Non-positively curved -manifolds with non-Kähler π1

Which 3-manifold groups are Kähler groups?

“…Then G = π 1 (M ) is not a Kähler group. The same conclusion was reached by Hernández-Lamoneda in [19], under the assumption that M is a geometrizable 3-manifold, with all pieces hyperbolic.…”

Three-manifolds and Kähler groups

“…If M has a finite orientable covering that is not an R-homology sphere, then π 1 (M) is not a Kähler group. 1 A first step in this direction was soon taken by Hernández-Lamoneda, although his paper [10] was only published much later. Theorem 4 is more general than the Corollary because not every group whose real cohomology satisfies 3-dimensional Poincaré duality is the fundamental group of an aspherical three-manifold.…”