On Some Examples in Symplectic Topology

Article is devoted to the Examples 2 and 3 of the symplectic solvable Lie groups R with some special cohomological properties, which have been constructed by Benson and Gordon. But they are not succeeded in constructing corresponding compact forms for symplectic structures on these Lie groups. Recently A.Tralle proved that there is no compact form in the Example 3. But his proof is rather complicated and uses some very special topological methods. We propose much more simpler (and purely algebraic) method to prove the main result of the Tralle’s paper. Moreover we prove that for Example 2 there is no compact form too. But it appears that some modification of the construction of the Example 2 gives some other example of a solvable Lie group R with the same cohomological properties as R, but with a compact form. Let (M,ω) be a compact symplectic manifold. This article is devoted to some examples, which have been constructed in [1], where some problems about Kählerian structures on solvmanifolds (i.e. homogeneous spaces R/Γ of solvable Lie groups R with discrete stationary subgroups Γ) are studied. There are Examples 2 and 3 of the symplectic solvable Lie groups R in [1] with some special properties (closely related to the properties of Kählerian Lie groups). But the authors of [1] are not succeeded in constructing corresponding compact forms R/Γ for this symplectic structures on Lie groups. These examples in [1] have been constructed for the purpose of illustration of the conditions of the main result (see Theorem 2 there about a structure of compact solvmanifolds R/Γ which admit a Kähler structure) of this paper. Recently A.Tralle [2] proved that there is no compact form (i.e. a compact solvmanifold R/Γ for the Lie group R) in the Example 3 from [1]. But the proof in [2] is rather complicated and uses some special topological methods (rational models ets.). We propose much more simpler (and purely algebraic) method to prove the main result from [2]. Moreover we prove that for Example 2 from [1] there is no compact form too. But it appears that some modification of the construction of the Example 2 from [1] gives some other example of a solvable Lie group R with a compact form M = R/Γ. The author express his thanks to A.Tralle for giving information about some modern problems in symplectic topology. Firstly we are going to describe the Lie algebras from the Examples 2 and 3 in [1]. Partially supported by Russian Foundation of Basic Research (grant N 01-01-00709) 1991 Mathematics Subject Classification. 53C40, 53C55.