Quadratic Presentations and Nilpotent Kähler Groups

It has been known for at least thirty years that certain nilpotent groups cannot be Kähler groups, i.e., fundamental groups of compact Kähler manifolds. The best known examples are lattices in the three-dimensional real or complex Heisenberg groups. It is also known that lattices in certain other standard nilpotent Lie groups, e.g., the full group of upper triangular matrices and the free k-step nilpotent Lie groups, k > 1, are not Kähler. The Heisenberg case was known to J-P. Serre in the early 1960’s, and unified proofs of the above statements follow readily from Sullivan’s theory of minimal models [6],[15], [19], or from Chen’s theory of iterated integrals [4], [10], or from more recent developments such as [9].