This article gives a description, by means of functorial intrinsic fibrations, of the geometric structure (and conjecturally also of the Kobayashi pseudometric, as well as of the arithmetic in the projective case) of compact Kähler manifolds. We first define special manifolds as being the compact Kähler manifolds with no meromorphic map onto an orbifold of general type, the orbifold structure on the base being given by the divisor of multiple fibres. We next show that rationally connected Kähler...

For any compact Kähler manifold $X$ and for any equivalence relation generated by a symmetric binary relation with compact analytic graph in $X\times X$, the existence of a meromorphic quotient is known from Inv. Math. 63 (1981). We give here a simplified and detailed proof of the existence of such quotients, following the approach of that paper. These quotients are used in one of the two constructions of the core of $X$ given in the previous paper of this fascicule, as well as in many other questions.

We show global existence and convergence results for the pluriclosed flow on manifolds for which certain naturally associated tensor bundles are globally generated.

Let $X$ be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let $G$ be a connected complex reductive affine algebraic group equipped with a real form ${\sigma }_G$. We define pseudo-real principal $G$-bundles on $X$. These are generalizations of real algebraic principal $G$-bundles over a real algebraic variety. Next we define stable, semistable and polystable pseudo-real principal $G$-bundles. Their relationships with the usual stable, semistable...

In this paper we study the balanced metrics on some Hartogs triangles of exponent $\gamma \in {\mathbb{Z}}^{+}$, i.e., $${\Omega }_n\left(\gamma \right)=\{z=(z_1,\cdots ,z_n)\in {ℂ}^n:|z_1{|}^{1/\gamma }<|z_2|<\cdots <|z_n|<1\}$$ equipped with a natural Kähler form ${\omega }_{g\left(\mu \right)}:=\frac{1}{2}(i/\pi )\partial \overline{\partial }{\Phi }_n$ with $${\Phi }_n\left(z\right)=-{\mu }_{1}ln\left(\right|z_2{|}^{2\gamma }-\left|z_1{|}^2\right)-\sum _{i=2}^{n-1}{\mu }_{i}ln\left(\right|z_{i+1}{|}^2-\left|z_i{|}^2\right)-{\mu }_{n}ln(1-|z_n{|}^2),$$ where $\mu =({\mu }_1,\cdots ,{\mu }_n)$, ${\mu }_i>0$, depending on $n$ parameters. The purpose of this paper is threefold. First, we compute the explicit expression for the weighted Bergman kernel function for $({\Omega }_n\left(\gamma \right),g\left(\mu \right))$ and we prove that $g\left(\mu \right)$ is balanced if and only if ${\mu }_1>1$ and $\gamma {\mu }_1$ is an integer, ${\mu }_i$ are integers such that ${\mu }_i\ge 2$ for all $i=2,...,n-1$, and ${\mu }_n>1$. Second, we prove that $g\left(\mu \right)$ is Kähler-Einstein if and only if ${\mu }_1={\mu }_2=\cdots ={\mu }_n=2\lambda$, where $\lambda$ is a nonzero...

Dans cette note nous établissons le résultat suivant, annoncé dans [CCE13] : si $G\subset {\mathrm{GL}}_n\left(ℂ\right)$ est l’image d’une représentation linéaire d’un groupe kählérien ${\pi }_1\left(X\right)$, il admet un sous-groupe d’indice fini qui est l’image d’une représentation linéaire du groupe fondamental d’une variété projective complexe lisse $X^{\text{'}}$.Il s’agit donc de la solution (à indice fini près) pour les représentations linéaires d’une question usuelle demandant si le groupe fondamental d’une variété kählérienne compacte est aussi celui d’une variété...

This work is devoted to the study of Einstein equations with a special shape of the energy-momentum tensor. Our results continue Stepanov’s classification of Riemannian manifolds according to special properties of the energy-momentum tensor to Kähler manifolds. We show that in this case the number of classes reduces.

Soit $M=\mathbb{P}\left(ℰ\right)$ une surface complexe réglée. Nous introduisons des métriques de volume fini sur $M$ dons les singularités sont paramétrisées par une structure parabolique sur le fibré $ℰ$. Nous généralisons alors un résultat de Burns-deBartolomeis et Le Brun, en montrant que l’existence de métriques kählériennes singulières, de volume fini, à courbure scalaire constante négative ou nulle sur $M$ est équivalente à une condition de polystabilité parabolique sur $ℰ$ ; de plus ces métriques proviennent toutes de quotients...

It is proved that if $M$ is a weakly 1-complete Kähler manifold with only one end, then $H_c^1(M,𝒪)=0$ or there exists a proper holomorphic mapping of $M$ onto a Riemann surface.

We give a simple proof of a result originally due to Dimca and Suciu: a group that is both Kähler and the fundamental group of a closed three-manifold is finite. We also prove that a group that is both the fundamental group of a closed three-manifold and of a non-Kähler compact complex surface is $\mathbb{Z}$ or $\mathbb{Z}\oplus {\mathbb{Z}}_2$.

Based on the results of the first two parts to this paper, we prove that the canonical bundle of a minimal Kähler threefold (i.e.$K_X$ is nef) is good,i.e.its Kodaira dimension equals the numerical Kodaira dimension, (in particular some multiple of $K_X$ is generated by global sections); unless $X$ is simple. “Simple“ means that there is no compact subvariety through the very general point of $X$ and $X$ not Kummer. Moreover we show that a compact Kähler threefold with only terminal singularities whose canonical...

We prove that every Kähler solvmanifold has a finite covering whose holomorphic reduction is a principal bundle. An example is given that illustrates the necessity, in general, of passing to a proper covering. We also answer a stronger version of a question posed by Akhiezer for homogeneous spaces of nonsolvable algebraic groups in the case where the isotropy has the property that its intersection with the radical is Zariski dense in the radical.