On the existence of Levi Foliations
Autor(a) principal: | |
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Data de Publicação: | 2001 |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Anais da Academia Brasileira de Ciências (Online) |
Texto Completo: | http://old.scielo.br/scielo.php?script=sci_arttext&pid=S0001-37652001000100002 |
Resumo: | 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}$"> : df = 0. This problem was proposed by D. Cerveau in a meeting (see Fernandez 1997). |
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On the existence of Levi FoliationsLevi foliationsholomorphic foliationssingularitiesLevi varietiesLet L <img src="http:/img/fbpe/aabc/v73n1/0059c.gif"> <img src="http:/img/fbpe/aabc/v73n1/0059c2.gif"> be a real 3 dimensional analytic variety. For each regular point p <img src="http:/img/fbpe/aabc/v73n1/0059e.gif"> L there exists a unique complex line l p on the space tangent to L at p. When the field of complex line p <img ALIGN="MIDDLE" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img4.gif" ALT="$\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).Academia Brasileira de Ciências2001-03-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersiontext/htmlhttp://old.scielo.br/scielo.php?script=sci_arttext&pid=S0001-37652001000100002Anais da Academia Brasileira de Ciências v.73 n.1 2001reponame:Anais da Academia Brasileira de Ciências (Online)instname:Academia Brasileira de Ciências (ABC)instacron:ABC10.1590/S0001-37652001000100002info:eu-repo/semantics/openAccessOSTWALD,RENATA N.eng2001-03-09T00:00:00Zoai:scielo:S0001-37652001000100002Revistahttp://www.scielo.br/aabchttps://old.scielo.br/oai/scielo-oai.php||aabc@abc.org.br1678-26900001-3765opendoar:2001-03-09T00:00Anais da Academia Brasileira de Ciências (Online) - Academia Brasileira de Ciências (ABC)false |
dc.title.none.fl_str_mv |
On the existence of Levi Foliations |
title |
On the existence of Levi Foliations |
spellingShingle |
On the existence of Levi Foliations OSTWALD,RENATA N. Levi foliations holomorphic foliations singularities Levi varieties |
title_short |
On the existence of Levi Foliations |
title_full |
On the existence of Levi Foliations |
title_fullStr |
On the existence of Levi Foliations |
title_full_unstemmed |
On the existence of Levi Foliations |
title_sort |
On the existence of Levi Foliations |
author |
OSTWALD,RENATA N. |
author_facet |
OSTWALD,RENATA N. |
author_role |
author |
dc.contributor.author.fl_str_mv |
OSTWALD,RENATA N. |
dc.subject.por.fl_str_mv |
Levi foliations holomorphic foliations singularities Levi varieties |
topic |
Levi foliations holomorphic foliations singularities Levi varieties |
description |
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). |
publishDate |
2001 |
dc.date.none.fl_str_mv |
2001-03-01 |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://old.scielo.br/scielo.php?script=sci_arttext&pid=S0001-37652001000100002 |
url |
http://old.scielo.br/scielo.php?script=sci_arttext&pid=S0001-37652001000100002 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
10.1590/S0001-37652001000100002 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
text/html |
dc.publisher.none.fl_str_mv |
Academia Brasileira de Ciências |
publisher.none.fl_str_mv |
Academia Brasileira de Ciências |
dc.source.none.fl_str_mv |
Anais da Academia Brasileira de Ciências v.73 n.1 2001 reponame:Anais da Academia Brasileira de Ciências (Online) instname:Academia Brasileira de Ciências (ABC) instacron:ABC |
instname_str |
Academia Brasileira de Ciências (ABC) |
instacron_str |
ABC |
institution |
ABC |
reponame_str |
Anais da Academia Brasileira de Ciências (Online) |
collection |
Anais da Academia Brasileira de Ciências (Online) |
repository.name.fl_str_mv |
Anais da Academia Brasileira de Ciências (Online) - Academia Brasileira de Ciências (ABC) |
repository.mail.fl_str_mv |
||aabc@abc.org.br |
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1754302855447052288 |