Symplectic structures on nilmanifolds: an obstruction for their existence (Q2927943)

A nilmanifold is a compact quotient of a simply connected nilpotent Lie group by a lattice. From the point of view of symplectic geometry, nilmanifolds are very interesting since a symplectic nilmanifold which is not a torus does not carry any Kähler metric (this is due to results of \textit{C. Benson} and \textit{C. S. Gordon} [Topology 27, No. 4, 513--518 (1988; Zbl 0672.53036)] and \textit{K. Hasegawa} [Proc. Am. Math. Soc. 106, No. 1, 65--71 (1989; Zbl 0691.53040)]).NEWLINENEWLINEA result of Nomizu implies that every symplectic structure on a nilmanifold is cohomologous to a left invariant symplectic structure. In particular, the study of symplectic structures on nilpotent Lie algebras is of great interest.NEWLINENEWLINENilpotent Lie algebras have been classified up to dimension 7; the classification in higher dimension is a difficult problem, since there are too many of them. In this sense, it is important to find obstructions on the existence of a symplectic structure on an arbitrary nilpotent Lie algebra. Notice that some results on filiform nilpotent Lie algebras or nilpotent Lie algebras of Heisenberg type exist in the literature.NEWLINENEWLINEIn the paper under review, the author applies some previous results of hers [the author, J. Algebra Appl. 14, No. 1, Article ID 1450078, 17 p. (2015; Zbl 1329.17011)] in order to find a necessary condition for such existence problem. This obstruction is of cohomological type and comes from a spectral sequence associated to a certain canonical filtration on the dual of a nilpotent Lie algebra.NEWLINENEWLINEApplications are then given to the study of nilpotent Lie algebras which appear as nilradicals of minimal parabolic subalgebras associated to the real split Lie algebra of classical complex simple Lie algebras. For such nilpotent Lie algebras, the necessary condition on the existence of symplectic structures, provided by the cohomological obstruction, turns out to be also sufficient.