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

TL;DR

This paper proves that closed almost K"ahler 4-manifolds with constant non-negative Hermitian holomorphic sectional curvature are necessarily K"ahler, using self-duality, Chern-Weil theory, and Seiberg--Witten invariants.

Contribution

It establishes a rigidity result linking constant Hermitian holomorphic sectional curvature to K"ahler structure in four dimensions.

Findings

Such manifolds are self-dual.

Integral formulas relate curvature and topology.

Results extend to negative curvature with Ricci invariance.

Abstract

We show that a closed almost K\"ahler 4-manifold of globally constant holomorphic sectional curvature $k\geq 0$ with respect to the canonical Hermitian 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 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.