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Uniformizable foliated projective structures along singular foliations

Bertrand Deroin, Adolfo Guillot

TL;DR

The paper develops a comprehensive framework for foliated projective structures along singular holomorphic foliations, distinguishing parabolic and non-parabolic singularities and analyzing leafwise uniformizability. It proves that a strongly uniformizable structure on a compact Kähler manifold with a single non-degenerate non-parabolic singularity forces meromorphic complete integrability of the foliation, and it derives a Baum–Bott type index theorem imposing cohomological constraints when all singularities are non-degenerate and parabolic. The authors supply explicit global and local results, including linearization near non-parabolic singularities, a global finiteness constraint on even-dimensional manifolds, and a nonexistence result for strongly uniformizable structures on degree $ ext{d}\ge 2$ foliations of projective spaces. They illustrate the theory with concrete examples (Hilbert modular foliations and complex geodesic foliations) and derive broad consequences for the geometry of foliations, such as meromorphic first integrals and restrictions in the general type setting. Overall, the work clarifies when foliations can support strongly uniformizable projective structures and links complex projective geometry with foliation theory in a precise, quantitative way.

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 Kähler 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.

Uniformizable foliated projective structures along singular foliations

TL;DR

The paper develops a comprehensive framework for foliated projective structures along singular holomorphic foliations, distinguishing parabolic and non-parabolic singularities and analyzing leafwise uniformizability. It proves that a strongly uniformizable structure on a compact Kähler manifold with a single non-degenerate non-parabolic singularity forces meromorphic complete integrability of the foliation, and it derives a Baum–Bott type index theorem imposing cohomological constraints when all singularities are non-degenerate and parabolic. The authors supply explicit global and local results, including linearization near non-parabolic singularities, a global finiteness constraint on even-dimensional manifolds, and a nonexistence result for strongly uniformizable structures on degree foliations of projective spaces. They illustrate the theory with concrete examples (Hilbert modular foliations and complex geodesic foliations) and derive broad consequences for the geometry of foliations, such as meromorphic first integrals and restrictions in the general type setting. Overall, the work clarifies when foliations can support strongly uniformizable projective structures and links complex projective geometry with foliation theory in a precise, quantitative way.

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 Kähler 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.
Paper Structure (14 sections, 22 theorems, 43 equations)

This paper contains 14 sections, 22 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Curves, foliations, projective structures and uniformizability
  3. Curves
  4. Foliations
  5. Examples
  6. Hilbert modular foliations
  7. The complex geodesic foliation of a complex-hyperbolic manifold
  8. Local normal forms
  9. Foliated projective structures
  10. Foliated affine structures
  11. Global consequences
  12. The index theorem
  13. Foliations on projective spaces
  14. On the analytic subspaces invariant by foliations on Kähler manifolds

Key Result

Theorem 1

Let $\mathcal{F}$ be a foliation by curves on $\mathbb{P}^n$, of degree $d\geq 2$, all of whose singularities are non-degenerate. Then $\mathcal{F}$ does not admit a strongly uniformizable foliated projective structure.

Theorems & Definitions (46)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Proposition 6
  • Proposition 7
  • proof : Sketch of proof
  • Lemma 8
  • proof
  • ...and 36 more