Minimal invariant varieties and first integrals for algebraic foliations

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Let X be an irreducible algebraic variety over C, endowed with an algebraic foliation F. In this paper, we introduce the notion of minimal invariant variety V(F,Y) with respect to (F,Y), where Y is a subvariety of X. If Y={x} is a smooth point where the foliation is regular, its minimal invariant variety is simply the Zariski closure of the leaf passing through x. First we prove that for very generic x, the varieties V(F,x) have the same dimension p. Second we generalize a result due to X. Gomez-Mont. More precisely, we prove the existence of a dominant rational map F:X→ Z, where Z has dimension (n-p), such that for every very generic x, the Zariski closure of F⁻¹(F(x)) is one and only one minimal invariant variety of a point. We end up with an example illustrating both results.