Three-manifolds and Kaehler groups

We give a simple proof of a result originally due to Dimca and Suciu: a group that is both Kaehler 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-Kaehler compact complex surface is infinite cyclic or the direct product of an infinite cyclic group and a group of order two.Comment: 6 pages; corrected statement of Theorem 6; final version to appear in Ann. Inst. Fourie

On products of harmonic forms

We prove that manifolds admitting a Riemannian metric for which products of harmonic forms are harmonic satisfy strong topological restrictions, some of which are akin to properties of flat manifolds. Others are more subtle, and are related to symplectic geometry and Seiberg-Witten theory. We also prove that a manifold admits a metric with harmonic forms whose product is not harmonic if and only if it is not a rational homology sphere.Comment: Revised to include flatness of formal metrics on tori of arbitrary dimensio