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

Abstract: 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.