## Anais da Academia Brasileira de Ciências

*Print version* ISSN 0001-3765*On-line version* ISSN 1678-2690

#### 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 *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 q is real a 1-form on ; then the pull-back of q by *f* defines a Levi foliation : *f*^{*}q = 0 which is tangent to the holomorphic foliation : *df* = 0. This problem was proposed by D. Cerveau in a meeting (see Fernandez 1997).

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