2023/11/30 by Zhang, Shiyu, Zhang, Xi · 3 citations
#53C07 #58E15 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2311.18779
In this paper, we establish a "pseudo-effective" version of the holonomy principle for compact Kähler manifolds with nonnegative holomorphic sectional curvature. As applications, we prove that if a compact complex manifold M admits a Kähler metric ω with nonnegative holomorphic sectional curvature and (M,ω) has no nonzero truly flat tangent vector at some point (which is satisfied when the holomorphic sectional curvature is quasi-positive), then M must be projective and rationally connected. This answers a problem raised by Matsumura and Yang and extends Yau's conjecture. We also prove that a compact simply connected Kähler manifold with nonnegative holomorphic sectional curvature is projective and rationally connected. Additionally, we classify non-projective Kähler 3-dimensional manifolds with nonnegative holomorphic sectional curvature. Furthermore, we show that a compact Kähler manifold admits a Hermitian metric with positive real bisectional curvature is a projective and rationally connected manifold.