Construct a variety with fundamental group a lattice whose Lie algebra is L5,9

Construct a smooth complex algebraic variety M with basepoint x such that the fundamental group π1(M, x) is a lattice in a simply connected nilpotent Lie group whose Lie algebra is isomorphic to the five-dimensional complex nilpotent Lie algebra L5,9, defined by the nonzero brackets [X1, X2] = X3, [X2, X3] = X4, and [X1, X3] = X5.

Background

Known examples of smooth complex algebraic varieties include C \ {0}, (C \ {0}) × (C \ {0}), and (H3(R)/Γ) × R>0, whose fundamental groups have Malcev completions isomorphic to the abelian Lie algebras L1,1 and L2,1, and the Heisenberg Lie algebra L3,2, respectively. This suggests that certain nilpotent Lie algebras can arise as the Lie algebras of the Malcev completion of fundamental groups of smooth complex algebraic varieties.

The question asks to extend these constructions to the specific five-dimensional nilpotent Lie algebra L5,9, which in the classification (Table 1) is defined by [X1, X2] = X3, [X2, X3] = X4, and [X1, X3] = X5. Establishing such an example would clarify which low-dimensional nilpotent Lie algebras can be realized via Morgan's mixed Hodge structures and the associated Malcev completions, complementing the negative results provided in the paper for many other Lie algebras.

Addressing this problem would provide a positive instance in contrast to the many obstructions shown in the paper for filiform and certain non-filiform Lie algebras, and would refine our understanding of the realizability of lattices in nilpotent Lie groups as fundamental groups of smooth complex algebraic varieties.