Invariant differential forms on compact nilmanifolds
Let \(N=G/\Gamma\) be a compact nilmanifold, where \(G\) is a connected, simply connected nilpotent Lie group with Lie algebra \(\mathfrak g\) and \(\Gamma\) is a discrete subgroup of \(G\). Let \(H^*({\mathfrak g})\) be the cohomology ring of the complex \(\Lambda ({\mathfrak g}^*)\) of left-invariant differential forms on \(G\), and \(I^*({\mathfrak g})\) be the invariant differential forms on \(\mathfrak g\). Using orientable fibrations and spectral sequences the author proves that, if \(I^*({\mathfrak g}) \to H^*({\mathfrak g})\) is an injective map, then \(G\) is abelian and \(N\) is a torus. This statement has the following corollaries: (1) A compact nilmanifold \(N=G/\Gamma\) has a formal minimal model if and only if \(I^*({\mathfrak g}) \to H^*({\mathfrak g})\) is injective. (2) If \(N=G/\Gamma\) is an even-dimensional compact nilmanifold, \(N\) has an invariant symplectic structure and a Kähler structure if and only if \(I^*({\mathfrak g}) \to H^*({\mathfrak g})\) is injective. The second corollary is interesting since there are many compact symplectic nilmanifolds which do not admit a Kähler structure [cf. e.g. \textit{M. Fernández, M. J. Gotay} and \textit{A. Gray}: Proc. Am. Math. Soc. 103, 1209-1212 (1988; Zbl 0656.53034)].
scientific article; zbMATH DE number 942737