Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the integrability of singular differential forms in two complex variables

Author: A. Meziani
Journal: Trans. Amer. Math. Soc. 321 (1990), 595-620
MSC: Primary 32L05; Secondary 58F18
MathSciNet review: 989582
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this work we study the integrability of a germ at $ 0 \in {{\mathbf{C}}^2}$ of a singular differential form for which the closure of the integral curves are analytic varieties that pass through 0. The focii of this paper are the existence of pure meromorphic integrals, the linearization and the nonexistence of a topological criterion for transcendental integrability.

References [Enhancements On Off] (What's this?)

  • [1] César Camacho and Paulo Sad, Pontos singulares de equações diferenciais analíticas, 16^{𝑜} Colóquio Brasileiro de Matemática. [16th Brazilian Mathematics Colloquium], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1987 (Portuguese). MR 953780
  • [2] D. Cerveau and J.-F. Mattei, Formes intégrables holomorphes singulières, Astérisque, vol. 97, Société Mathématique de France, Paris, 1982 (French). With an English summary. MR 704017
  • [3] Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
  • [4] L. Kaup and B. Kaup, Holomorphic functions of several complex variables, De Gruyter Stud. in Math., vol. 3, De Gruyter, 1983.
  • [5] B. Malgrange, Frobenius avec singularités. I. Codimension un, Inst. Hautes Études Sci. Publ. Math. 46 (1976), 163–173 (French). MR 0508169
  • [6] J.-F. Mattei and R. Moussu, Holonomie et intégrales premières, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 4, 469–523 (French). MR 608290
  • [7] Raghavan Narasimhan, Introduction to the theory of analytic spaces, Lecture Notes in Mathematics, No. 25, Springer-Verlag, Berlin-New York, 1966. MR 0217337
  • [8] H. Poincaré, Sur les propriétés des functions définies par les equations aux différences partielles, Thèse, Paris, 1879.
  • [9] Kyoji Saito, On a generalization of de-Rham lemma, Ann. Inst. Fourier (Grenoble) 26 (1976), no. 2, vii, 165–170 (English, with French summary). MR 0413155
  • [10] A. Seidenberg, Reduction of singularities of the differential equation 𝐴𝑑𝑦=𝐵𝑑𝑥, Amer. J. Math. 90 (1968), 248–269. MR 0220710
  • [11] George Springer, Introduction to Riemann surfaces, Addison-Wesley Publishing Company, Inc., Reading, Mass., 1957. MR 0092855
  • [12] Masakazu Suzuki, Sur les relations d’équivalence ouvertes dans les espaces analytiques, Ann. Sci. École Norm. Sup. (4) 7 (1974), 531–541 (1975) (French). MR 0372238
  • [13] Masakazu Suzuki, Sur les intégrales premières de certains feuilletages analytiques complexes, Fonctions de plusieurs variables complexes, III (Sém. François Norguet, 1975–1977) Lecture Notes in Math., vol. 670, Springer, Berlin, 1978, pp. 53–79, 394 (French). MR 521913
  • [14] Hassler Whitney, Complex analytic varieties, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972. MR 0387634

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32L05, 58F18

Retrieve articles in all journals with MSC: 32L05, 58F18

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society