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 globally constant holomorphic sectional curvature $k\ge 0$ 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.

13 pages