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

TL;DR

This paper investigates the structure of projective manifolds with semi-positive holomorphic sectional curvature, proving that their rationally connected fibrations have non-big canonical bundles and classifying certain surfaces.

Contribution

It generalizes Yang's solution to Yau's conjecture and explores the structure of manifolds with semi-positive curvature, providing new insights into their fibrations and classifications.

Findings

Canonical bundles of images are not big

Compact Kähler surfaces are either rationally connected, tori, or ruled over elliptic curves

Generalizes Yang's approach using RC positivity

Abstract

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.