Tohoku Mathematical Journal
2020
December
SECOND SERIES VOL. 72, NO. 4

Tohoku Math. J.
72 (2020), 581-594

Title CLOSED ALMOST K\"AHLER 4-MANIFOLDS OF CONSTANT NON-NEGATIVE HERMITIAN HOLOMORPHIC SECTIONAL CURVATURE ARE K\"AHLER

Author Mehdi Lejmi and Markus Upmeier

(Received March 19, 2018)
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.

Mathematics Subject Classification. Primary 53C25; Secondary 53C15.
Key words and phrases. Holomorphic sectional curvature, almost-Kähler geometry, canonical Hermitian connection.

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