Unfoldings of foliations with multiform first integrals

Tatsuo Suwa

Annales de l'institut Fourier (1983)

  • Volume: 33, Issue: 3, page 99-112
  • ISSN: 0373-0956

Abstract

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Let F = ( ω ) be a codim 1 local foliation generated by a germ ω of the form ω = f 1 ... f p ∑ i = 1 p λ i d f i f i for some complex numbers λ i and germs f i of holomorphic functions at the origin in C n . We determine, under some conditions, the set of equivalence classes of first order unfoldings and construct explicitly a universal unfolding of F . Special cases of this include foliations with holomorphic or meromorphic first integrals. We also show that the unfolding theory for F is equivalent to the unfolding theory for the multiform function f = f 1 λ 1 ... f p λ p . Thus a universal unfolding of f is also given explicity.

How to cite

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Suwa, Tatsuo. "Unfoldings of foliations with multiform first integrals." Annales de l'institut Fourier 33.3 (1983): 99-112. <http://eudml.org/doc/74602>.

@article{Suwa1983,
abstract = {Let $F=(\omega )$ be a codim 1 local foliation generated by a germ $\omega $ of the form $\omega =f_1\ldots f_p\sum ^p_\{i=1\}\lambda _i \{df_i\over f_i\}$ for some complex numbers $\lambda _i$ and germs $f_i$ of holomorphic functions at the origin in $\{\bf C\}^n$. We determine, under some conditions, the set of equivalence classes of first order unfoldings and construct explicitly a universal unfolding of $F$. Special cases of this include foliations with holomorphic or meromorphic first integrals. We also show that the unfolding theory for $F$ is equivalent to the unfolding theory for the multiform function $f=f^\{\lambda _1\}_1\ldots f^\{\lambda _p\}_p$. Thus a universal unfolding of $f$ is also given explicity.},
author = {Suwa, Tatsuo},
journal = {Annales de l'institut Fourier},
keywords = {unfoldings of foliations; complex codim 1 local foliation; foliations with holomorphic or meromorphic first integrals; unfolding theory for the multiform function},
language = {eng},
number = {3},
pages = {99-112},
publisher = {Association des Annales de l'Institut Fourier},
title = {Unfoldings of foliations with multiform first integrals},
url = {http://eudml.org/doc/74602},
volume = {33},
year = {1983},
}

TY - JOUR
AU - Suwa, Tatsuo
TI - Unfoldings of foliations with multiform first integrals
JO - Annales de l'institut Fourier
PY - 1983
PB - Association des Annales de l'Institut Fourier
VL - 33
IS - 3
SP - 99
EP - 112
AB - Let $F=(\omega )$ be a codim 1 local foliation generated by a germ $\omega $ of the form $\omega =f_1\ldots f_p\sum ^p_{i=1}\lambda _i {df_i\over f_i}$ for some complex numbers $\lambda _i$ and germs $f_i$ of holomorphic functions at the origin in ${\bf C}^n$. We determine, under some conditions, the set of equivalence classes of first order unfoldings and construct explicitly a universal unfolding of $F$. Special cases of this include foliations with holomorphic or meromorphic first integrals. We also show that the unfolding theory for $F$ is equivalent to the unfolding theory for the multiform function $f=f^{\lambda _1}_1\ldots f^{\lambda _p}_p$. Thus a universal unfolding of $f$ is also given explicity.
LA - eng
KW - unfoldings of foliations; complex codim 1 local foliation; foliations with holomorphic or meromorphic first integrals; unfolding theory for the multiform function
UR - http://eudml.org/doc/74602
ER -

References

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  1. [1] D. CERVEAU, Contribution à l'étude des formes intégrables singulières, Thèse, Université de Dijon, 1981. 
  2. [2] D. CERVEAU et A. LINS NETO, Formes intégrables tangentes à des actions commutatives, C.R.A.S., Paris, 291 (1980), 647-649. Zbl0458.53021MR82b:58003
  3. [3] D. CERVEAU et J.F. MATTEI, Intégrales premières, to appear in Astérisque. 
  4. [4] D. CERVEAU et R. MOUSSU, Extension de facteurs intégrants et applications, C.R.A.S., Paris, 294 (1982), 17-19. Zbl0504.58005MR83m:58006
  5. [5] J.F. MATTEI et R. MOUSSU, Holonomie et intégrales premières, Ann. Scient. Ec. Norm. Sup., 13 (1980), 469-523. Zbl0458.32005MR83b:58005
  6. [6] T. SUWA, Unfoldings of complex analytic foliations with singularities, to appear in Japanese J. of Math., 9 (1983). Zbl0591.32020MR85h:32036
  7. [7] T. SUWA, A theorem of versality for unfoldings of complex analytic foliation singularities, Invent. Math., 65 (1981), 29-48. Zbl0487.32016MR83e:32025
  8. [8] T. SUWA, Singularities of complex analytic foliations, to appear in the Proceedings of Symposia in Pure Math. 40, Amer. Math. Soc. Zbl0552.32008MR85a:32029
  9. [9] T. SUWA, Kupka-Reeb phenomena and universal unfoldings of certain foliation singularities, to appear in Osaka J. of Math., 20. Zbl0522.58008MR85g:58073
  10. [10] T. SUWA, Unfoldings of meromorphic functions, Math. Ann., 262 (1983), 215-224. Zbl0491.58009MR85g:32030

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