Geodesic
On the resolution of singularities of one-dimensional foliations on three-manifolds
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 76 (2021) no. 2, pp. 291-355 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper is devoted to the resolution of singularities of holomorphic vector fields and one-dimensional holomorphic foliations in dimension three, and it has two main objectives. First, within the general framework of one-dimensional foliations, we build upon and essentially complete the work of Cano, Roche, and Spivakovsky (2014). As a consequence, we obtain a general resolution theorem comparable to the resolution theorem of McQuillan–Panazzolo (2013) but proved by means of rather different methods. The other objective of this paper is to consider a special class of singularities of foliations containing, in particular, all the singularities of complete holomorphic vector fields on complex manifolds of dimension three. We then prove that a much sharper resolution theorem holds for this class of holomorphic foliations. This second result was the initial motivation for this paper. It relies on combining earlier resolution theorems for (general) foliations with some classical material on asymptotic expansions for solutions of differential equations. Bibliography: 34 titles.
Keywords: complex manifolds of dimension three, complete holomorphic vector fields, resolution of singularities, persistently nilpotent singularities, asymptotic expansions for solutions of ordinary differential equations, formal curve
Mots-clés : valuation. 
@article{RM_2021_76_2_a2,
     author = {J. C. Rebelo and H. Reis},
     title = {On the resolution of singularities of one-dimensional foliations on three-manifolds},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {291--355},
     year = {2021},
     volume = {76},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2021_76_2_a2/}
}
TY  - JOUR
AU  - J. C. Rebelo
AU  - H. Reis
TI  - On the resolution of singularities of one-dimensional foliations on three-manifolds
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2021
SP  - 291
EP  - 355
VL  - 76
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/RM_2021_76_2_a2/
LA  - en
ID  - RM_2021_76_2_a2
ER  - 
%0 Journal Article
%A J. C. Rebelo
%A H. Reis
%T On the resolution of singularities of one-dimensional foliations on three-manifolds
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2021
%P 291-355
%V 76
%N 2
%U http://geodesic.mathdoc.fr/item/RM_2021_76_2_a2/
%G en
%F RM_2021_76_2_a2
J. C. Rebelo; H. Reis. On the resolution of singularities of one-dimensional foliations on three-manifolds. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 76 (2021) no. 2, pp. 291-355. http://geodesic.mathdoc.fr/item/RM_2021_76_2_a2/

[1] M. Abate, “The residual index and the dynamics of holomorphic maps tangent to the identity”, Duke Math. J., 107:1 (2001), 173–207 | DOI | MR | Zbl

[2] V. I. Arnol'd, Yu. S. Il'yashenko, “Ordinary differential equations”, Dynamical systems I, Encyclopaedia Math. Sci., 1, Springer-Verlag, Berlin, 1988, 1–148 | MR | MR | Zbl | Zbl

[3] M. Atiyah, N. Hitchin, The geometry and dynamics of magnetic monopoles, M. B. Porter Lectures, 11, Princeton Univ. Press, Princeton, NJ, 1988, viii+134 pp. | DOI | MR | MR | Zbl | Zbl

[4] F. E. Brochero Martínez, F. Cano, L. López-Hernanz, “Parabolic curves for diffeomorphisms in $\mathbb{C}^2$”, Publ. Mat., 52:1 (2008), 189–194 | DOI | MR | Zbl

[5] C. Camacho, A. Lins Neto, P. Sad, “Topological invariants and equidesingularization for holomorphic vector fields”, J. Differential Geom., 20:1 (1984), 143–174 | DOI | MR | Zbl

[6] C. Camacho, P. Sad, “Invariant varieties through singularities of holomorphic vector fields”, Ann. of Math. (2), 115:3 (1982), 579–595 | DOI | MR | Zbl

[7] F. Cano, “Reduction of the singularities of codimension one singular foliations in dimension three”, Ann. of Math. (2), 160:3 (2004), 907–1011 | DOI | MR | Zbl

[8] F. Cano, C. Roche, “Vector fields tangent to foliations and blow-ups”, J. Singul., 9 (2014), 43–49 | DOI | MR | Zbl

[9] F. Cano, C. Roche, M. Spivakovsky, “Reduction of singularities of three-dimensional line foliations”, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 108:1 (2014), 221–258 | DOI | MR | Zbl

[10] F. Cano Torres, Desingularization strategies for three-dimensional vector fields, Lecture Notes in Math., 1259, Springer-Verlag, Berlin, 1987, x+194 pp. | DOI | MR | Zbl

[11] R. Conte (ed.), The Painlevé property. One century later, CRM Ser. Math. Phys., Springer-Verlag, New York, 1999, xxvi+810 pp. | DOI | MR | Zbl

[12] G. Dloussky, K. Oeljeklaus, M. Toma, “Surfaces de la classe $\mathrm{VII}_0$ admettant un champ de vecteurs”, Comment. Math. Helv., 75:2 (2000), 255–270 | DOI | MR | Zbl

