Three-manifolds and Kähler groups

Kotschick, D.

Annales de l'Institut Fourier, Tome 62 (2012) no. 3, pp. 1081-1090.

On donne une preuve simple d’un résultat dû à Dimca et Suciu : un groupe de Kähler qui est aussi le groupe fondamental d’une variété trois-dimensionelle est fini. On montre également qu’un groupe qui est le groupe fondamental d’une variété trois-dimensionelle et en même temps le groupe fondamental d’une surface complexe compacte non-kählerienne est soit $\mathbb{Z}$ soit $\mathbb{Z}\oplus {\mathbb{Z}}_2$.

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 $\mathbb{Z}\oplus {\mathbb{Z}}_2$.

Classification : 32Q15, 57M05, 14F35, 32J15, 57M50
Keywords: three-manifold groups, Kähler groups
Mot clés : groupes fondamentaux des variétés trois-dimensionelles, groupes de Kähler

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