Burger et Mozes ont construit des exemples de groupes simples infinis, qui sont des réseaux dans le groupe des automorphismes d’un immeuble cubique. On montre qu’il n’existe pas de morphisme d’un groupe kählérien vers l’un de ces groupes dont le noyau soit finiment engendré. On en déduit que ces groupes ne sont pas kählériens.

We prove that one can obtain natural bundles of Lie algebras on rank two $s$-Kähler manifolds, whose fibres are isomorphic respectively to $\mathbf{so}(s+1,s+1)$, $\mathbf{su}(s+1,s+1)$ and $\mathbf{sl}(2s+2,ℝ)$. These bundles have natural flat connections, whose flat global sections generalize the Lefschetz operators of Kähler geometry and act naturally on cohomology. As a first application, we build an irreducible representation of a rational form of $\mathbf{su}(s+1,s+1)$ on (rational) Hodge classes of Abelian varieties with rational period matrix.

A primary Hopf surface is a compact complex surface with universal cover ${ℂ}^2-\left\{\right(0,0\left)\right\}$ and cyclic fundamental group generated by the transformation $(u,v)\mapsto (\alpha u+\lambda v^m,\beta v)$, $m\in \mathbb{Z}$, and $\alpha ,\beta ,\lambda \in ℂ$ such that $\mid \alpha \mid \ge \mid \beta \mid \>1$ and $(\alpha -{\beta }^m)\lambda =0$. Being diffeomorphic with $S^3\times S^1$ Hopf surfaces cannot admit any Kähler metric. However, it was known that for $\lambda =0$ and $\mid \alpha \mid =\mid \beta \mid$ they admit a locally conformally Kähler metric with parallel Lee form. We here provide the construction of a locally conformally Kähler metric with parallel Lee form for all primary Hopf surfaces of class $1$ ($\lambda =0$). We also show...

In 1981 J. Noguchi proved that in a logarithmic algebraic manifold, having logarithmic irregularity strictly bigger than its dimension, any entire curve is algebraically degenerate.In the present paper we are interested in the case of manifolds having logarithmic irregularity equal to its dimension. We restrict our attention to Brody curves, for which we resolve the problem completely in dimension 2: in a logarithmic surface with logarithmic irregularity $2$ and logarithmic Kodaira dimension $2$, any...