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 -Kähler manifolds, whose fibres are isomorphic respectively to , and . 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 on (rational) Hodge classes of Abelian varieties with rational period matrix.
A primary Hopf surface is a compact complex surface with universal cover and cyclic fundamental group generated by the transformation , , and such that and . Being diffeomorphic with Hopf surfaces cannot admit any Kähler metric. However, it was known that for and 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 (). 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 and logarithmic Kodaira dimension , any...
We prove the existence of non-positively curved Kähler-Einstein metrics with cone singularities along a given simple normal crossing divisor of a compact Kähler manifold, under a technical condition on the cone angles, and we also discuss the case of positively-curved Kähler-Einstein metrics with cone singularities. As an application we extend to this setting classical results of Lichnerowicz and Kobayashi on the parallelism and vanishing of appropriate holomorphic tensor fields.
Un des problèmes les plus intéressants de la géométrie différentielle complexe consiste à comprendre les classes de Kähler de variétés complexes admettant des métriques à courbure scalaire constante. La question de l’unicité a été récemment résolue par Donaldson, Mabuchi, Chen–Tian. Des liens forts avec la stabilité algébrique des variétés ont été mis en évidence. L’exposé s’attachera à exposer les idées nouvelles qui ont mené à ces résultats.
In this paper we obtain all invariant, anti-invariant and submanifolds in endowed with a globally conformal Kähler structure which are minimal and tangent or normal to the Lee vector field of the g.c.K. structure.