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

Abstract

We show that a closed almost Kähler 4-manifold of pointwise constant holomorphic sectional curvature $k\geq 0$ 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.