Related papers: Higher regularity for singular K\"ahler-Einstein m…

In this note we prove convexity, in the sense of Colding-Naber, of the regular set of solutions to some complex Monge-Ampere equations with conical singularities along simple normal crossing divisors. In particular, any two points in the…

We study two different natural notions of singular K\"ahler-Einstein metrics on normal complex varieties. In the setting of singular Ricci flat K\"ahler cone metrics that arise as non-collapsed limits of sequences of K\"ahler-Einstein…

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…

The subject of this paper is the explicit momentum construction of complete Einstein metrics by ODE methods. Using the Calabi ansatz, further generalized by Hwang-Singer, we show that there are non-trivial complete conformally K\"ahler…

We consider the metric space of all toric K\"ahler metrics on a compact toric manifold; when "looking at it from infinity" (following Gromov), we obtain the tangent cone at infinity, which is parametrized by equivalence classes of complete…

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 consider constant scalar curvature K\"{a}hler metrics on a smooth minimal model of general type in a neighborhood of the canonical class, which is the perturbation of the canonical class by a fixed K\"{a}hler metric. We show that…

We prove continuity results for new stability thresholds related to uniform K-stability and deduce that uniform K-stability is an open condition in the K\"ahler cone of any compact K\"ahler manifold, thus establishing an algebro-geometric…

This is the first of a series of three papers which provide proofs of results announced recently in arXiv:1210.7494.

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…

We construct infinitely many complete Calabi-Yau metrics on $\mathbf{C}^n$ for $n \geq 3$, with maximal volume growth, and singular tangent cones at infinity. In addition we construct Calabi-Yau metrics in neighborhoods of certain isolated…

In this short note, we prove that on a compact K\"ahler variety $X$ with log terminal singularities and $c_1(X)=0$, any singular Ricci-flat K\"ahler metric has orbifold singularities in restriction to the orbifold locus of $X$.

Let $(M,g)$ be a compact K\"ahler-Einstein manifold with $c_1 > 0$. Denote by $K\to M$ the canonical line-bundle, with total space $X$, and $X_0$ the singular space obtained by blowing down $X$ along its zero section. We employ a…

After a review of the general properties of holomorphic spheres in complex surfaces we describe the local geometry in the vicinity of a CP^1 embedded with a negative normal bundle. As a by-product, we build (asymptotically locally…

Consider a compact K\"ahler manifold which either admits an extremal K\"ahler metric, or is a small deformation of such a manifold. We show that the blowup of the manifold at a point admits an extremal K\"ahler metric in K\"ahler classes…

We discover a "no semistability at infinity" phenomenon for complete Calabi-Yau metrics asymptotic to cones, by eliminating the possible appearance of an intermediate K-semistable cone in the 2-step degeneration theory developed by…

In this paper we prove that, under natural assumptions, the scalar curvature of a Kaehler-Einstein metric on a compactification of C^n is strictly positive.

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.…

Let $X$ be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let $L$ be an ample line bundle on $X$. Assume that the pair $(X,L)$ is the flat limit of a family of smooth polarized Calabi-Yau…

We show that the K\"ahler-Einstein metrics on the four families of examples of symmetric toric Fano manifolds presented by Batyrev and Selivanova cannot be realized as metrics induced by immersions into projective spaces equipped with…