I would like to know the intuition behind the holomorphic bisectional curvature of Hermitian manifolds. I already know that the classical sectional curvature of a Riemannian (not necessarily complex) manifold roughly tells us how geodesics spread apart. What is the analogue for holomorphic bisectional curvature? Do you know any reference, where I can read more about this?
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$\begingroup$ (For a compact Kahler manifold). If the hbc is strictly positive, your manifold is $CP^n$; if it is non-negative, your manifold is (conjecturally) homogeneous. HBC measures positivity of the tangent bundle. $\endgroup$– Misha VerbitskyCommented Oct 30, 2013 at 10:00
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Ngaiming Mok, The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature, J. Differential Geom. Volume 27, Number 2 (1988), 179-214.
Mok improves Mori's and Siu-Yau's proof of Fraenkel's conjecture claiming that a compact Kahler manifold with positive HBC is biholomorphic to ${\Bbb C} P^n$. He shows that any compact Kahler manifold with non-negative HBC is locally isometric to a product of symmetric spaces and ${\Bbb C} P^n$. It is interesting that Mok uses Hamilton's evolution equation (essentially the same as used by Perelman 15 years later).
answered Oct 30, 2013 at 10:11