Closed almost Kähler 4-manifolds of constant non-negative Hermitian holomorphic sectional curvature are Kähler

We show that a closed almost Kähler 4-manifold of pointwise constant holomorphic sectional curvature k≥0 with respect to the canonical Hermitian connection is automatically Kähler. The same result holds for k<0 if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observation that such manifolds are self-dual, so that Chern–Weil theory implies useful integral formulas, which are then combined with results from Seiberg–Witten theory.