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The fundamental group, rational connectedness and the positivity of Kähler manifolds. (English) Zbl 1470.53063

Summary: Firstly, we confirm a conjecture asserting that any compact Kähler manifold \(N\) with \(\operatorname{Ric}^{\perp}>0\) 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 \(\operatorname{Ric}_k>0\) (for some \(k\in\{1,\dots,n\} \), with \(\operatorname{Ric}_n\) being the Ricci curvature), generalizing a well-known result of Campana, and independently of Kollár, Miyaoka and Mori, for the Fano manifolds. The proof utilizes both the above comass consideration and a second variation consideration of the first author and F. Zheng [“Positivity and Kodaira embedding theorem”, Preprint, arXiv:1804.09696]. Thirdly, motivated by \(\operatorname{Ric}^{\perp}\) and the classical work of E. Calabi and E. Vesentini [Ann. Math. (2) 71, 472–507 (1960; Zbl 0100.36002)], we propose two new curvature notions. The cohomology vanishing \(H^q(N,T^{\prime}N)=\{0\}\) for any \(1\leq q\leq n\) 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 homogeneous Kähler manifolds related to a generalized Hartshone conjecture.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
57M05 Fundamental group, presentations, free differential calculus
14M22 Rationally connected varieties
14J45 Fano varieties

Citations:

Zbl 0100.36002
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References:

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