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.
Bibliographical noteThe first author was supported in part by a PSC-CUNY award #60053-00 48, jointly funded by The Professional Staff Congress and The City University of New York. The second author was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –UP 85/2-1, UP 85/3-1.
- Holomorphic sectional curvature
- almost-Kähler geometry
- canonical Hermitian connection