Related papers: Variation of singular K\"ahler-Einstein metrics: p…

We extend Tsuji's iterative construction of complete K\"ahler--Einstein metrics with negative scalar curvature to noncompact K\"ahler manifolds with bounded geometry, using Berndtsson's method from the compact setting. Consequently, given a…

Given a family $f:\mathcal X \to S$ of canonically polarized manifolds, the unique K\"ahler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle $\mathcal K_{\mathcal X/S}$. We use a global elliptic…

Let $X$ be a canonically polarized variety, i.e. a complex projective variety such that its canonical class $K_{X}$ defines an ample $\Q-$line bundle, and satisfying the conditions $G_1$ and $S_2$. Our main result says that $X$ admits a…

Let $p:X\to Y$ be an holomorphic surjective map between compact K\"ahler manifolds and let $D$ be an effective divisor on $X$ with generically simple normal crossings support and coefficients in $(0,1)$. Provided that the adjoint canonical…

Given an effectively parameterized family $f:X\to S$ of canonically polarized manifolds, the K\"ahler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle $K_{X/S}$. We use a global elliptic equation to…

For projective varieties with definite first Chern class we have one type of canonical metric which is called K\"ahler-Einstein metric. But for varieties with an intermidiate Kodaira dimension we can have several different types of…

Let $M=P(E)$ be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle $E \to \Sigma$ over a compact complex curve $\Sigma$ of genus $\ge 2$. Building on ideas of Fujiki, we prove that $M$…

Let f : X --> Y be a holomorphic map of complex manifolds, which is proper, Kahler, and surjective with connected fibers, and which is smooth over Y-Z the complement of an analytic subset Z. Let E be a Nakano semi-positive vector bundle on…

In this paper, we show that along $\mathbb Q$-Fano fibration, when general fibres, base and central fiber (with at worst Kawamata log terminal singularities)are K-poly stable then there exists a relative K\"ahler-Einstein metric. We…

We study several questions involving relative Ricci-flat K\"ahler metrics for families of log Calabi-Yau manifolds. Our main result states that if $p:(X,B)\to Y$ is a K\"ahler fiber space such that $\displaystyle (X_y, B|_{X_y})$ is…

Let $L$ be a (semi)-positive line bundle over a Kahler manifold, $X$, fibered over a complex manifold $Y$. Assuming the fibers are compact and non-singular we prove that the hermitian vector bundle $E$ over $Y$ whose fibers over points $y$…

Let $p:X\rightarrow Y$ be a surjective holomorphic mapping between K\"ahler manifolds. Let $D$ be a bounded smooth domain in $X$ such that every generic fiber $D_y:=D\cap p^{-1}(y)$ for $y\in Y$ is a strongly pseudoconvex domain in…

Let $X$ be a compact K\"ahler manifold and $D$ be a simple normal crossing divisor on $X$ such that $K_X+D$ is big and nef. We first prove that the singular K\"ahler--Einstein metric constructed by Berman--Guenancia is almost-complete on $X…

This paper is a sequel to \cite{Berndtsson}. In that paper we studied the vector bundle associated to the direct image of the relative canonical bundle of a smooth K\"ahler morphism, twisted with a semipositive line bundle. We proved that…

Given an effectively parameterized family of canonically polarized manifolds the Kaehler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle. We use a global elliptic equation to show that this metric…

We prove the existence of non-positively curved K\"ahler-Einstein metrics with cone singularities along a given simple normal crossing divisor on a compact K\"ahler manifold, under a technical condition on the cone angles, and we also…

Kodaira embedding theorem provides an effective characterization of projectivity of a K\"ahler manifold in terms the second cohomology. Recently X. Yang [21] proved that any compact K\"ahler manifold with positive holomorphic sectional…

Let $X$ be a compact complex manifold of dimension $n$ and let $m$ be a positive integer with $m\leq n$. Assume that $X$ admits a K\"ahler metric $\omega$ and a weakly positive, $\partial\bar\partial$-closed, smooth $(n-m,\,n-m)$-form…

In this paper, I give a new construction of a K\"{a}hler-Einstein metrics on a smooth projective variety with ample canonical bundle. This result can be generalized to the construction of a singular K\"{a}hler-Einstein metric on a smooth…

We established a Yau--Tian--Donaldson type correspondence, expressed in terms of a single Delzant polytope, concerning the existence of extremal K\"ahler metrics on a large class of toric fibrations, introduced by…