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arXiv · 1709.05210

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

Mehdi Lejmi·Markus Upmeier
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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.

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Mehdi Lejmi, Markus Upmeier. 2017-09-15. Closed almost-K\"ahler 4-manifolds of constant non-negative Hermitian holomorphic sectional curvature are K\"ahler. https://arxiv.org/abs/1709.05210

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