler-Ricci flow, in which the Ricci form is twisted by a closed, non-negative (1,1)-form. We show that when a twisted Kahler-Einstein metric exists, then this twisted flow converges…

In this paper, we extend the existence and regularity theorems for K\"ahler-Einstein metrics having conic singularities along a simple normal crossing divisor to the case of normal crossing divisor, i.e. when components of the divisor are…

We construct a braided structure on the algebra of K\"ahler differential forms of a commutative algebra twisted by an endomorphism. This generalises the construction done in M. Karoubi, Quantum Methods in Algebraic Topology, see…

We give a complete criterion for the existence of generalized K\"ahler Einstein metrics on toric Fano manifolds from view points of a uniform stability in a sense of GIT and the properness of a functional on the space of K\"ahler metrics.

We prove an integral inequality and two stability results for the ADM mass on AE K\"ahler manifolds of all complex dimensions. The inequality bounds the ADM mass from below by an integral of the scalar curvature and the Hessian of certain…

We prove the existence of tilting objects on generalized Brauer--Severi varieties, some relative flags and some twisted forms of relative flags. As an application we obtain tilting objects on certain homogeneous varieties of classical type…