Projective manifolds with pseudo-effective tangent bundle admit a smooth fibration to a flat projective manifold with rationally connected general fiber, and all minimal such surfaces are classified.
Beauville, Vari\'et\'es K\"ahleriennes dont la premi\`ere classe de Chern est nulle, Geometry and analysis on complex manifolds, 39--50, J
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On projective manifolds with pseudo-effective tangent bundle
Projective manifolds with pseudo-effective tangent bundle admit a smooth fibration to a flat projective manifold with rationally connected general fiber, and all minimal such surfaces are classified.