Hypersurfaces quasi-invariant by codimension one foliations

Abstract

We present a variant of the classical Darboux-Jouanolou Theorem. Our main result provides a characterization of foliations which are pull-backs of foliations on surfaces by rational maps. As an application, we provide a structure theorem for foliations on 3-folds admitting an infinite number of extremal rays. Mathematics Subject Classification 37F75 • 14E30 1.1 Statement of the main result Rational pull-backs of foliations on projective surfaces provide natural examples with infinitely many quasi-invariant divisors. Our main result shows that the existence of sufficiently many quasi-invariant hypersurfaces characterizes this class of foliations.

Pull-back of singular Levi-flat hypersurfaces

“…every leaf of F | S is algebraic. The concept of quasi-invariant subvarieties was introduced by Pereira-Spicer [19] for codimension one holomorphic foliations on complex projective manifolds to prove a variant of the classical Darboux-Jouanolou Theorem. Here we shall use this concept for Levi foliations to prove our main result: Theorem 1.1.…”

“…The paper is organized as follows: in Section 2, we define the concept of quasiinvariant subvarieties of a foliation with complex leaves and state the main result of [19], such a result is key to prove Theorem 1.1. Section 3 is devoted to the study of real analytic Levi-flat subset in complex manifolds, using some results of [3] and [2], we prove the algebraic extension of the intrinsic complexification of H. In Section 4, we prove Theorem 1.1 and in Section 5 we prove Corollary 1.2.…”

“…Motivated by [19], we define the concept of a subvariety quasi-invariant by a real analytic foliation with complex leaves. Definition 2.2.…”

“…We note that the restriction foliation G | S is a codimension one foliation on S and when G | S is an algebraically integrable foliation, we have that every leaf of G | S are projective complex hypersurfaces in S. Codimension one holomorphic foliations on Z which admit infinitely many quasi-invariant hypersurfaces have been studied in [19] and its main result is the following. Theorem 2.3.…”

“…Theorem 2.3. (Pereira-Spicer [19]) Let F be a codimension one holomorphic foliation on a projective manifold Z. If F admits infinitely many quasi-invariant hypersurfaces then either F is an algebraically integrable foliation, or F is a pull-back of a foliation of dimension one on a projective surface under a dominant rational map.…”