[13] P. M. Elizarov, Yu. S. Il'yashenko, “Remarks on the orbital analytic classification of germs of vector fields”, Math. USSR-Sb., 49:1 (1984), 111–124 | DOI | MR | Zbl

[14] A. È. Eremenko, “Meromorphic solutions of algebraic differential equations”, Russian Math. Surveys, 37:4 (1982), 61–95 | DOI | MR | Zbl

[15] É. Ghys, J. C. Rebelo, “Singularités des flots holomorphes. II”, Ann. Inst. Fourier (Grenoble), 47:4 (1997), 1117–1174 | DOI | MR | Zbl

[16] A. Guillot, “Sur les équations d'Halphen et les actions de $\operatorname{SL}_2(\mathbf C)$”, Publ. Math. Inst. Hautes Études Sci., 105:1 (2007), 221–294 | DOI | MR | Zbl

[17] A. Guillot, “The geometry of Chazy's homogeneous third-order differential equations”, Funkcial. Ekvac., 55:1 (2012), 67–87 | DOI | MR | Zbl

[18] A. Guillot, J. C. Rebelo, “Semicomplete meromorphic vector fields on complex surfaces”, J. Reine Angew. Math., 2012:667 (2012), 27–65 | DOI | MR | Zbl

[19] M. Hakim, Transformations tangent to the identity. Stable pieces of manifolds, Prépublication Orsay 97-30, Univ. de Paris-Sud, Orsay, 1997, 36 pp. https://www.imo.universite-paris-saclay.fr/~biblio/pub/1997/abs/ppo1997_30.html

[20] M. Hakim, “Analytic transformations of $(\mathbf{C}^p,0)$ tangent to the identity”, Duke Math. J., 92:2 (1998), 403–428 | DOI | MR | Zbl

[21] Yu. Ilyashenko, S. Yakovenko, Lectures on analytic differential equations, Grad. Stud. Math., 86, Amer. Math. Soc., Providence, RI, 2008, xiv+625 pp. | DOI | MR | Zbl

[22] E. L. Ince, Ordinary differential equations, Reprint of the 1st ed., Dover Publications, New York, 1956, viii+558 pp. | MR | Zbl

[23] J. Malmquist, “Sur l'étude analytique des solutions d'un système d'équations différentielles dans le voisinage d'un point singulier d'indétermination. I”, Acta Math., 73 (1941), 87–129 | DOI | MR | Zbl

[24] F. Martin, E. Royer, “Formes modulaires et périodes”, Formes modulaires et transcendance, Sémin. Congr., 12, Soc. Math. France, Paris, 2005, 1–117 | MR | Zbl

[25] J.-F. Mattei, R. Moussu, “Holonomie et intégrales premières”, Ann. Sci. École Norm. Sup. (4), 13:4 (1980), 469–523 | DOI | MR | Zbl

[26] M. McQuillan, D. Panazzolo, “Almost étale resolution of foliations”, J. Differential Geometry, 95:2 (2013), 279–319 ; preprint IHES/M/09/51, IHES, Bures-sur-Yvette, 2009, 31 pp. http://preprints.ihes.fr/2009/M/M-09-51.pdf | DOI | MR | Zbl

[27] D. Panazzolo, “Resolution of singularities of real-analytic vector fields in dimension three”, Acta Math., 197:2 (2006), 167–289 | DOI | MR | Zbl

[28] O. Piltant, “An axiomatic version of Zariski's patching theorem”, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 107:1 (2013), 91–121 ; preprint, Univ. de Valladolid, 2008 | DOI | MR | Zbl

[29] J.-P. Ramis, Y. Sibuya, “Hukuhara domains and fundamental existence and uniqueness theorems for asymptotic solutions of Gevrey type”, Asymptotic Anal., 2:1 (1989), 39–94 | DOI | MR | Zbl

[30] J. C. Rebelo, “Singularités des flots holomorphes”, Ann. Inst. Fourier (Grenoble), 46:2 (1996), 411–428 | DOI | MR | Zbl

[31] H. Reis, “Equivalence and semi-completude of foliations”, Nonlinear Anal., 64:8 (2006), 1654–1665 | DOI | MR | Zbl

[32] H. Reis, “Semi-complete vector fields of saddle-node type in $\mathbb C^n$”, Trans. Amer. Math. Soc., 360:12 (2008), 6611–6630 | DOI | MR | Zbl

[33] A. Seidenberg, “Reduction of singularities of the differential equation $A\,dy=B\,dx$”, Amer. J. Math., 90 (1968), 248–269 | DOI | MR | Zbl

[34] B. J. Weickert, “Attracting basins for automorphisms of $\mathbf{C}^2$”, Invent. Math., 132:3 (1998), 581–605 | DOI | MR | Zbl