On real bisectional curvature for Hermitian manifolds

Abstract

Motivated by the recent work of Wu and Yau on the ampleness of canonical line bundle for compact Kähler manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called $\text{real bisectional curvature}$ for Hermitian manifolds. When the metric is Kähler, this is just the holomorphic sectional curvature $H$, and when the metric is non-Kähler, it is slightly stronger than $H$. We classify compact Hermitian manifolds with constant non-zero real bisectional curvature, and also slightly extend Wu-Yau's theorem to the Hermitian case. The underlying reason for the extension is that the Schwarz lemma of Wu-Yau works the same when the target metric is only Hermitian but has nonpositive real bisectional curvature.

References in corpus (6)

• Negative Holomorphic curvature and positive canonical bundle
• An extension of a theorem of Wu-Yau
• Form-type Calabi-Yau equations on Kähler manifolds of nonnegative orthogonal bisectional curvature
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Cited by in corpus (3)

• Invariant metrics on negatively pinched complete Kähler manifolds
• RC-positivity, rational connectedness and Yau's conjecture
• RC-positivity and rigidity of harmonic maps into Riemannian manifolds