Related papers: Strict positivity of K\"ahler-Einstein currents
This paper is concerned with the existence of constant scalar curvature Kaehler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature Kaehler metrics. We also consider 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…
Consider a divisor D with simple normal crossings in a compact K\"ahler manifold X. We show in this article that a K\"ahler metric in an arbitrary class, with constant scalar curvature and cusp singularities along the divisor is unique in…
Let $X$ be a compact K\"ahler manifold and $\{\theta\}$ be a big cohomology class. We prove several results about the singularity type of full mass currents, answering a number of open questions in the field. First, we show that the Lelong…
We prove that a radial Kaehler metric g is Kaehler-Einstein if and only if one of the following conditions is satisfied: 1. g is extremal and it is associated to a Kaehler-Ricci soliton; 2. two different generalized scalar curvatures of g…
We study the existence of extremal K\"ahler metrics on K\"ahler manifolds. After introducing a notion of relative K-stability for K\"ahler manifolds, we prove that K\"ahler manifolds admitting extremal K\"ahler metrics are relatively…
In our previous paper math.DG/0010008, we develop some new techniques in attacking the convergence problems for the K\"ahler Ricci flow. The one of main ideas is to find a set of new functionals on curvature tensors such that the Ricci flow…
As part of a programme to classify quasi-Einstein metrics $(M,g,X)$ on closed manifolds and near-horizon geometries of extreme black holes, we study such spaces when the vector field $X$ is divergence-free but not identically zero. This…
In the present paper and the companion paper [9] a probabilistic (statistical-mechanical) approach to the construction of canonical metrics on a complex algebraic varieties X is introduced, by sampling "temperature deformed" determinantal…
We obtain a necessary and sufficient condition of existence of a K{\"a}hler-Einstein metric on a $G\times G$-equivariant Fano compactification of a complex connected reductive group $G$ in terms of the associated polytope. This condition is…
The goal of this work is give a precise numerical description of the K\"ahler cone of a compact K\"ahler manifold. Our main result states that the K\"ahler cone depends only on the intersection form of the cohomology ring, the Hodge…
The present paper aims to investigate $(m,\rho)$-quasi-Einstein metrices on almost co-K\"ahler manifolds $\mathcal{M}$. It is proven that if a $(\kappa,\mu)$-almost co-K\"ahler manifold with $\kappa<0$ is $(m,\rho)$-quasi-Einstein manifold,…
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…
In this short note, we prove the existence of constant scalar curvature K\"ahler metrics on compact K\"ahler manifolds with semi-ample canonical bundles.
We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kaehler metric. This generalises conjectures by Yau, Tian and Donaldson which relate to the case of Kaehler-Einstein and constant scalar…
In the present paper and the companion paper [8] a probabilistic (statistical mechanical) approach to the study of canonical metrics and measures on a complex algebraic variety X is introduced. On any such variety with positive Kodaira…
Given a collection of K\"ahler forms and a continuous weight on a compact complex manifold we show that it is possible to define natural new notions of extremal potentials and equilibrium measures which coincide with classical notions when…
We prove that if a compact smooth polarized complex manifold admits in the corresponding Hodge K\"ahler class a conformally K\"ahler, Einstein--Maxwell metric, or more generally, a K\"ahler metric of constant $(\xi, a, p)$-scalar curvature,…
We show that any compact convex simple lattice polytope is the moment polytope of a K\"ahler-Einstein orbifold, unique up to orbifold covering and homothety. We extend the Wang-Zhu Theorem \cite{WZ} giving the existence of a K\"ahler-Ricci…
In this short note, we investigate the existence of orbifold K\"ahler-Einstein metrics on toric varieties. In particular, we show that every $\mathbb{Q}$-factorial normal projective toric variety allows an orbifold K\"ahler-Einstein metric.…