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Abstract

OSTWALD, RENATA N.. On the existence of Levi Foliations. An. Acad. Bras. Ciênc. [online]. 2001, vol.73, n.1, pp. 07-13. ISSN 0001-3765.  http://dx.doi.org/10.1590/S0001-37652001000100002.

Let L be a real 3 dimensional analytic variety. For each regular point p L there exists a unique complex line lp on the space tangent to L at p. When the field of complex line p $\displaystyle \mapsto$ lp 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 $ \cal {L}$ be a Levi foliation and let $ \cal {F}$ be the induced holomorphic foliation. Then, $ \cal {F}$ admits a Liouvillian first integral. In other words, if $ \cal {L}$ is a 3 dimensional analytic foliation such that the induced complex distribution defines an holomorphic foliation $ \cal {F}$; that is, if $ \cal {L}$ is a Levi foliation; then $ \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 q is real a 1-form on $ \mathbb {R}$; then the pull-back of q by f defines a Levi foliation $ \cal {L}$ : f*q = 0 which is tangent to the holomorphic foliation $ \cal {F}$ : df = 0. This problem was proposed by D. Cerveau in a meeting (see Fernandez 1997).

Keywords : Levi foliations; holomorphic foliations; singularities; Levi varieties.

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