The holomorphic sectional curvature of a Kähler metric is its holomorphic bisectional
curvature evaluated on two equal complex tangent
directions. With the same curvature convention, for a nonzero vector
It depends only on the complex line spanned by Positive holomorphic
bisectional curvature implies positive holomorphic
sectional curvature, but the two positivity conditions are distinct. The Yau
uniformization conjecture assumes the stronger bisectional condition.
See also Holomorphic Bisectional Curvature ,
Kähler Metric ,
Sectional
Curvature ,
Yau Uniformization Conjecture
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References Diverio, S. "Kobayashi Hyperbolicity, Negativity of the Curvature and Positivity of the Canonical Bundle." In Hyperbolicity Properties
of Algebraic Varieties (Ed. S. Diverio). Paris: Soc. Math. France, pp. 209-252,
2021. https://arxiv.org/abs/2011.11379 .
Cite this as:
Weisstein, Eric W. "Holomorphic Sectional Curvature."
From MathWorld https://mathworld.wolfram.com/HolomorphicSectionalCurvature.html
Subject classifications
of type 
it is
. Positive holomorphic
bisectional curvature implies positive holomorphic
sectional curvature, but the two positivity conditions are distinct. The Yau
uniformization conjecture assumes the stronger bisectional condition.
