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The notion of Kodaira dimension has recently been extended to general almostcomplex manifolds. In this paper we focus on the Kodaira dimension of almostK\"ahler manifolds, providing an explicit computation for a family of almostK\"ahler threefolds on the differentiable manifold underlying a Nakamuramanifold. We concentrate also on the link between Kodaira dimension and thecurvature of the canonical connection of an almost K\"ahler manifold, and showthat in the previous example (and in another one obtained from a Kodairasurface) the Ricci curvature of the almost K\"ahler metric vanishes for all themembers of the family.

Kodaira dimension of almost K\"ahler manifolds and curvature of the canonical connection