Kodaira dimension of almost Kahler manifolds and curvature of the canonical connection

We study the Kodaira dimension of almost Kahler manifolds, providing an explicit computation for a family of almost Kahler threefolds on the differentiable manifold underlying a Nakamura manifold. We concentrate also on the link between Kodaira dimension and the curvature of the canonical connection of an almost Kahler manifold and show that in the previous example the Ricci curvature of the almost Kahler metric vanishes for all the members of the family.