Sobre distribuição e folheações holomorfas de codimensão maior do que um

Abstract

Let w be a holomorphic LDS r-form on a complex manifold M. In the case M = Cn, we show that if ker(w) admits a trivial subbundle of rank k, then there exists a holomrphic LDS (r - k)-form n on Cn such that ! is the exterior product of k with the product of k linearly independent global sections of ker(w). In the case that M is compact and connected we approach the classical Darboux-Jouanolou problem and we prove that if w has a suficiently large number of invariant analytic hypersurfaces, then w admits a meromorphic first integral. Next, we prove that if k >= r and w has k infinite families of w-invariant analytic hypersurfaces whose members intersect transversely, then w admits a meromorphic first integral of rank k. In particular, if k = r, thenw! is integrable. Continuing in this direction we prove that in the integrable case ! has a transversal structure by translations if and only if w is a multiples of a product of closed 1-forms. We conclude this work by showing that in the presence of a Kupka type singularity, there exists a coordinate system around the singularity such that w reduces to r+1 variables. In particular, w is integrable and the foliation induced by w has the product struture of a foliation by curves in Cr+1 multiplied by a regular foliation.