Uniformizable foliated projective structures along singular foliations

TL;DR

This paper investigates the existence of holomorphic projective structures along leaves of singular foliations on compact complex manifolds, revealing how singularities restrict such structures and establishing conditions for their existence.

Contribution

It introduces new restrictions on foliated projective structures imposed by singularities and proves non-existence results for certain foliations on projective spaces.

Findings

Singularities divide foliated projective structures into parabolic and non-parabolic types.

Presence of a non-degenerate, non-parabolic singularity implies foliation is completely integrable.

Foliations of degree at least two with only non-degenerate singularities on projective space do not admit strongly uniformizable structures.

Abstract

We consider holomorphic foliations by curves on compact complex manifolds, for which we investigate the existence of projective structures along the leaves varying holomorphically (foliated projective structures), that satisfy particular uniformizability properties. Our results show that the singularities of the foliation impose severe restrictions for the existence of such structures. A foliated projective structure separates the singularities of a foliation into parabolic and non-parabolic ones. For a strongly uniformizable foliated projective structure on a compact K\"ahler manifold, the existence of a single non-degenerate, non-parabolic singularity implies that the foliation is completely integrable. We establish an index theorem that imposes strong cohomological restrictions on the foliations having only non-degenerate singularities that support foliated projective structures making all of them parabolic. As an application of our results, we prove that, on a projective space of any dimension, a foliation by curves of degree at least two, with only non-degenerate singularities, does not admit a strongly uniformizable foliated projective structure.