2019/02/03 by Lei Ni, Ni, Lei · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1902.00974
openalex publication_date 2019/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
First we confirm a conjecture asserting that any compact Kähler manifold N with \Ric^⊥>0 must be simply-connected by applying a new viscosity consideration to Whitney's comass of (p, 0)-forms. Secondly we prove the projectivity and the rational connectedness of a Kähler manifold of complex dimension n under the condition \Rick>0 (for some k∈ \1, ⋯, n\, with \Ricn being the Ricci curvature), generalizing a well-known result of Campana, and independently of Kollár-Miyaoka-Mori, for the Fano manifolds. The proof utilizes both the above comass consideration and a second variation consideration of \citeNi-Zheng2. Thirdly, motivated by \Ric^⊥ and the classical work of Calabi-Vesentini \citeCV, we propose two new curvature notions. The cohomology vanishing Hq(N, T'N)=\0\ for any 1≤ q≤ n and a deformation rigidity result are obtained under these new curvature conditions. In particular they are verified for all classical Kähler C-spaces with b2=1. The new conditions provide viable candidates for a curvature characterization of homogenous Kähler manifolds related to a generalized Hartshone conjecture.