The holomorphic bisectional curvature of a Kähler metric measures the curvature interaction of two complex
tangent directions. For nonzero vectors 
of type 
, it is
 |
using the curvature sign in which the Fubini-Study metric has positive holomorphic bisectional curvature.
Setting 
gives holomorphic
sectional curvature. Positivity means 
for every such pair, without requiring a uniform
positive lower bound
on a noncompact manifold. The Euclidean Kähler
metric has zero holomorphic bisectional curvature. The Yau
uniformization conjecture uses pointwise positive
bisectional curvature.
