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On the image of MRC fibrations of projective manifolds with semi-positive holomorphic sectional curvature
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In this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semi-positive holomorphic sectional curvature. Toward these conjectures, we prove that the canonical bundle of images of such fibrations is not big. Our proof gives a generalization of Yang's solution using RC positivity for Yau's conjecture. As an application, we show that any compact K\"ahler surface with semi-positive holomorphic sectional curvature is rationally connected, or a complex torus, or a ruled surface over an elliptic curve.
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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.
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