Rational homotopy theory implies that if a nilpotent group group is Kahler, then it's Malcev Lie algebra must be quadratically presented. The usual Heisenberg group is not quadratically presented, hence is not Kahler. However, the higher Heisenberg groups are quadratically presented. We show that the Heisenberg groups of ranks 4 and 6 are not Kahler. This result is sharp by the results of Campana, [C. R. Acad. Sci. Paris Ser. I Math. 317 (1993), no. 8, 777--780; MR 94k:32048].
The Heisenberg groups are 2-step nilpotent groups. We exhibit infinitely many 3-step quadratically presented nilpotent Lie algebras, and we classify quadratically presented nilpotent Lie algebras whose abelianization has dimension at most 5.
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Last modified by jac at 15:52 on 12/27/1997.