Morgan's mixed Hodge structures on $p$-filiform Lie algebras and low-dimensional nilpotent Lie algebras

The aim of this paper is to show that the fundamental group of any smooth complex algebraic variety cannot be realized as a lattice of any simply connected nilpotent Lie group whose Lie algebra is $p$-filiform Lie algebra such that neither abelian nor $2$-step nilpotent. Moreover, we provide a sufficient condition for a lattice in a simply connected nilpotent Lie group of dimension up to $6$ not to be isomorphic to the fundamental group of any smooth complex algebraic variety.