Related papers: The Space of Kaehler metrics

We introduce mu-scalar curvature for a K"ahler metric with a moment map mu and start up a study on constant mu-scalar curvature K"ahler metric as a generalization of both cscK metric and K"ahler-Ricci soliton and as a continuity path to…

We establish a uniform Sobolev inequality for K\"ahler metrics, which only require an entropy bound and no lower bound on the Ricci curvature. We further extend our Sobolev inequality to singular K\"ahler metrics on K\"ahler spaces with…

This is a continuation of the previous articles on Kahler cone metrics. In this article, we introduce weighted function spaces and provide a self-contained treatment on cone angles in the whole interval $(0,1]$. We first construct geodesics…

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 present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal K\"{a}hler manifold with constant nonpositive holomorphic sectional curvature is…

A compactness theorem is proved for a family of K\"{a}hler surfaces with constant scalar curvature and volume bounded from below, diameter bounded from above, Ricci curvature bounded and the signature bounded from below. Furthermore, a…

We construct two types of Eguchi-Hanson metrics with the negative constant scalar curvature. The type I metrics are K\"{a}hler. The type II metrics are ALH whose total energy can be negative.

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…

In this paper, we shall show that a polarized algebraic manifold is K-stable if the polarization class admits a Kaehler metric of constant scalar curvature. This generalizes the results of Chen-Tian, Donaldson and Stoppa. (Parts of the…

This paper is the first step in a systematic project to study examples of K\"ahler manifolds with positive holomorphic sectional curvature ($H > 0$). Previously Hitchin proved that any compact K\"ahler surface with $H>0$ must be rational…

We show that on Kahler manifolds with negative first Chern class, the sequence of algebraic metrics introduced by H. Tsuji converges uniformly to the Kahler-Einstein metric. For algebraic surfaces of general type and orbifolds with isolated…

We prove that an admissible manifold (as defined by Apostolov, Calderbank, Gauduchon and T{\o}nnesen-Friedman), arising from a base with a local K\"ahler product of constant scalar curvature metrics, admits Generalized Quasi-Einstein…

For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen's metric and interpret it as a local index theorem for the…

Over a compact K\"ahler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a K\"ahler metric with conic singularities along a divisor. We deduce several existence results of constant scalar…

In this paper, assuming that a polarized algebraic manifold $(X,L)$ is strongly K-stable, we shall show that the polarization class $c_1(L)$ admits a constant scalar curvature Kaehler metric.

We show the theorem which provides some sufficient condition to the non-existence of a complete K\"ahler--Einstein metric of negative scalar curvature whose holomorphic sectional curvature is negatively pinched: Let $\Omega$ be a bounded…

Let $\cM_{g,n}$ be the moduli space of Riemann surfaces of genus $g$ with $n$ punctures. From a complex perspective, moduli space is hyperbolic. For example, $\cM_{g,n}$ is abundantly populated by immersed holomorphic disks of constant…

Given $(X,\omega)$ compact K\"ahler manifold and $\psi\in\mathcal{M}^{+}\subset PSH(X,\omega)$ a model type envelope with non-zero mass, i.e. a fixed potential determing some singularities such that $\int_{X}(\omega+dd^{c}\psi)^{n}>0$, we…

On a compact $n$-dimensional manifold $M$, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume, is Einstein. It has been conjectured that a critical metric of the total…

We study a system of equations on a compact complex manifold, that couples the scalar curvature of a Kaehler metric with a spectral function of a first-order deformation of the complex structure. The system comes from an…