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On manifolds with nonnegative curvature on totally isotropic 2-planes. (English) Zbl 0785.53034
Let $$\widehat{R}$$ be the $$\mathbb{C}$$-linear curvature operator on $$\bigwedge^ 2V^{\mathbb{C}}\cdot K(\sigma) = \langle \widehat{R}(z\wedge w, z\wedge w)\rangle/\| z\wedge w\|^ 2$$ is the number which depends only on the complex subspace $$\sigma$$ spanned by $$z$$ and $$w$$. The curvature tensor $$R$$ has nonnegative (positive) curvature on totally isotropic 2-planes if $$K(\sigma) \geq 0$$ ($$> 0$$) for all totally isotropic (complex) two-dimensional subspaces of $$V^{\mathbb{C}}$$.
The main purpose of this paper is to prove that a compact orientable $$2n$$-dimensional Riemannian manifold $$(M,g)$$ with second Betti number nonzero, nonnegative Riemannian curvature on totally isotropic 2-planes, and satisfying a positivity-type condition at one point, is necessarily Kähler, with second Betti number 1. Using the methods of Siu and Yau the author proves if the positivity condition is satisfied at every point then the manifold is biholomorphic to complex projective space.