Quadratic presentations and nilpotent Kähler groups (Q1904556)

A Kähler group is the fundamental group of a compact Kähler manifold. The Lie algebra of the Mal'tsev completion of the fundamental group is quadratically presented if it is the quotient of the free Lie algebra on its abelianization by an ideal generated in degree two. It is known that the Lie algebras of such groups do not admit quadratic presentations. One purpose of this paper is to obtain an infinite family of examples of quadratically presented three-step nilpotent Lie algebras and to classify quadratically presented complex nilpotent Lie algebras with abelianization of dimension at most five. The second purpose of this article is to prove that if \(M\) is a compact Kähler manifold with nilpotent fundamental group which is not almost abelian and if \(\omega\) is a non-zero integral element of type (1,1) in its characteristic subspace \(\mathcal C\), then the rank of \(\omega\) is at least 8. Several applications of the above results are also given.