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

Abstract

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-)K\"ahler groups posed by Aguilar and Campana.