Weak spectral positivity implies rational connectedness and simple connectedness for compact Kähler manifolds and simple connectedness for even-dimensional Riemannian manifolds.
Boucksom, Divisorial Zariski decompositions on compact complex manifolds,Annales scientifiques de l’Ecole normale sup´ erieure.Vol
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Simply connectedness of K\"ahler and Riemannian manifolds via spectral estimates (with an appendix by Shiyu Zhang)
Weak spectral positivity implies rational connectedness and simple connectedness for compact Kähler manifolds and simple connectedness for even-dimensional Riemannian manifolds.