Comparison and vanishing theorems for Kähler manifolds

In this paper, we consider orthogonal Ricci curvature R i c ⊥ for Kähler 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 Kähler 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 Kähler manifolds with R i c ⊥ > 0 is carried in Ni et al. (Manifolds with positive orthogonal Ricci curvature. arXiv:1806.10233 ).