Abstract
In this paper, we consider orthogonal Ricci curvature $$Ric^{\perp }$$ for Kahler manifolds, which is a curvature condition closely related to Ricci curvature and holomorphic sectional curvature. We prove comparison theorems and a vanishing theorem related to these curvature conditions, and construct various examples to illustrate subtle relationship among them. As a consequence of the vanishing theorem, we show that any compact Kahler manifold with positive orthogonal Ricci curvature must be projective. This result complements a recent result of Yang (RC-positivity, rational connectedness, and Yau’s conjecture. arXiv:1708.06713 ) on the projectivity under the positivity of holomorphic sectional curvature. The simply-connectedness is shown when the complex dimension is smaller than five. Further study of compact Kahler manifolds with $$Ric^{\perp }>0$$ is carried in Ni et al. (Manifolds with positive orthogonal Ricci curvature. arXiv:1806.10233 ).
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