A theorem of versality for unfoldings of complex analytic foliation singularities

1981 ◽  
Vol 65 (1) ◽  
pp. 29-48 ◽  
Author(s):  
Tatsuo Suwa
Keyword(s):  
Analytic Foliation

Minimality and ergodicity of a generic analytic foliation of ℂ2

2008 ◽  
Vol 28 (5) ◽  
pp. 1533-1544
Author(s):  
T. GOLENISHCHEVA-KUTUZOVA ◽  
V. KLEPTSYN
Keyword(s):  
Analogous Result ◽  
Generic Polynomial ◽  
Analytic Foliation

AbstractIt is well known that a generic polynomial foliation of ℂ2 is minimal and ergodic. In this paper we prove an analogous result for analytic foliations.

scholarly journals On the structure of the singular set of a complex analytic foliation

1998 ◽  
Vol 50 (4) ◽  
pp. 837-857 ◽  
Author(s):  
Junya YOSHIZAKI
Keyword(s):  
Singular Set ◽  
Analytic Foliation

A generic analytic foliation in ℂ2 has infinitely many cylindrical leaves

2006 ◽  
Vol 254 (1) ◽  
pp. 180-183 ◽  
Author(s):  
T. I. Golenishcheva-Kutuzova
Keyword(s):  
Analytic Foliation

scholarly journals Determinacy of Analytic Foliation Germs

Foliations ◽  
2018 ◽  
Author(s):  
Tatsuo Suwa
Keyword(s):  
Analytic Foliation

scholarly journals Residues of complex analytic foliation singularities

1984 ◽  
Vol 36 (1) ◽  
pp. 37-45 ◽  
Author(s):  
Tatsuo SUWA
Keyword(s):  
Analytic Foliation

scholarly journals Determinacy of complex analytic foliation germs without integrating factors

1991 ◽  
Vol 112 (4) ◽  
pp. 989-989
Author(s):  
Dominique Cerveau ◽  
Tatsuo Suwa
Keyword(s):  
Integrating Factors ◽  
Analytic Foliation

scholarly journals On the existence of Levi Foliations

2001 ◽  
Vol 73 (1) ◽  
pp. 07-13
Author(s):  
RENATA N. OSTWALD
Keyword(s):  
Regular Point ◽  
Rational Functions ◽  
First Integral ◽  
Holomorphic Foliation ◽  
Complex Line ◽  
3 Dimensional ◽  
Prove Theorem ◽  
Integrable Distribution ◽  
Line P ◽  
Analytic Foliation

Let L <img src="http:/img/fbpe/aabc/v73n1/0059c.gif"> <img src="http:/img/fbpe/aabc/v73n1/0059c2.gif"> be a real 3 dimensional analytic variety. For each regular point p <img src="http:/img/fbpe/aabc/v73n1/0059e.gif"> L there exists a unique complex line l p on the space tangent to L at p. When the field of complex line p <img ALIGN="MIDDLE" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img4.gif" ALT="$\displaystyle \mapsto$"> l p is completely integrable, we say that L is Levi variety. More generally; let L <img src="http:/img/fbpe/aabc/v73n1/0059c.gif"> M be a real subvariety in an holomorphic complex variety M. If there exists a real 2 dimensional integrable distribution on L which is invariant by the holomorphic structure J induced by M, we say that L is a Levi variety. We shall prove: Theorem. Let <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img5.gif" ALT="$ \cal {L}$"> be a Levi foliation and let <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$"> be the induced holomorphic foliation. Then, <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$"> admits a Liouvillian first integral. In other words, if <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img5.gif" ALT="$ \cal {L}$"> is a 3 dimensional analytic foliation such that the induced complex distribution defines an holomorphic foliation <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$">; that is, if <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img5.gif" ALT="$ \cal {L}$"> is a Levi foliation; then <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$"> admits a Liouvillian first integral--a function which can be constructed by the composition of rational functions, exponentiation, integration, and algebraic functions (Singer 1992). For example, if f is an holomorphic function and if theta is real a 1-form on <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img8.gif" ALT="$ \mathbb {R}$">; then the pull-back of theta by f defines a Levi foliation <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img5.gif" ALT="$ \cal {L}$"> : f*theta = 0 which is tangent to the holomorphic foliation <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$"> : df = 0. This problem was proposed by D. Cerveau in a meeting (see Fernandez 1997).

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