Koszul-Sullivan models and the cohomology of certain solvmanifolds
The computation of the de Rham cohomology of nilmanifolds is a very simple matter, taking into account Nomizu's theorem which states that this cohomology is isomorphic with the Chevalley-Eilenberg cohomology of the Lie group. Another very useful tool for investigating the existence of Kähler structures on manifolds is the well-known result by \textit{P. Deligne}, \textit{P. Griffiths}, \textit{J. Morgan} and \textit{D. Sullivan} [Invent. Math. 29, 245-274 (1975; Zbl 0312.55011)] who proved that the minimal model of a Kähler manifold is formal. Using these results and the hard Lefschetz theorem, several authors have proved that the only nilmanifolds which carry Kähler structures are tori. Also, it was possible to obtain a lot of examples of non-Kähler symplectic manifolds. Although Nomizu's theorem was extended by Hattori to completely solvable solvmanifolds, it does not work for general solvmanifolds. Thus, the problem to compute the de Rham cohomology is frequently a very hard job. In this paper, the authors introduce a fundamental tool which permits to compute the cohomology of certain classes of solvmanifolds. Indeed, they use a theorem of Felix and Thomas concerning the Koszul-Sullivan model of a fibration. Using this result, the authors prove the following conjecture for solvmanifolds of dimension four: Conjecture. The only solvmanifolds which carry Kähler structures are tori.
scientific article; zbMATH DE number 1080034