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

Firstly, we confirm a conjecture asserting that any compact Kähler manifold N with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>Ric</m:mi> <m:mo>⊥</m:mo> </m:msup> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {Ric⊥>0} must be simply-connected by applying a new viscosity consideration to Whitney’s comass of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mn>0</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(p,0)} -forms. Secondly we prove the projectivity and the rational connectedness of a Kähler manifold of complex dimension n under the condition <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>Ric</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {Ricₖ>0} (for some <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> {k∈\{1,,n\}} , with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>Ric</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {Ricₙ} 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>Ric</m:mi> <m:mo>⊥</m:mo> </m:msup> </m:math> {Ric⊥} 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mi>q</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>N</m:mi> <m:mo>,</m:mo> <m:mrow> <m:msup> <m:mi>T</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mi>N</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>0</m:mn> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> {Hq(N,T^{}N)=\{0\}} for any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>q</m:mi> <m:mo>≤</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> {1≤ q≤ 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>b</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {b₂=1} . The new conditions