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

Abstract

First a conjecture asserting that any compact K\"ahler manifold \(N\) with \(Ric^\perp>0\) must be simply-connected is confirmed by adapting the comass of \((p, 0)\)-forms into a maximum principle via the viscosity consideration. Secondly the projectivity and the rational connectedness of a K\"ahler manifold of complex dimension \(n\) under the condition \(Ric_k>0\) (for some \(k\in \{1, \cdots, n\}\)) is proved, generalizing the previous result of Campana, and Koll\'ar-Miyaoka-Mori independently, for the Fano manifolds. Thirdly we show that under the assumption of Picard number one a manifold with \(Ric^\perp>0\) is Fano. Then via a new curvature notion motivated by \(Ric^\perp\), the cohomology vanishing \(H^q(N, T'N)=\{0\}\) for any \(1\le q\le n\) (as well as a deformation rigidity result) for classical K\"ahler C-spaces with \(b_2=1\) is proved, which generalizes the classical result of Calabi-Vesentini. This new curvature (which is quadratic in terms of linear maps from \(T'N\) to \(T''N\)) leads to a related notion of Ricci curvature, \(Ric^+\). We also showed that a compact K\"ahler manifold with \(Ric^+>0\) is projective and simply-connected.