Radial Foliations in Dimension Three

• Felipe Cano ^[1] ; Beatriz Molina-Samper ^[1]
  1. [1] Universidad de Valladolid

     Universidad de Valladolid

     Valladolid, España

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 2, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • Radial germs of holomorphic foliations in dimension two have a characteristic property: they are the only singular foliations whose reduction of singularities has no singular points. We also know that they are desingularized by a single dicritical blowing-up. Let us say that a foliated space ((C3, 0), E, F) is almost radial when it has a reduction of singularities without singular points; it will be “radial” under a certain additional condition on the morphism of reduction of singularities. We show that the radial condition corresponds to the “open book” situation. We end the paper with a discussion on the general almost radial case.

• Referencias bibliográficas
  • 1. Brunella, M.: Birational Geometry of Foliations Publicações Matemáticas. IMPA, (2003). Reedited as Volumen 1 de IMPA Monographs. Springer,...
  • 2. Camacho, C., Sad, P.: Invariant varieties through singularities of vector fields. Ann. of Math. 115, 579–595 (1982)
  • 3. Cano, F.: Reduction of singularities of codimension one foliations in dimension three. Ann. of Math. (2) 160(3), 907–1011 (2004)
  • 4. Cano, F., Cerveau, D.: Desingularization of non-dicritical holomorphic foliations and existence of separatrices. Acta Math. 169, 1–103...
  • 5. Cano, F., Cerveau, D., Déserti, J.: Théorie élémentaire des feuilletages holomorphes singuliers. Belin Education Editions. ISBN-13: 978–2701174846...
  • 6. Cano, F., Mattei, J.F.: Hypersurfaces intégrales des feuilletages holomorphes. Annales de l’institut Fourier 42(1–2), 49–72 (1992)
  • 7. Cano, F., Molina-Samper, B.: Invariant Surfaces for Toric Type Foliations in Dimension Three. Pub. Matemàtiques 65, 291–307 (2021)
  • 8. Cano, F., Molina-Samper, B.: Invariant Hypersurfaces of Codimension one Singular Foliations. Handbook Singularities VI, Springer (2024)
  • 9. Cano, F., Ravara-Vago, M.: Local Brunella’s alternative II. Partial separatrices. Int. Math. Res. Not. IMRN, 23, pp. 12840–12876 (2015)
  • 10. Cano, F., Ravara-Vago, M., Soares, M.: Local Brunella’s alternative I. RICH foliations. Int. Math. Res. Notices, 2015(9), 2525–2575 (2015)
  • 11. Cerveau, D.: Pinceaux linéaires de feuilletages sur CP(3) et conjecture de Brunella. Publications Mathématiques de l’IHES 46, 441–451...
  • 12. Galindo, C., Montserrat, F., Olivares, J.: Foliations with Isolated Singularities on Hirzebruch SurfacesForum Math. (2021); aop. De Gruyter....
  • 13. Jouanolou, J.P.: Équations de Pfaff algébriques. Lecture Notes in Mathematics, vol. 708. SpringerVerlag ISBN-10: 3–540-09239-0 (1979)
  • 14. Mattei, J.F.: Modules de feuillages holomorphes singuliers: I équisingularité. Invent. Math. 103(2), 297–326 (1991)
  • 15. Mattei, J.F., Moussu, R.: Holonomie et intégrales premières Annales scientifiques de l’É.N.S. 4e série, tome 13(4), 469–523 (1980)