Three-manifolds and Kähler groups

Abstract

ABSTRACT. We give a simple proof of a result originally due to Dimca and Suciu [6]: 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 Z or Z ⊕ Z 2 .

Kählerian three-manifold groups

“…The proof of Theorem 2 given above has the pleasant feature of dealing with the cases that M is closed or with non-empty boundary uniformly. In particular, it gives yet another treatment of closed three-manifolds that is different from [4,12]. Now, taking for granted the closed case, an alternative -and much more high-tech -treatment of manifolds with non-empty and non-spherical boundary is implicit in my recent paper [13], where I discussed Kähler groups of positive deficiency.…”

Kähler groups and subdirect products of surface groups

“…While a general answer seems out of reach for the moment, it is fruitful to consider Serre's question in the context of more specific classes of groups. For instance, it has been shown that if the fundamental group of a compact 3-manifold without boundary is Kähler then it is finite [20] (see also [6] and [26]) and that a Kähler group with non-trivial first L2-Betti number is commensurable to a surface group (i.e. the fundamental group of a closed Riemann surface) [23].…”

Quasiprojective Three-Manifold Groups and Complexification of Three-Manifolds

“…Remark 3.12. In order to recover the main Theorems of [DiSu] or [Ko1] from Theorem 3.4 with the modifications mentioned in Remark 3.11, it remains to show that fundamental groups of circle bundles N over closed surfaces of positive genus are not Kähler. If the bundle is trivial, then b 1 (N ) is odd.…”

“…A group is called quasiprojective (respectively, Kähler) if it is the fundamental group of a smooth complex quasiprojective variety (respectively, compact Kähler manifold). Kähler and quasiprojective 3manifold groups have attracted much attention of late [DiSu,Ko1,BMS,DPS,FrSu,Ko2]. In this paper we characterize quasiprojective 3-manifold groups.…”

“…As a consequence of our results we deduce the restrictions on quasiprojective 3-manifold groups obtained by the authors of [DPS,FrSu,Ko2] and the restrictions on good complexifications of 3-manifolds deduced in [To] (this is done in in Section 3.1.1). We also indicate, in Remark 3.12, how to deduce the classification of (closed) 3-manifold Kähler groups [DiSu,Ko1,BMS] using the techniques of Theorem 1.1, thus providing a unified treatment of known results.…”