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.
Albanese maps of projective manifolds with nef anticanonical bundles
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abstract
Let $X$ be a projective manifold such that the anticanonical bundle $-K_X$ is nef. We prove that the Albanese map $p: X \rightarrow Y$ is locally isotrivial. In particular, $p$ is a submersion.
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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.