Toward the structure of fibered fundamental groups of projective varieties

Arapura, Donu

Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 595-611.

Le groupe fondamental d’une variété projective lisse est dit fibré s’il s’envoie surjectivement sur celui d’une courbe de genre $2$ ou plus. Le but de cet article est d’établir des restrictions fortes sur ces groupes, et en particulier sur ceux des surfaces de Kodaira. Dans le cas spécifique d’une surface de Kodaira, ces résultats se présentent sous la forme de restrictions sur la représentation de monodromie dans le ‘mapping class group’. Lorsque la représentation de monodromie se compose de certaines représentations standard, les images sont Zariski denses dans un groupe semi-simple de type hermitien.

The fundamental group of a smooth projective variety is fibered if it maps onto the fundamental group of a smooth curve of genus $2$ or more. The goal of this paper is to establish some strong restrictions on these groups, and in particular on the fundamental groups of Kodaira surfaces. In the specific case of a Kodaira surface, these results are in the form of restrictions on the monodromy representation into the mapping class group. When the monodromy is composed with certain standard representations, the images are Zariski dense in a semisimple group of Hermitian type.

Classification : 14H30
Keywords: Kähler group, Mumford-Tate group
Mot clés : Groupe de Kähler, groupe de Mumford-Tate

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