Almost-K\"ahler four manifolds with harmonic self-dual Weyl curvature

Abstract

We show that a compact almost-K\"ahler four manifold $(M, g, \omega)$ with harmonic self-dual Weyl curvature and constant scalar curvature is K\"ahler if $c_{1}\cdot\omega\geq 0$. We also prove an integral curvature inequality for compact almost-K\"ahler four manifolds with harmonic self-dual Weyl curvature.