Rationally connected foliations on surfaces (Q2389155)

Foliations on smooth projective varieties over \({\mathbb C}\) with rationally connected leaves can be constructed from the Harder-Narasimhan filtration of the tangent bundle of the variety. This construction depends on a chosen polarization. A fibration with rationally connected fibers can also be constructed from the maximal rationally connected quotient which is a rational map whose fibers are rationally connected. In this paper, which is a part of the future Ph. D. thesis of the author, he proves that, on surfaces, there always exists a polarization such that the Harder-Narasimhan filtration yields the maximal rationally connected quotient. Similar results to those in this paper has been obtained independently by L. Solá and M. Toma.