Almost-Kähler four manifolds with harmonic self-dual Weyl curvature

We show that a compact almost-Kähler four manifold $(M,g,\omega )$ with harmonic self-dual Weyl curvature and constant scalar curvature is Kähler if $c_1\cdot \omega \ge 0$. We also prove an integral curvature inequality for compact almost-Kähler four manifolds with harmonic self-dual Weyl curvature.