  First we confirm a conjecture asserting that any compact K\"ahler manifold
$N$ with $\Ric^\perp>0$ 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 K\"ahler manifold of complex
dimension $n$ under the condition $\Ric_k>0$ (for some $k\in \{1, \cdots, n\}$,
with $\Ric_n$ being the Ricci curvature), generalizing a well-known result of
Campana, and independently of Koll\'ar-Miyaoka-Mori, for the Fano manifolds.
The proof utilizes both the above comass consideration and a second variation
consideration of \cite{Ni-Zheng2}. Thirdly, motivated by $\Ric^\perp$ and the
classical work of Calabi-Vesentini \cite{CV}, we propose two new curvature
notions. The cohomology vanishing $H^q(N, T'N)=\{0\}$ for any $1\le q\le n$ and
a deformation rigidity result are obtained under these new curvature
conditions. In particular they are verified for all classical K\"ahler C-spaces
with $b_2=1$. The new conditions provide viable candidates for a curvature
characterization of homogenous K\"ahler manifolds related to a generalized
Hartshone conjecture.
