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

First we confirm a conjecture asserting that any compact Kähler 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 Kähler 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 Kollár-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 Kähler C-spaces with $b_2=1$. The new conditions provide viable candidates for a curvature characterization of homogenous Kähler manifolds related to a generalized Hartshone conjecture.