4-dimensional almost Kähler manifolds and L2-scalar curvature functional

In this paper, we consider the integrability of compact almost Kähler 4-manifolds for which a certain part of the divergence of the Weyl conformal tensor vanishes. Proof of our main theorem gives another one of the theorem of M. Itoh [Intern. J. Math. 15 (2004) 573-580] which gives partial answer to the Goldberg conjecture in case of 4-dimensional negative scalar curvature.