On Singular Holomorphic Foliations with Projective Transverse Structure

  • Bruno ScárduaEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 222)


In this paper we study holomorphic foliations with singularities having a homogeneous transverse structure of projective model (i.e., \(\mathrm{I\!P}SL(2,\mathbb {C})\) model). Our basic situation is the case of a foliation with singularities \(\mathcal F\) on a complex analytic space M of dimension two and the structure exists in the complement of some analytic subset \(S \subset M\) of codimension one. The main case occurs, as we shall see, when the analytic set is invariant by the foliation. We address both, the local and the global cases. This means two basic situations: (i) M is a projective surface (like \(M=\mathbb {C}P (2)\) or \(\overline{\mathbb {C}} \times \overline{\mathbb {C}}\)) and (ii) \(M=(\mathbb {C}^2,0)\) which means the case of germs of foliations at the origin \(0 \in \mathbb {C}^2\), having an isolated singularity at the origin. Our focus is the extension of the structure in a suitable sense. After performing a characterization of the existence of the structure in terms of suitable triples of differential forms, we consider the problem of extension of such structures to the analytic invariant set for germs of foliations and for foliations in complex projective spaces. Basic examples of this situation are given by logarithmic foliations and Riccati foliations. We also study the holonomy of such invariant sets, as a consequence of a strict link between this holonomy and the monodromy of a projective structure. These holonomy groups are proved to be solvable. Our final aim is the classification of such object under some mild conditions on the singularities they exhibit. In this work we perform this classification in the case where the singularities of the foliation are supposed to be non-dicritical and non-degenerate (more precisely, generalized curves). This case, we will see, corresponds to the transversely affine case and therefore to the class of logarithmic foliations. The more general case, which has to do with Riccati foliations, is dealt with by some extension results we prove and evoking results from Loray-Touzet-Vitorio.


Holomorphic foliation Projective transverse structure Holonomy group Riccati foliation 


