Cohomologically Kähler manifolds with no Kähler metrics
It is well known that the existence of a Kähler structure on a compact differentiable manifold imposes cohomological restrictions, not only on the Betti numbers but also on the cohomology ring, such as the hard Lefschetz property and formality. The authors give six- and eight-dimensional examples of compact solvmanifolds, which satisfy these restrictions but do not admit any Kähler structure. As they point out, most of their examples are not new [see \textit{C. Benson} and \textit{C. S. Gordon}, Proc. Am. Math. Soc. 108, No. 4, 971--980 (1990; Zbl 0689.53036); \textit{L. C. de Andrés}, \textit{M. Fernández}, \textit{M. de León} and \textit{J. J. Mencía}, Rend. Mat. Appl., VII. Ser. 12, No. 1, 59--67 (1992; Zbl 0758.53018)] but it was not known before whether they carry any Kähler structure. This is proven here using results of Arapura-Nori [\textit{D. Arapura} and \textit{M. Nori}, Compos. Math. 116, No. 2, 173--188 (1999; Zbl 0971.14020)] and Campana [\textit{F. Campana}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 28, No. 3, 307--316 (1995; Zbl 0829.32006)] on the fundamental group of compact Kähler manifolds. It follows also from the Benson-Gordon conjecture, which states that any compact Kähler solvmanifold of a completely solvable Lie group is a torus. That conjecture can be obtained as a corollary of results of Benson-Gordon and Arapura-Nori. It is a special case of Arapuras classification of compact Kähler solvmanifolds up to biholomorphism [\textit{D. Arapura}, Kähler solvmanifolds, Int. Math. Res. Not. 2004, No. 3, 131--137 (2004)].
scientific article; zbMATH DE number 2012619