On the existence of Levi Foliations

Anais da Academia Brasileira de Ciências

Renata N. Ostwald¹ 

[1] ,Instituto de Matemática Pura e AplicadaRio de Janeiro,Brasil

关键词: Levi foliations;    holomorphic foliations;    singularities;    Levi varieties;    folheações de Levi;    folheações holomorfas;    singularidades;    variedades de Levi;   

DOI  :  10.1590/S0001-37652001000100002

来源: SciELO

【 摘 要 】

Let L be a real 3 dimensional analytic variety. For each regular point p L there exists a unique complex line l p on the space tangent to L at p. When the field of complex line p l p is completely integrable, we say that L is Levi variety. More generally; let L 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 be a Levi foliation and let be the induced holomorphic foliation. Then, admits a Liouvillian first integral. In other words, if is a 3 dimensional analytic foliation such that the induced complex distribution defines an holomorphic foliation ; that is, if is a Levi foliation; then 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 ; then the pull-back of theta by f defines a Levi foliation : f*theta = 0 which is tangent to the holomorphic foliation : df = 0. This problem was proposed by D. Cerveau in a meeting (see Fernandez 1997).

【 授权许可】

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