Every compact Kähler manifold with nef anti-canonical bundle admits a locally trivial fibration over a Calabi-Yau manifold with rationally connected fibers.
An introduction to the K \"a hler-Ricci flow , volume 2086 of Lecture Notes in Mathematics
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Compact K\"ahler manifolds with nef anti-canonical bundle
Every compact Kähler manifold with nef anti-canonical bundle admits a locally trivial fibration over a Calabi-Yau manifold with rationally connected fibers.