Rationally connected foliations on surfaces

TL;DR

This paper investigates foliations with rationally connected leaves on surfaces, establishing a polarization that maximizes the rationally quotient via the Harder-Narasimhan filtration of the tangent bundle.

Contribution

It introduces a polarization on surfaces that aligns the Harder-Narasimhan filtration with the maximal rationally quotient, advancing understanding of foliations with rationally connected leaves.

Findings

Existence of a specific polarization on surfaces

Harder-Narasimhan filtration yields maximal rationally quotient

Enhanced understanding of foliations with rationally connected leaves

Abstract

In this short note we study foliations with rationally connected leaves on surfaces. Our main result is that on surfaces there exists a polarisation such that the Harder-Narasimhan filtration of the tangent bundle with respect to this polarisation yields the maximal rationally quotient of the surface.