Weak spectral positivity implies rational connectedness and simple connectedness for compact Kähler manifolds and simple connectedness for even-dimensional Riemannian manifolds.
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Compact Kähler manifolds of algebraic dimension zero are isogenous to products of Kummer and simple manifolds; four-dimensional strictly simple ones are étale tori quotients or holomorphically symplectic.
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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.
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Bogomolov decomposition and compact K{\"a}hler manifolds of algebraic dimension zero
Compact Kähler manifolds of algebraic dimension zero are isogenous to products of Kummer and simple manifolds; four-dimensional strictly simple ones are étale tori quotients or holomorphically symplectic.