A splitting theorem for good complexifications

Biswas, Indranil; Mj, Mahan; Parameswaran, A. J.

Annales de l'Institut Fourier, Tome 66 (2016) no. 5, pp. 1965-1985.

Le but de cet article est de montrer qu’il existe des restrictions aux groupes fondamentaux que peuvent avoir les variétés admettant une bonne complexification, en démontrant le théorème suivant de décomposition, de type Cheeger–Gromoll : Toute variété fermée $M$ admettant une bonne complexification a un recouvrement fini $M_1$, possédant un structure de fibré de base ${\left(S^1\right)}^k$ et de fibre $N$ ayant une bonne complexification et un premier nombre de Betti virtuel nul. On donne plusieurs applications de ce théorème aux variétés de dimension au plus $5$.

The purpose of this paper is to produce restrictions on fundamental groups of manifolds admitting good complexifications by proving the following Cheeger-Gromoll type splitting theorem: Any closed manifold $M$ admitting a good complexification has a finite-sheeted regular covering $M_1$ such that $M_1$ admits a fiber bundle structure with base ${\left(S^1\right)}^k$ and fiber $N$ that admits a good complexification and also has zero virtual first Betti number. We give several applications to manifolds of dimension at most 5.

Classification : 14P25, 57M05, 14F35, 20F65, 57M50, 57M07, 20F67
Keywords: Good complexification, splitting theorem, affine variety, fundamental group, Geometrization theorem, virtual Betti number.
Mot clés : bonne complexification, théorème de décomposition, variété affine, groupe fondamental, théorème de géométrisation, nombre de Betti virtuel.

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