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Anais da Academia Brasileira de Ciências
On-line version ISSN 1678-2690
OSTWALD, RENATA N.. On the existence of Levi Foliations. An. Acad. Bras. Ciênc. [online]. 2001, vol.73, n.1, pp. 07-13. ISSN 1678-2690. 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 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 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.