Firstly, we confirm a conjecture asserting that any compact Kähler manifold N with Ric ⊥ > 0 {\operatorname{Ric}^{\perp}>0} must be simply-connected by applying a new viscosity consideration to Whitney’s comass of ( p , 0 ) {(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 {\operatorname{Ric}_{k}>0} (for some k ∈ { 1 , … , n } {k\in\{1,\dots,n\}} , with Ric n {\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 [L. Ni and F. Zheng, Positivity and Kodaira embedding theorem, preprint 2020, https://arxiv.org/abs/1804.09696]. Thirdly, motivated by Ric ⊥ {\operatorname{Ric}^{\perp}} and the classical work of Calabi and Vesentini [E. Calabi and E. Vesentini, On compact, locally symmetric Kähler manifolds, Ann. of Math. (2) 71 1960, 472–507], we propose two new curvature notions. The cohomology vanishing H q ⁢ ( N , T ′ ⁢ N ) = { 0 } {H^{q}(N,T^{\prime}N)=\{0\}} for any 1 ≤ q ≤ n {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 {b_{2}=1} . The new conditions provide viable candidates for a curvature characterization of homogeneous Kähler manifolds related to a generalized Hartshone conjecture.

中文翻译：

---

Kähler流形的基本群，有理连通性和正性

首先，我们证实一个猜想，即任何对Ric⊥> 0 {\ operatorname {Ric} ^ {\ perp}> 0}的紧致Kähler流形N都必须通过对（p， 0）{（p，0）}-形式。其次，我们证明在条件Ric k> 0 {\ operatorname {Ric} _ {k}> 0}（对于某些k∈{1，…，n} { k \ in \ {1，\ dots，n \}}，其中Ric n {\ operatorname {Ric} _ {n}}是Ricci曲率），概括了Campana的一个著名结果，并且独立于宫冈市的Kollár和森，用于Fano流形。该证明既利用了上述指南针考虑因素，又利用了[L. Ni和F. Zheng，《正定性和Kodaira嵌入定理》，预印本2020，https：//arxiv.org/abs/1804.09696]。第三，受里克（Ric）⊥{\ operatorname {Ric} ^ {\ perp}}以及卡拉比（Calabi）和维森蒂尼（Vesentini）[E. Calabi和E. Vesentini，在紧凑的局部对称Kähler流形上，Ann。数学。（2）71 1960，472–507]，我们提出了两个新的曲率概念。对于任何1≤q≤n {在这些新的曲率条件下获得1 \ leq q \ leq n}并获得变形刚度结果。特别是，对所有经典kählerC空间的b 2 = 1 {b_ {2} = 1}进行了验证。新条件为与广义Hartshone猜想有关的齐次Kähler流形的曲率表征提供了可行的候选方案。局部对称Kähler流形，安。数学。（2）71 1960，472–507]，我们提出了两个新的曲率概念。对于任何1≤q≤n {在这些新的曲率条件下获得1 \ leq q \ leq n}并获得变形刚度结果。特别是，对所有经典kählerC空间的b 2 = 1 {b_ {2} = 1}进行了验证。新条件为与广义Hartshone猜想有关的齐次Kähler流形的曲率表征提供了可行的候选方案。局部对称Kähler流形，安。数学。（2）71 1960，472–507]，我们提出了两个新的曲率概念。对于任何1≤q≤n {在这些新的曲率条件下获得1 \ leq q \ leq n}并获得变形刚度结果。特别是，对所有经典kählerC空间的b 2 = 1 {b_ {2} = 1}进行了验证。新条件为与广义Hartshone猜想有关的齐次Kähler流形的曲率表征提供了可行的候选方案。对于任何1≤q≤n {1 \ leq q \ leq n}的T ^ {\ prime} N）= \ {0 \}}，并在这些新曲率条件下获得了变形刚度结果。特别是，对所有经典kählerC空间的b 2 = 1 {b_ {2} = 1}进行了验证。新条件为与广义Hartshone猜想有关的齐次Kähler流形的曲率表征提供了可行的候选方案。对于任何1≤q≤n {1 \ leq q \ leq n}的T ^ {\ prime} N）= \ {0 \}}，并在这些新曲率条件下获得了变形刚度结果。特别是，对所有经典kählerC空间的b 2 = 1 {b_ {2} = 1}进行了验证。新条件为与广义Hartshone猜想有关的齐次Kähler流形的曲率表征提供了可行的候选方案。