On the image of MRC fibrations of projective manifolds with semi-positive holomorphic sectional curvature

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 Kähler surface with semi-positive holomorphic sectional curvature is rationally connected, or a complex torus, or a ruled surface over an elliptic curve.

Keywords

holomorphic sectional curvatures, maximal rationally connected fibrations, Abelian varieties, ruled surfaces, RC positivity, minimal models

2010 Mathematics Subject Classification

Primary 53C55. Secondary 14M22, 32Q10.

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The author is supported by the Grant-in-Aid for Young Scientists (A) #17H04821, Fostering Joint International Research (A) #19KK0342, Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, from JSPS.

Received 17 November 2019

Accepted 30 May 2020

Published 17 February 2021