Uniformizable foliated projective structures along singular foliations
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911366731792384 |
|---|---|
| author | Deroin, Bertrand Guillot, Adolfo |
| author_facet | Deroin, Bertrand Guillot, Adolfo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04626 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniformizable foliated projective structures along singular foliations Deroin, Bertrand Guillot, Adolfo Complex Variables 57M50, 32M25, 34M35, 34M45, 32S65 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. |
| title | Uniformizable foliated projective structures along singular foliations |
| topic | Complex Variables 57M50, 32M25, 34M35, 34M45, 32S65 |
| url | https://arxiv.org/abs/2501.04626 |