Essential manifolds with extra structures

We consider classes of algebraic manifolds A, of symplectic manifolds S, of symplectic manifolds with the hard Lefschetz property HS and the class of cohomologically symplectic manifolds CS. For every class of manifolds C we denote by EC(π,n) a subclass of n-dimensional rationally essential manifolds with fundamental group π. In this paper we prove that for all the above classes with symplectically aspherical form the condition EC(π,2n)≠∅ implies that EC(π,2n−2)≠∅ for every n>2. Also we prove that all the inclusions EA⊂EHS⊂ES⊂ECS are proper.