On the torsion-free nilpotent fundamental groups of smooth quasi-projective varieties of rank up to seven

Let $X$ be a smooth quasi-projective variety. Assume that the (topological) fundamental group ${\pi }_1(X,x)$ is torsion-free nilpotent. We show that if the first Betti number $b_1(X)\le 3$, then ${\pi }_1(X,x)$ is isomorphic to either ${\mathbb{Z}}^n$ for $n=1,2,3$, a lattice in the Heisenberg group $H_3(\mathbb{R})$ or $\mathbb{R}\times H_3(\mathbb{R})$. Moreover, we prove that ${\pi }_1(X,x)$ is abelian or $2$-step nilpotent if its rank is less than or equal to seven. More precisely, we determine the real nilpotent Lie groups in which torsion-free nilpotent fundamental groups can be embedded as lattices for ranks up to six and seven, respectively. Our main results are a partial positive answer to a question on nilpotent (quasi-)Kähler groups posed by Aguilar and Campana.