Related papers: On the uniqueness of conical K\"ahler-Einstein met…

In this paper is to extend the Cheeger-Colding Theory to the class of conic Kahler-Einstein metrics. This extension provides a technical tool for [LTW] in which we prove a version of the Yau-Tian-Donaldson conjecture for Fano varieties with…

In this paper, we introduce the "coupled Ricci iteration", a dynamical system related to the Ricci operator and twisted K\"ahler-Einstein metrics as an approach to the study of coupled K\"ahler-Einstein (CKE) metrics. For negative first…

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…

Given a compact K\"ahler manifold $(X,\omega)$, due to the work of Darvas-Di Nezza-Lu, the space of singularity types of $\omega$-psh functions admits a natural pseudo-metric $d_\mathcal S$ that is complete in the presence of positive mass.…

We prove the existence and uniqueness of K\"ahler-Einstein metrics on Q-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on…

Robert Bryant (Theorie des varietes minimales et applications, 1988, 154: 321-347) proved that an isolated singularity of a conformal metric of positive constant curvature on a Riemann surface is a conical one. Using Complex Analysis, we…

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…

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 prove a regularity result for Monge-Amp\`ere equations degenerate along smooth divisor on Kaehler manifolds in Donaldson's spaces of $\beta$-weighted functions. We apply this result to study the curvature of Kaehler metrics with conical…

In this paper, we study the stability of the conical K\"ahler-Ricci flows on Fano manifolds. That is, if there exists a conical K\"ahler-Einstein metric with cone angle $2\pi\beta$ along the divisor, then for any $\beta'$ sufficiently close…

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…

We introduce a compactification of the space of simple positive divisors on a Riemann surface, as well as a compactification of the universal family of punctured surfaces above this space. These are real manifolds with corners. We then…

This thesis results from an intensive study on the canonical metrics on the Teichm\"{u}ller spaces and the moduli spaces of Riemann surfaces. There are several renowned classical metrics on $T_g$ and $\mathcal{M}_g$, including the…

In this expository article we review the problem of finding Einstein metrics on compact K\"ahler manifolds and Sasaki manifolds. In the former half of this article we see that, in the K\"ahler case, the problem fits better with the notion…

We construct a family of K\"ahler-Einstein edge metrics on all Hirzebruch surfaces using the Calabi ansatz and study their angle deformation. This allows us to verify in some special cases a conjecture of Cheltsov-Rubinstein that predicts…

We use twistor methods to promote Namikawa's universal Poisson deformations of conic affine symplectic singularities to families of hyperk\"ahler structures deforming hyperk\"ahler cone metrics. The metrics we produce are generally…

We focus on the classical open problem of the classification of K\"ahler-Einstein manifolds that can be K\"ahler immersed into a complex projective space endowed with the Fubini-Study metric. In particular, we will deal with such problem in…

We find a new obstruction for a real Einstein 4-orbifold with an A1-singularity to be a limit of smooth Einstein 4-manifolds. The obstruction is a curvature condition at the singular point. For asymptotically hyperbolic metrics, with…

We discuss how metric limits and rescalings of K\"ahler-Einstein metrics connect with Algebraic Geometry, mostly in relation to the study of moduli spaces of varieties, and singularities. Along the way, we describe some elementary examples,…

Let $E\to M$ be a holomorphic vector bundle over a compact Kaehler manifold $(M, \omega)$. We prove that if $E$ admits a $\omega$-balanced metric (in X. Wang's terminology) then it is unique. This result together with a result of L.…