Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 5

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

Pages: 1419 – 1439

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n5.a3


Shin-ichi Matsumura (Mathematical Institute, Tohoku University, Sendai, Japan)


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.


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