Compact Kähler manifolds with pseudo-effective tangent bundle admit a smooth fibration whose base is an étale quotient of a torus and whose fibers are rationally connected.
Campana, Remarques sur les groupes de K\"ahler nilpotents, Ann
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On compact K\"ahler manifolds with pseudo-effective tangent bundle
Compact Kähler manifolds with pseudo-effective tangent bundle admit a smooth fibration whose base is an étale quotient of a torus and whose fibers are rationally connected.