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.
A decomposition theorem for projective manifolds with nef anticanonical bundle
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abstract
Let X be a simply connected projective manifold with nef anticanonical bundle. We prove that X is a product of a rationally connected manifold and a manifold with trivial canonical bundle. As an application we describe the MRC fibration of any projective manifold with nef anticanonical bundle.
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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.