  1. 1.
    Baum, P., Bott, R.: On the zeros of meromorphic vector fields, Essais en l’honeur de G. De Rham, 29–47 (1970)Google Scholar
  2. 2.
    Baum, P., Bott, R.: Singularities of holomorphic foliations. J. Differ. Geom. 7, 279–342 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berthier, M., Touzet, F.: Sur l’intégration des équations différentielles holomorphes réduites en dimension deux. Bol. Soc. Bras. Mat. 30(3), 247–286 (1999)CrossRefzbMATHGoogle Scholar
  4. 4.
    Blumenthal, R.: Transversely homogeneous foliations. Ann. Inst. Fourier t. 29(4), 143–158 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Camacho, C., Scárdua, B.: Foliations on complex projective spaces with algebraic limit sets. Géométrie complexe et systèmes dynamiques (Orsay, Astérisque 261, 57–88 (2000); Soc. Math. France, Paris (2000)Google Scholar
  6. 6.
    Camacho, C., Lins Neto, A.: Geometric Theory of Foliations. Birkhauser Inc, Boston (1985)CrossRefzbMATHGoogle Scholar
  7. 7.
    Camacho, C., Sad, P.: Invariant varieties through singularities of holomorphic vector fields. Ann. Math. 115, 579–595 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Camacho, C., Scárdua, B.: Holomorphic foliations with Liouvillian first integrals. Ergod. Theory Dyn. Syst. 21, 717–756 (2001)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Camacho, C., Scárdua, B.: Extension theorems for analytic objects associated to foliations. Bulletin des Sciences Mathmatiques (Paris. 1885) 136, 54–71 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Camacho, C., Lins Neto, A., Sad, P.: Topological invariants and equidesingularization for holomorphic vector fields. J. of Differ. Geom. 20(1), 143–174 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Camacho, C., Lins Neto, A., Sad, P.: Foliations with algebraic limit sets. Ann. Math. 136, 429–446 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cerveau, D., Lins Neto, A.: Holomorphic foliations in \(C P^ 2\) having an invariant algebraic curve. Ann. Inst. Fourier, Grenoble 41(4), 883–903 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cerveau, D., Moussu, R.: Groupes d’automorphismes de \(({\mathbb{C}},0)\) et équations différentielles \(ydy+\cdots =0\). Bull. Soc. Math. Fr. 116, 459–488 (1988)CrossRefzbMATHGoogle Scholar
  14. 14.
    Cerveau, D., Sad, P.: Liouvillian integration and Bernoulli foliations. Trans. Am. Math. Soc. 350(8), 3065–3081 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Deligne, P.: Lecture Notes in Mathematics. Équations différentielles à points singuliers réguliers, vol. 163. Springer, Berlin (1970)Google Scholar
  16. 16.
    Dulac, H.: Solutions d’un système de équations différentiale dans le voisinage des valeus singulières. Bull. Soc. Math. Fr. 40, 324–383 (1912)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Godbillon, C.: Progress in Mathematics. Feuilletages. Études géométriques. With a preface by G. Reeb, vol. 98. Birkhäuser, Basel (1991)Google Scholar
  18. 18.
    Gunning, R.C.: Introduction to holomorphic functions of several variables. Function Theory. Wadsworth & Brooks/Cole Advanced Books & Software, vol. I. Pacific Grove, CA (1990)Google Scholar
  19. 19.
    Gunning, R.C.: Introduction to holomorphic functions of several variables. Local Theory. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, vol. II. Monterey, CA (1990)Google Scholar
  20. 20.
    Licanic, S.: Logarithmic foliations on compact algebraic surfaces. Bol. Soc. Brasil. Mat. (N.S.) 31(1), 113–125 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Loray, F., Touzet, F., Vitrio, J.: Representations of quasiprojective groups, flat connections and transversely projective foliations (2015). arXiv:1402.1382 [math.AG]
  22. 22.
    Martinet, J., Ramis, J.-P.: Problème de modules pour des équations différentielles non lineaires du premier ordre. Publ. Math. Inst. Hautes Études Scientifiques 55, 63–124 (1982)Google Scholar
  23. 23.
    Martinet, J., Ramis, J.-P.: Classification analytique des équations différentielles nonlineaires resonnants du premier ordre. Ann. Sc. Ec. Norm. Sup. 16, 571–621 (1983)Google Scholar
  24. 24.
    Mattei, J.F., Moussu, R.: Holonomie et intégrales premières. Ann. Sci. École Norm. Sup. 13(4), 469–523 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Nakai, I.: Separatrices for nonsolvable dynamics on \({\mathbb{C}},0\). Ann. Inst. Fourier (Grenoble) 44, 569–599 (1994)Google Scholar
  26. 26.
    Nicolau, M., Paul, E.: A geometric proof of a Galois differential theory theorem. In: Mozo Fernandez, J. (ed.) Ecuaciones diferenciales y singularidades, Universidad de Valladolid, pp. 277–290 (1997)Google Scholar
  27. 27.
    Scárdua, B.: Transversely affine and transversely projective foliations. Ann. Sc. École Norm. Sup. 4\(^{e}\) sériet 30, 169–204 (1997)Google Scholar
  28. 28.
    Scárdua, B.: Integration of complex differential equations. J. Dyn. Control Syst. 1(5), 1–50 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Scárdua, B.: Complex vector fields having orbits with bounded geometry. Tohoku Math. J. (2) 54(3), 367–392 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Scárdua, B.: Differential algebra and Liouvillian first integrals of foliations. J. Pure Appl. Algebra 215(5), 764–788 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Seidenberg, A.: Reduction of singularities of the differential equation \(Ady=Bdx\). Am. J. Math. 90, 248–269 (1968)Google Scholar
  32. 32.
    Seke, B.: Sur les structures transversalement affines des feuilletages de codimension un. Ann. Inst. Fourier, Grenoble 30(1), 1–29 (1980)Google Scholar
  33. 33.
    Singer, M.F.: Liouvillian first integrals of differential equations. Trans. Am. Math. Soc. 333(2), 673–688 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Siu, Y.: Techniques of Extension of Analytic Objects. Marcel Dekker, N.Y. (1974)zbMATHGoogle Scholar
  35. 35.
    Touzet, F.: Sur les feuilletages holomorphes transversalement projectifs. Ann. Inst. Fourier 53(3), 815–846 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Instituto de MatemáticaUniversidade Federal do Rio de JaneiroRio de Janeiro-RJBrazil

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