AbstractFirstly, we confirm a conjecture asserting that any compact Kähler manifold N with {\operatorname{Ric}^{\perp}>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 {\operatorname{Ric}_{k}>0} (for some {k\in\{1,\dots,n\}}, with {\operatorname{Ric}_{n}} being the Ricci curvature), generalizing a well-known result of Campana, and independently of Kollár, Miyaoka and Mori, for the Fano manifolds. The proof utilizes both the above comass consideration and a second variation consideration of [L. Ni and F. Zheng,
Positivity and Kodaira embedding theorem,
preprint 2020, https://arxiv.org/abs/1804.09696].
Thirdly, motivated by {\operatorname{Ric}^{\perp}} and the classical work of Calabi and Vesentini [E. Calabi and E. Vesentini,
On compact, locally symmetric Kähler manifolds, Ann. of Math. (2) 71 1960, 472–507], we propose two new curvature notions. The cohomology vanishing {H^{q}(N,T^{\prime}N)=\{0\}} for any {1\leq q\leq 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 {b_{2}=1}. The new conditions provide viable candidates for a curvature characterization of homogeneous Kähler manifolds related to a generalized Hartshone conjecture.