Skip to main content Accessibility help
×
Home

A Darboux-type theorem for germs of holomorphic one-dimensional foliations

  • CÉSAR CAMACHO (a1) and BRUNO SCÁRDUA (a2)

Abstract

We show that a germ of a holomorphic one-dimensional foliation at a singularity in a space of dimension two admits a holomorphic first integral if and only if there are infinitely many closed leaves and a finite number of separatrices, with each separatrix having linearizable holonomy. Indeed, if there are infinitely many closed leaves and the set of separatrices is finite, then the foliation admits either a holomorphic first integral or a formal simple integrating factor of Darboux type.

Copyright

References

Hide All
[1]Abate, M.. Discrete holomorphic local dynamical systems. Holomorphic Dynamical Systems (Lecture Notes in Mathematics, 1998). Eds. Gentili, G., Guenot, J. and Patrizio, G.. Springer, Berlin, 2010, pp. 155.
[2]Alexander, J. C. and Verjovsky, A.. First integrals for singular holomorphic foliations with leaves of bounded volume. Holomorphic Dynamics (Mexico, 1986) (Lecture Notes in Mathematics, 1345). Springer, Berlin, 1988, pp. 110.
[3]Bracci, F.. Local dynamics of holomorphic diffeomorphisms. Boll. Unione Mat. Ital. 7-B (2004), 609–636.
[4]Biswas, K.. Simultaneous linearization of germs of commuting holomorphic diffeomorphisms. Ergod. Th. & Dynam. Sys. 32 (2012), 12161225.
[5]Camacho, C.. On the local structure of conformal maps and holomorphic vector fields in ℂ2. Journées Singuliéres de Dijon (12–16 Juin 1978) (Astérisque, 59). Société Mathématique de France, Paris, 1978, pp. 8394.
[6]Camacho, C., Kuiper, N. and Palis, J.. The topology of holomorphic flows with singularity. Inst. Hautes Études Sci. Publ. Math. 48 (1978), 538.
[7]Camacho, C. and Lins Neto, A.. Geometric Theory of Foliations. Birkhauser, Boston, 1985.
[8]Camacho, C., Lins Neto, A. and Sad, P.. Foliations with algebraic limit sets. Ann. of Math. (2) 136 (1992), 429446.
[9]Camacho, C. and Sad, P.. Invariant varieties through singularities of holomorphic vector fields. Ann. of Math. (2) 115 (1982), 579595.
[10]Camacho, C. and Sad, P.. Topological classification and bifurcations of holomorphic flows with resonances in ℂ2. Invent. Math. 67 (1982), 447472.
[11]Camacho, C. and Scárdua, B.. Foliations on complex projective spaces with algebraic limit sets. Géométrie Complexe et Systèmes Dynamiques (Orsay, 1995) (Astérisque, 261). Société Mathématique de France, Paris, 2000, pp. 5788.
[12]Camacho, C. and Scárdua, B.. On the existence of stable compact leaves for transversely holomorphic foliations. Topology Appl. 160(13) (2013), 18021808.
[13]Cerveau, D. and Moussu, R.. Groupes d’automorphismes de (ℂ, 0) et équations différentielles y dy + ⋯ = 0. Bull. Soc. Math. France 116 (1988), 459488.
[14]Cremer, H.. Zum Zentrumproblem. Math. Ann. 98 (1927), 151163.
[15]Darboux, G.. Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré. Bull. Sci. Math. 2 (1878), 6096 123–144; 151–200.
[16]Dulac, H.. Solutions d’un système de équations différentiale dans le voisinage des valeus singulières. Bull. Soc. Math. France 40 (1912), 324383.
[17]Gunning, R. C.. Introduction to Holomorphic Functions of Several Variables, Volume I: Function Theory. Wadsworth & Brooks/Cole, Pacific Grove, CA, 1990.
[18]Gunning, R. C.. Introduction to Holomorphic Functions of Several Variables, Volume II: Local Theory. Wadsworth & Brooks/Cole, Monterey, CA, 1990.
[19]Jouanolou, J. P.. Équations de Pfaff algèbriques (Lecture Notes in Mathematics, 708). Springer, Berlin, 1979.
[20]Mattei, J. F. and Moussu, R.. Holonomie et intégrales premières. Ann. Sci. École Norm. Sup. (4) 13 (1980), 469523.
[21]Martinet, J. and Ramis, J.-P.. Classification analytique des équations différentielles non linéaires resonnants du premier ordre. Ann. Sci. École Norm. Sup. (4) 16 (1983), 571621.
[22]Martinet, J. and Ramis, J.-P.. Problème de modules pour des équations différentielles non linéaires du premier ordre. Publ. Math. Inst. Hautes Études Sci. 55 (1982), 63124.
[23]Nakai, I.. Separatrices for nonsolvable dynamics on ℂ,0. Ann. Inst. Fourier (Grenoble) 44 (1994), 569599.
[24]Pérez-Marco, R.. Sur les dynamiques holomorphes non linéarisables et une conjecture de V. I. Arnold. Ann. Sci. École Norm. Sup. 26 (1993), 565644.
[25]Pérez-Marco, R.. Sur une question de Dulac et Fatou. C.R. Acad. Sci. Paris 321 (1995), 10451048.
[26]Pérez-Marco, R.. Fixed points and circle maps. Acta Math. 179 (1997), 243294.
[27]Pérez-Marco, R. and Yoccoz, J.-C.. Germes de feuilletages holomorphes à holonomie prescrite. Complex Analytic Methods in Dynamical Systems 7 (Rio de Janeiro, 1992) (Astérisque, 222). Société Mathématique de France, Paris, 1994, pp. 345371.
[28]Poincaré, H.. Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré. Rend. Circ. Mat. Palermo 5 (1881), 161191.
[29]Raissy, J.. Geometrical methods in the normalization of germs of biholomorphisms. PhD Thesis, Università di Pisa, 2010.
[30]Scárdua, B.. Complex vector fields having orbits with bounded geometry. Tohoku Math. J. (2) 54(3) (2002), 367392.
[31]Scárdua, B.. Integration of complex differential equations. J. Dynam. Control Sys. 5(1) (1999), 150.
[32]Seidenberg, A.. Reduction of singularities of the differential equation A dy = B dx. Amer. J. Math. 90 (1968), 248269.

A Darboux-type theorem for germs of holomorphic one-dimensional foliations

  • CÉSAR CAMACHO (a1) and BRUNO SCÁRDUA (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed