Maximal rationally connected fibrations and movable curves

TL;DR

This paper explores the relationship between movable curves and the maximal rationally connected fibration of complex projective manifolds, linking geometric properties with the Harder-Narasimhan filtration of the tangent bundle.

Contribution

It demonstrates that a suitable movable curve can be used to recover the maximal rationally connected fibration via the Harder-Narasimhan filtration.

Findings

A movable curve can determine the maximal rationally connected fibration.

The Harder-Narasimhan filtration encodes the structure of rationally connected fibrations.

The approach generalizes Miyaoka's criterion for uniruledness.

Abstract

A well known result of Miyaoka asserts that a complex projective manifold is uniruled if its cotangent bundle restricted to a general complete intersection curve is not nef. Using the Harder-Narasimhan filtration of the tangent bundle, it can moreover be shown that the choice of such a curve gives rise to a rationally connected foliation of the manifold. In this note we show that, conversely, a movable curve can be found so that the maximal rationally connected fibration of the manifold may be recovered as a term of the associated Harder-Narasimhan filtration of the tangent bundle.