  Among other results, a compact almost K\"ahler manifold is proved to be
K\"ahler if the Ricci tensor is semi-negative and its length coincides with
that of the star Ricci tensor or if the Ricci tensor is semi-positive and its
first order covariant derivatives are Hermitian. Moreover, it is shown that
there are no compact almost K\"ahler manifolds with harmonic Weyl tensor and
non-parallel semi-positive Ricci tensor. Stronger results are obtained in
dimension 4.
