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 , 2 n ) ≠ ∅ implies that EC ( π , 2 n − 2 ) ≠ ∅ for every n > 2 . Also we prove that all the inclusions EA ⊂ EHS ⊂ ES ⊂ ECS are proper.