Truncated Local Uniformization of Formal Integrable Differential Forms

  1. Fernández-Duque, M. 1
  2. Cano, F. 1
  1. 1 Universidad de Valladolid
    info

    Universidad de Valladolid

    Valladolid, España

    ROR https://ror.org/01fvbaw18

Journal:
Qualitative theory of dynamical systems

ISSN: 1575-5460

Year of publication: 2022

Volume: 21

Issue: 1

Type: Article

DIALNET GOOGLE SCHOLAR lock_openOpen access editor

More publications in: Qualitative theory of dynamical systems

Abstract

We prove the existence of Local Uniformization for rational codimension one foliations along rational rank one valuations, in any ambient dimension. This result is consequence of the Truncated Local Uniformization of integrable formal differential 1-forms, that we also state and prove in the paper. Thanks to the truncated approach, we perform a classical inductive procedure, based both in the control of the Newton Polygon and in the possibility of avoiding accumulations of values, given by the existence of suitable Tschirnhausen transformations.

€ View funding

Funding information

The authors are grateful with O. Piltant, M. Spivakovsky and B. Teissier for many fruitful conversations on the subject. We specially thank the careful referees’ work and helpful comments. This work has been supported by the Spanish Research Project MTM2016-77642-C2-1-P (Grant No. PID2019-105621GB-I00 Spain) and the second author has been supported by the Post-Doctoral Program DGAPA at IM-UNAM (Mexico).

Funders

Bibliographic References

  • 1. Abramovich, D.: Resolution of singularities of complex algebraic varieties and their families. arXiv:1711.09976. To appear in Proc. Int. Cong. of Math. Rio de Janeiro (2018)
  • 2. Belotto, A.: Local monomialization of a system of first integrals of Darboux type. arXiv:1411.5333. To appear in Revista Matemática Iberoamericana 34 (2018)
  • 3. Belotto, A.: Global resolution of singularities subordinated to a 1-dimensional foliation. J. Algebra 447, 397–423 (2016)
  • 4. Belotto, A.: Local resolution of ideals subordinated to a foliation. Rev. Real Acad. Ciencias Exactas Fís. Nat. Ser. A Mat. 110(2), 841–862 (2016)
  • 5. Cano, F.: Desingularization Strategies of Three-Dimensional Vector Fields. Lecture Notes in Mathematics 1259, Springer, Berlin (1987)
  • 6. Cano, F.: Reduction of the singularities of codimension one singular foliations in dimension three. Ann. Math. (2) 160(3), 907–1011 (2004)
  • 7. Cano, F., Cerveau, D.: Desingularization of non-dicritical holomorphic foliations and existence of separatrices. Acta Math. 169(1–2), 1–103 (1992)
  • 8. Cano, F., Moussu, R., Sanz, F.: Oscillation, spiralement, tourbillonnement. Comment. Math. Helv. 75(2), 284–318 (2000)
  • 9. Cano, F., Moussu, R., Sanz, F.: Nonoscillating projections for trajectories of vector fields. J. Dyn. Control Syst. 13(2), 173–176 (2007)
  • 10. Cano, F., Moussu, R., Rolin, J.-P.: Non-oscillating integral curves and valuations. J. Reine Angew. Math. 2005(582), 107–141 (2005)
  • 11. Cano, F., Roche, C., Spivakovsky, M.: Reduction of singularities of three-dimensional line foliations. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 108(1), 221–258 (2014)
  • 12. Cerveau, D., Mattei, J.-F.: Formes holomorphes intégrables singulières. Astérisque 97 (1982)
  • 13. Cossart, V., Piltant, O.: Resolution of singularities of threefolds in positive characteristic I. J. Algebra 320(7), 1051–1082 (2008)
  • 14. Cossart, V., Piltant, O.: Resolution of singularities of threefolds in positive characteristic II. J. Algebra 321(7), 1836–1976 (2009)
  • 15. Cutkosky, D.: Local monomialization and factorization of morphisms. Astérisque 260 (1999)
  • 16. Cutkosky, D., El Hitti, S.: Formal prime ideals of infinite value and their algebraic resolution. Ann. Fac. Sci. Toulouse Math. 19, 635–649 (2010)
  • 17. Decaup, J., Mahboub, W., Spivakovsky, M.: Abstract key polynomials and comparison theorems with the key polynomials of Mac Lane–Vaquie. arxiv:1611.06392v1
  • 18. Eisenbud, D.: Commutative Algebra: With a View Toward Algebraic Geometry, p. 150. Springer, Berlin (2013)
  • 19. Fernández-Duque, M.: Elimination of resonances in codimension one foliations. Publ. Mat. 59(1), 75–97 (2015)
  • 20. Herrera Govantes, F.-J., Olalla Acosta, M.A., Spivakovsky, M., Teissier, B.: Extending a valuation centered in a local domain to the formal completion. Proc. LMS 105(3), 571–621 (2012)
  • 21. Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. I & II. Ann. Math. (2) 79, 109–203, 205–326 (1964)
  • 22. Mattei, J.-F.: Modules de feuilletages holomorphes singuliers: I équisingularité. Invent. Math. 103(1), 297–325 (1991)
  • 23. Mattei, J.-F., Moussu, R.: Holonomie et intégrales premières. Ann. Sci. École Norm. Sup. (4) 13(4), 469–523 (1980)
  • 24. McQuillan, M., Pannazolo, D.: Almost étale resolution of foliations. J. Differ. Geom. 95(2), 279–319 (2013)
  • 25. Novacoski, J., Spivakovsky, M.: Reduction of local uniformization to the rank one case. Valuation Theory in Interaction, pp. 404–431. European Mathematical Society, EMS Series of Congress Reports (2014)
  • 26. Panazzolo, D.: Resolution of singularities of real-analytic vector fields in dimension three. Acta Math. 197(2), 167–289 (2006)
  • 27. Saito, K.: On a generalization of de-Rham lemma. Ann. Inst. Fourier (Grenoble) 26(2), 165–170 (1976)
  • 28. Seidenberg, A.: Reduction of singularities of the differential equation Ady = Bdx. Am. J. Math. 90, 248–269 (1968)
  • 29. Spivakovsky, M.A.: Solution to Hironaka’s Polyhedra Game Arithmetic and Geometry, II. Papers Dedicated to I. R. Shafarevich on the Occasion of His Sixtieth Birthday, M. Artin and J. Tate, editors. Birkhäuser, pp. 419–432 (1983)
  • 30. Teissier, B.: Overweight deformations of affine toric varieties and local uniformization“Valuation theory in interaction”. European Math. Soc. Publishing House, Congress Reports Series, pp. 474–565 (2014)
  • 31. Zariski, O.: Local uniformization on algebraic varieties. Ann. Math. (2) 41, 852–896 (1940)
Data source: Dialnet