Journal of the European Mathematical Society

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Volume 11, Issue 3, 2009, pp. 521–528

DOI: 10.4171/JEMS/158

Which 3-manifold groups are Kähler groups?

Alexandru Dimca (1) and Alexander I. Suciu (2)

(1) Laboratoire J. A. Dieudonné, UMR CNRS 6621, Université de Nice Sophia Antipolis, Parc Valrose, 06108, NICE, FRANCE
(2) Department of Mathematics, Northeastern University, 360 Huntington Avenue, MA 02115, BOSTON, UNITED STATES

The question in the title, first raised by Goldman and Donaldson, was partially answered by Reznikov. We give a complete answer, as follows: if G can be realized as both the fundamental group of a closed 3-manifold and of a compact Kähler manifold, then G must be finite—and thus belongs to the well-known list of finite subgroups of O(4), acting freely on S³.

Keywords: Kähler manifold, 3-manifold, fundamental group, cohomology ring, resonance variety, isotropic subspace