Related papers: Mabuchi geometry of big cohomology classes with pr…

Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$, and $\theta$ be a closed smooth real $(1,1)$-form representing a big and nef cohomology class. We introduce a metric $d_p, p\geq 1$, on the finite energy space…

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…

Let $X$ be a compact K\"ahler manifold. Given a big cohomology class $\{\theta\}$, there is a natural equivalence relation on the space of $\theta$-psh functions giving rise to $\mathcal S(X,\theta)$, the space of singularity types of…

Let $(X,\omega)$ be a compact K\"ahler manifold and $\theta$ be a smooth closed real $(1,1)$-form that represents a big cohomology class. In this paper, we show that for $p\geq 1$, the high energy space $\mathcal{E}^{p}(X,\theta)$ can be…

Suppose $(X,\omega)$ is a compact K\"ahler manifold. Following Mabuchi, the space of smooth K\"ahler potentials $\mathcal H$ can be endowed with a Riemannian structure, which induces an infinite dimensional path length metric space…

Suppose $(X,\omega)$ is a compact K\"ahler manifold of dimension $n$, and $\theta$ is closed $(1,1)$-form representing a big cohomology class. We introduce a metric $d_1$ on the finite energy space $\mathcal{E}^1(X,\theta)$, making it a…

The space of K\"ahler potentials in a compact K\"ahler manifold, endowed with Mabuchi's metric, is an infinite dimensional Riemannian manifold. We characterize local isometries between spaces of K\"ahler potentials, and prove existence and…

On a compact complex manifold $(M, J)$ endowed with a holomorphic Poisson tensor $\pi_J$ and a deRham class $\alpha\in H^2(M, \mathbb R)$, we study the space of generalized K\"ahler (GK) structures defined by a symplectic form $F\in \alpha$…

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,…

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…

In this note, we shall prove geodesic convexity of the space of K\"ahler potentials on an ALE K\"ahler manifold. This extends earlier results in the compact case proved in the fundamental work of X-X. Chen. We further prove the boundedness…

We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop K{\"a}hler geometry on these varieties, with…

We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kahler potentials on a compact Kahler manifold thus confirming a conjecture of Chen and give some applications in Kahler geometry, including a…

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 $Y$ 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…

Let $X$ be a compact K\"ahler manifold and $\a \in H^{1,1}(X,\R)$ a K\"ahler class. We study the metric completion of the space $\HH_\a$ of K\"ahler metrics in $\a$, when endowed with the Mabuchi $L^2$-metric $d$. Using recent ideas of…

Given any two K\"ahler manifolds $X_1$ and $X_2$, L. Lempert recently proved that if their spaces of K\"ahler potentials are isometric with respect to the Mabuchi metric, then $X_1$ and $X_2$ must be diffeomorphic. We prove that this is no…

We construct the hyper-K\"ahler moduli space of framed monopoles over $\mathbb{R}^3$ for any connected, simply connected, compact, semisimple Lie group and arbitrary mass and charge, and hence symmetry breaking. In order to do so, we define…

We study degenerate complex Monge-Amp\`ere equations of the form $(\omega+dd^c \varphi)^n = e^{t \varphi} \mu$ where $\omega$ is a big semi-positive form on a compact K\"ahler manifold $X$ of dimension $n$, $t \in \R^+$, and $\mu=f\omega^n$…

Let $(X,\omega)$ be a K\"ahler manifold and $\psi: \Bbb R \to \Bbb R_+$ be a concave weight. We show that $\mathcal H_\omega$ admits a natural metric $d_\psi$ whose completion is the low energy space $\mathcal E_\psi$, introduced by…