Related papers: Singular cscK metrics on smoothable varieties

We run the continuity method for Mabuchi's generalization of K\"{a}hler-Einstein metrics, assuming the existence of an extremal K\"{a}hler metric. It gives an analytic proof (without minimal model program) of the recent existence result…

In this paper we extend recent breakthrough of Chen-Cheng \cite{CC1, CC2, CC3} on existence of constant scalar K\"ahler metric on a compact K\"ahler manifold to Calabi's extremal metric. Our argument follows \cite{CC3} and there are no new…

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 $X$.

In this paper, we propose a coupled system of complex Hessian equations which generalizes the equation for constant scalar curvature K\"ahler (cscK) metrics. We show this system can be realized variationally as the Euler-Lagrange equation…

Consider a polarized complex manifold (X,L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X,L). For most of the common functionals in K\"ahler geometry, we prove that the slope at infinity…

In this paper we study K-polystability of arbitrary (possibly non-projective) compact K\"ahler manifolds admitting holomorphic vector fields. As a main result, we show that existence of a constant scalar curvature K\"ahler (cscK) metric…

In 1984 Gauduchon conjectured that one can find Gauduchon metrics with prescribed Ricci curvature on all compact complex manifolds. This conjecture was settled by Sz\'ekelyhidi-Tosatti-Weinkove (TW17, TW19, STW17) by the study of the…

The article is devoted to the investigation of smoothness of functions $f(x_1,...,x_m)$ of variables $x_1,...,x_m$ in infinite fields with non-trivial multiplicative ultra-norms, where $m\ge 2$. Theorems about classes of smoothness $C^n$ or…

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…

We prove a compactness theorem for K\"ahler metrics with various bounds on Ricci curvature and the $\mathcal I$ functional. We explore applications of our result to the continuity method and the Calabi flow.

This article finds constant scalar curvature Kahler metrics on certain compact complex surfaces. The surfaces considered are those admitting a holomorphic submersion to a curve, with fibres of genus at least 2. The proof is via an adiabatic…

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…

Existence of strong K\"ahler with torsion metrics, shortly SKT metrics, on complex manifolds has been shown to be unstable under small deformations. We find necessary conditions under which the property of being SKT is stable for a smooth…

Let $X$ be a compact K\"ahler manifold. In this paper we study the existence of constant weighted scalar curvature K\"ahler (weighted cscK) metrics on $X$. More precisely, we establish a priori $C^{k}$-estimates ($k\geq 0$) for the K\"ahler…

We partially confirm an old conjecture of Donaldson that if there exists a cscK metrics in a given K\"ahler class, then there is no degenerated geodesic ray which is tamed by a bounded ambient geometry unless it parallels to a holomorphic…

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,L) be a polarized projective complex manifold. We show, by a simple toric one-dimensional example, that Mabuchi's K-energy functional on the geodesically complete space of bounded positive (1,1)-forms in the first Chern class of L,…

We formulate a notion of K-stability for K\"ahler manifolds, and prove one direction of the Yau-Tian-Donaldson conjecture in this setting. More precisely, we prove that the Mabuchi functional being bounded below (resp. coercive) implies…

We prove a criterion for the existence of harmonic metrics on Higgs bundles that are defined on smooth loci of klt varieties. As one application, we resolve the quasi-etale uniformisation problem for minimal varieties of general type to…

We prove that constant scalar curvature K\"ahler (cscK) manifolds with transcendental cohomology class are K-semistable, naturally generalising the situation for polarised manifolds. Relying on a very recent result by R. Berman, T. Darvas…