## Journal of the Mathematical Society of Japan

### Local polar invariants and the Poincaré problem in the dicritical case

#### Abstract

We develop a study on local polar invariants of planar complex analytic foliations at $(\mathbb{C}^{2},0)$, which leads to the characterization of second type foliations and of generalized curve foliations, as well as to a description of the $GSV$-index. We apply it to the Poincaré problem for foliations on the complex projective plane $\mathbb{P}^{2}_{\mathbb{C}}$, establishing, in the dicritical case, conditions for the existence of a bound for the degree of an invariant algebraic curve $S$ in terms of the degree of the foliation $\mathcal{F}$. We characterize the existence of a solution for the Poincaré problem in terms of the structure of the set of local separatrices of $\mathcal{F}$ over the curve $S$. Our method, in particular, recovers the known solution for the non-dicritical case, $\deg(S) \leq \deg (\mathcal{F}) + 2$.

#### Note

This work was supported by MATH-AmSud Project CNRS/CAPES/Concytec. The first author was supported by a grant ANR-13-JS01-0002-0. The second author was supported by Pronex/FAPERJ and Universal/CNPq.

#### Article information

Source
J. Math. Soc. Japan, Volume 70, Number 4 (2018), 1419-1451.

Dates
Received: 11 October 2016
Revised: 1 April 2017
First available in Project Euclid: 1 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1538380983

Digital Object Identifier
doi:10.2969/jmsj/76227622

Mathematical Reviews number (MathSciNet)
MR3868212

Zentralblatt MATH identifier
07009707

Subjects
Primary: 32S65: Singularities of holomorphic vector fields and foliations

#### Citation

GENZMER, Yohann; MOL, Rogério. Local polar invariants and the Poincaré problem in the dicritical case. J. Math. Soc. Japan 70 (2018), no. 4, 1419--1451. doi:10.2969/jmsj/76227622. https://projecteuclid.org/euclid.jmsj/1538380983

#### References

• M. Berthier, R. Meziani and P. Sad, On the classification of nilpotent singularities, Bull. Sci. Math., 123 (1999), 351–370.
• E. Brieskorn and H. Kn örrer, Plane algebraic curves, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1986, Translated from the German original by John Stillwell, reprint of the 1986 edition, 2012.
• M. Brunella, Feuilletages holomorphes sur les surfaces complexes compactes, Ann. Sci. École Norm. Sup. (4), 30 (1997), 569–594.
• M. Brunella, Some remarks on indices of holomorphic vector fields, Publ. Mat., 41 (1997), 527–544.
• C. Camacho, A. Lins Neto and P. Sad, Topological invariants and equidesingularization for holomorphic vector fields, J. Differential Geom., 20 (1984), 143–174.
• C. Camacho and P. Sad, Invariant varieties through singularities of holomorphic vector fields, Ann. of Math. (2), 115 (1982), 579–595.
• A. Campillo and M. Carnicer, Proximity inequalities and bounds for the degree of invariant curves by foliations of $\mathbf{P}^{\mathbf{2}}_{\mathbf{C}}$, Trans. Amer. Math. Soc., 349 (1997), 2211–2228.
• A. Campillo and J. Olivares, Assigned base conditions and geometry of foliations on the projective plane, In: Singularities–-Sapporo 1998, Adv. Stud. Pure Math., 29, Math. Soc. Japan, Kinokuniya, Tokyo, 2000, 97–113.
• F. Cano, N. Corral and R. Mol, Local polar invariants for plane singular foliations, to appear in Expo. Math., 2018.
• M. Carnicer, The Poincaré problem in the nondicritical case, Ann. of Math. (2), 140 (1994), 289–294.
• V. Cavalier and D. Lehmann, Localisation des résidus de Baum–Bott, courbes généralisées et $K$-théorie, I, Feuilletages dans $\mathbb{C}^2$, Comment. Math. Helv., 76 (2001), 665–683.
• V. Cavalier and D. Lehmann, On the Poincaré inequality for one-dimensional foliations, Compos. Math., 142 (2006), 529–540.
• D. Cerveau and A. Lins Neto, Holomorphic foliations in $\mathbf{C}\mathrm{P}(2)$ having an invariant algebraic curve, Ann. Inst. Fourier (Grenoble), 41 (1991), 883–903.
• D. Cerveau and R. Moussu, Groupes d'automorphismes de $(\mathbf{C},0)$ et équations différentielles $ydy+\cdots=0$, Bull. Soc. Math. France, 116 (1988), 459–488.
• N. Corral and P. Fernández, Isolated invariant curves of a foliation, Proc. Amer. Math. Soc., 134 (2006), 1125–1132.
• E. Esteves and S. Kleiman, Bounds on leaves of one-dimensional foliations, Bull. Braz. Math. Soc. (N.S.), 34 (2003), 145–169.
• C. Galindo and F. Monserrat, The Poincaré problem, algebraic integrability and dicritical divisors, J. Differential Equations, 256 (2014), 3614–3633.
• Y. Genzmer, Rigidity for dicritical germ of foliation in $\mathbb{C}^2$, Int. Math. Res. Not. IMRN, (19):Art. ID rnm072, 2007, 14 pp.
• X. Gómez-Mont, J. Seade and A. Verjovsky, The index of a holomorphic flow with an isolated singularity, Math. Ann., 291 (1991), 737–751.
• A. Lins Neto, Some examples for the Poincaré and Painlevé problems, Ann. Sci. École Norm. Sup. (4), 35 (2002), 231–266.
• J.-F. Mattei and R. Moussu, Holonomie et intégrales premières, Ann. Sci. École Norm. Sup. (4), 13 (1980), 469–523.
• J.-F. Mattei and E. Salem, Modules formels locaux de feuilletages holomorphes., 2004.
• M. McQuillan, An introduction to non-commutative Mori theory, In: European Congress of Mathematics, Vol. II (Barcelona, 2000), Progr. Math., 202, Birkhäuser, Basel, 2001, 47–53.
• R. Meziani, Classification analytique d'équations différentielles $y\,dy+\cdots=0$ et espace de modules, Bol. Soc. Brasil. Mat. (N.S.), 27 (1996), 23–53.
• R. Meziani and P. Sad, Singularités nilpotentes et intégrales premières, Publ. Mat., 51 (2007), 143–161.
• R. Mol, Meromorphic first integrals: some extension results, Tohoku Math. J. (2), 54 (2002), 85–104.
• H. Poncaré, Sur l'intégration algébrique des équations différentielles du premier ordre et du premier degré, Rend. Circ. Mat. Palermo, 5 (1891), 161–191.
• L. Puchuri, Degree of the first integral of a pencil in $\mathbb{P}^2$ defined by Lins Neto, Publ. Mat., 57 (2013), 123–137.
• A. Seidenberg, Reduction of singularities of the differential equation $A\,dy=B\,dx$, Amer. J. Math., 90 (1968), 248–269.