mixed Hodge structures on the strictly upper triangular Lie algebras

We show that, for $k\ge 3$, a lattice in the unitriangular Lie group $U_k(\mathbb{R})$ is the fundamental group of a smooth quasi-projective complex algebraic variety if and only if $k=3$. The proof uses the conjugation symmetry and weight conditions of mixed Hodge structure on the Malcev Lie algebra proven by Morgan. This result identifies a family of groups that must be excluded in the search for smooth quasi-projective varieties with torsion-free nilpotent fundamental groups of high nilpotency class.