Advertisement

Some Open Problems on Holomorphic Foliation Theory

  • Tien-Cuong DinhEmail author
  • Nessim Sibony
Article
  • 11 Downloads

Abstract

We present a list of open questions in the theory of holomorphic foliations, possibly with singularities. Some problems have been around for a while, others are very accessible.

Keywords

Singular foliation ddc-closed current Nevanlinna current Unique ergodicity 

Mathematics Subject Classification (2010)

37F75 37A 32U40 30F15 57R30 

Notes

Funding Information

The first author was supported by the NUS grants C-146-000-047-001 and AcRF Tier 1 R-146-000-248-114.

References

  1. 1.
    Bernal-González, L.: Universal entire functions for affine endomorphisms of \(\mathbb {C}^{N}\). (English summary). J. Math. Anal. Appl. 305(2), 690–697 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berndtsson, B., Sibony, N: The \(\overline \partial \)-equation on a positive current. Invent. Math. 147(2), 371–428 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Birkhoff, G.D.: Démonstration d’un théorème élémentaire sur les fonctions entières. C.R. Acad. Sci. Paris 189, 473–475 (1929)zbMATHGoogle Scholar
  4. 4.
    Bogomolov, F.: Complex manifolds and algebraic foliations. Publ. RIMS, vol. 1084. Kyoto. Unpublished (1996)Google Scholar
  5. 5.
    Brunella, M.: Inexistence of invariant measures for generic rational differential equations in the complex domain. Bol. Soc. Mat. Mexicana (3) 12(1), 43–49 (2006)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Buhovsky, L., Glucksam, A., Logunov, A., Sodin, M.: Translation-invariant probability measures on entire functions. Preprint (2017), 38pp. arXiv:1703.08101
  7. 7.
    Burns, D., Sibony, N.: Limit currents and value distribution of holomorphic maps. Ann. Inst. Fourier (Grenoble) 62(1), 145–176 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Camacho, C., Lins Neto, A., Sad, P.: Minimal sets of foliations on complex projective spaces. Inst. Hautes Études Sci. Publ. Math. 68, 187–203 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cerveau, D., Lins Neto, A.: Irreducible components of the space of foliations of degree two in \(\mathbb {P}^{n}\), n ≥ 3. Ann. Math. 143, 577–612 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Demailly, J. -P., Gaussier, H.: Algebraic embeddings of smooth almost complex structures. J. Eur. Math. Soc. (JEMS) 19(11), 3391–3419 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dinh, T.-C., Nguyen, V.-A., Sibony, N.: Heat equation and ergodic theorems for Riemann surface laminations. Math. Ann. 354(1), 331–376 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dinh, T.-C., Nguyen, V.-A., Sibony, N.: Entropy for Hyperbolic Riemann Surface Laminations I. Frontiers in Complex Dynamics, Princeton Math Ser., vol. 51, pp 569–592. Princeton Univ. Press, Princeton (2014)Google Scholar
  13. 13.
    Dinh, T.-C., Nguyen, V.-A., Sibony, N.: Unique Ergodicity for foliations on compact Kähler surfaces. Preprint (2018)Google Scholar
  14. 14.
    Dinh, T.-C., Sibony, N.: Unique ergodicity for foliations in \(\mathbb {P}^{2}\) with an invariant curve. Invent. Math. 211(1), 1–38 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fornæss, J.-E., Sibony, N.: Harmonic currents of finite energy and laminations. Geom. Funct. Anal. 15(5), 962–1003 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fornæss, J.-E., Sibony, N.: Riemann surface laminations with singularities. J. Geom. Anal. 18(2), 400–442 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fornæss, J.-E., Sibony, N., Wold, E.-F.: Examples of minimal laminations and associated currents. Math. Z. 269(1), 495–520 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Furstenberg, H.: Strict ergodicity and transformation of the torus. Am. J. Math. 83, 573–601 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Garnett, L.: Foliations, the ergodic theorem and Brownian motion. J. Funct. Anal. 51, 285–311 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ilyashenko, Yu: Some open problems in real and complex dynamical systems. Nonlinearity 21(7), T101–T107 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Jouanolou, J.P.: Feuilles compactes des feuilletages algébriques. (French). Math. Ann. 241(1), 69–72 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lins Neto, A.: Uniformization and the Poincaré metric on the leaves of a foliation by curves. Bol. Soc. Brasil. Mat. (N.S.) 31(3), 351–366 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lins Neto, A., Soares, M.: Algebraic solutions of one dimensional foliations. J. Diff. Geom. 43, 652–673 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nguyên, V.-A.: Oseledec multiplicative ergodic theorem for laminations. Mem. Am. Math. Soc. 246, 1164 (2017)MathSciNetGoogle Scholar
  25. 25.
    Nguyên, V.-A.: Singular holomorphic foliations by curves I: integrability of holonomy cocycle in dimension 2. Invent. Math. 212(2), 531–618 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Nguyên, V.-A.: Ergodic theory for Riemann surface laminations: a survey. In: Byun, J., Cho, H., Kim, S., Lee, K.H., Park, J.D. (eds.) Geometric Complex Analysis. Springer Proceedings in Mathematics & Statistics, vol. 246, pp 291–327. Springer, Singapore (2018)Google Scholar
  27. 27.
    Nishino, T.: Nouvelles recherches sur les fonctions entières de plusieurs variables complexes I-V. J. Math. Kyoto Univ. 8, 9, 10, 13, 15 (1968-1975)Google Scholar
  28. 28.
    Shcherbakov, A.A., Rosales-González, E., Ortiz-Bobadilla, L.: Countable set of limit cycles for the equation d w/d z = P n(z,w)/Q n(z,w). J. Dynam. Control Syst. 4(4), 539–581 (1998)CrossRefzbMATHGoogle Scholar
  29. 29.
    Sibony, N.: Pfaff systems, currents and hulls. Math. Z. 285(3–4), 1107–1123 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sibony, N., Wold, E.F.: Topology and complex structure of leaves of foliations by Riemann surfaces. J. Geom. Anal (2017)Google Scholar
  31. 31.
    Siu, Y.T.: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math. 27, 53–156 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Weiss, B.: Measurable entire functions. The heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday of T. J. Rivlin. Ann. Numer. Math. 4(1–4), 599–605 (1997)Google Scholar
  33. 33.
    Yamaguchi, H.: Parabolicité d’une fonction entière. J. Math. Kyoto Univ. 16, 71–92 (1976)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational University of SingaporeSingaporeSingapore
  2. 2.Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRSUniversité Paris-SaclayOrsayFrance
  3. 3.Korea Institute for Advanced StudySeoulSouth Korea

Personalised recommendations