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We show that a closed almost K\"ahler 4-manifold of globally constantholomorphic sectional curvature $k\geq 0$ with respect to the canonicalHermitian connection is automatically K\"ahler. The same result holds for $k<0$if we require in addition that the Ricci curvature is J-invariant. The proofsare based on the observation that such manifolds are self-dual, so thatChern-Weil theory implies useful integral formulas, which are then combinedwith results from Seiberg--Witten theory.

Closed almost-K\"ahler 4-manifolds of constant non-negative Hermitian holomorphic sectional curvature are K\"ahler