On symplectically aspherical manifolds with nontrivial pi_2

We construct closed symplectic manifolds for which spherical classes generate arbitrarily large subspaces in 2-homology, such that the first Chern class and cohomology class of the symplectic form both vanish on all spherical classes. We construct both Kaehler and non-Kaehler examples, and show independence of the conditions that these two cohomology classes vanish on spherical homology. In particular, we show that the symplectic form can pair trivially with all spherical classes even when the Chern class pairs nontrivially.