The fundamental group, rational connectedness and the positivity of Kaehler manifolds

First we confirm a conjecture asserting that any compact Kahler manifold $N$ with $Ric^perp>0$ must be simply-connected by applying a new viscosity consideration to Whitneys comass of $(p, 0)$-forms. Secondly we prove the projectivity and the rational connectedness of a Kahler manifold of complex dimension $n$ under the condition $Ric_k>0$ (for some $kin {1, cdots, n}$, with $Ric_n$ being the Ricci curvature), generalizing a well-known result of Campana, and independently of Kollar-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 $H^q(N, TN)={0}$ for any $1le qle n$ and a deformation rigidity result are obtained under these new curvature conditions. In particular they are verified for all classical Kahler C-spaces with $b_2=1$. The new conditions provide viable candidates for a curvature characterization of homogenous Kahler manifolds related to a generalized Hartshone conjecture.