Closed almost K�hler 4-manifolds of constant non-negative Hermitian holomorphic sectional curvature are K�hler
M Lejmi, M Upmeier - 2020 - projecteuclid.org
M Lejmi, M Upmeier
2020•projecteuclid.orgWe show that a closed almost K�hler 4-manifold of pointwise constant holomorphic
sectional curvature k≧0 with respect to the canonical Hermitian connection is automatically
K�hler. 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.
sectional curvature k≧0 with respect to the canonical Hermitian connection is automatically
K�hler. 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.
Abstract
We show that a closed almost K�hler 4-manifold of pointwise constant holomorphic sectional curvature with respect to the canonical Hermitian connection is automatically K�hler. The same result holds for if we require in addition that the Ricci curvature is -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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