On closed currents invariant by holomorphic foliations. I.

*(English)*Zbl 1304.37032All known examples of foliations admitting an invariant positive closed current can be divided in two categories: either they contain an algebraic invariant leaf or they admit a transversal Riemannian structure. It is conjectured that this dichotomy always occurs and the paper under review sheds some light on the first situation.

The idea is to use a real one-dimensional foliation \(\mathcal{H}\) on the leaves to obtain contractive elements in the holonomy pseudogroup of the foliation in the spirit of [C. Bonatti et al., Publ. Math., Inst. Hautes Étud. Sci. 75, 123–134 (1992; Zbl 0782.32023)]. Although contractive holonomy maps do not always exist, we can always define the one-dimensional foliation which leads us to the statement of the main results: Theorem A asserts that for every singular holomorphic foliation \(\mathcal{F}\) in an algebraic surface carrying an invariant closed positive current \(T\), if \(\mathcal{K}\) is a possibly singular minimal set in the support of \(T\) containing a trajectory of \(\mathcal{H}\) of infinite length, then \(\mathcal{K}\) is an algebraic curve invariant for \(\mathcal{F}\). This theorem is complemented by Theorem B which asserts that if \(T\) is not locally given by the integration on \(\mathcal{K}\) then \(\mathcal{F}\) admits a Liouvillean first integral in a neighbourhood of K.

The one-dimensional foliation \(\mathcal{H}\) is well defined on regular points as appearing in [loc. cit.]. Actually, in the non-singular case they can always find a path of infinite length, hence a contractive element in the holonomy pseudogroup based on a point \(p\). The leaf passing through \(p\) has to be algebraic.

However, the techniques involved in that article are not sufficient for the singular case and the author outlines three situations where the trajectories are affected by the singularities: there might be singular points in the divisor of zeros or poles, they might drastically reduce the domain of the holonomy and, as a specific case of the latter, ramified contractions may appear. Generalized Dulac transforms are introduced in the article and they play a crucial role in the argument provided by the author to overcome these difficulties.

The paper contains a very thorough introduction in Sections 1 and 2 and provides the necessary background in Section 3. The behaviour of \(\mathcal{H}\) around a Siegel singularity is studied in Section 4 where Dulac transforms are presented for the first time. In Section 5, the type of reduced singularities that might appear in a diffuse closed current are proven to be Siegel and irrational focus singularities, and the behaviour of \(\mathcal{H}\) is studied around the latter. Moreover, generalized Dulac transforms are introduced at the end of the section. In Section 6, a way of following trajectories through the singularities in order to find elements with contractive nature is presented. Once an infinite trajectory is assumed to exist in a minimal set contained in the support of a closed invariant positive current, in Section 7, using the contractive nature of the path, the author proves that either the minimal set is an algebraic curve or the trajectory is periodic passing through singularities of \(\mathcal{F}\). In Section 8, this last situation is studied and the proofs of the theorems are given.

The idea is to use a real one-dimensional foliation \(\mathcal{H}\) on the leaves to obtain contractive elements in the holonomy pseudogroup of the foliation in the spirit of [C. Bonatti et al., Publ. Math., Inst. Hautes Étud. Sci. 75, 123–134 (1992; Zbl 0782.32023)]. Although contractive holonomy maps do not always exist, we can always define the one-dimensional foliation which leads us to the statement of the main results: Theorem A asserts that for every singular holomorphic foliation \(\mathcal{F}\) in an algebraic surface carrying an invariant closed positive current \(T\), if \(\mathcal{K}\) is a possibly singular minimal set in the support of \(T\) containing a trajectory of \(\mathcal{H}\) of infinite length, then \(\mathcal{K}\) is an algebraic curve invariant for \(\mathcal{F}\). This theorem is complemented by Theorem B which asserts that if \(T\) is not locally given by the integration on \(\mathcal{K}\) then \(\mathcal{F}\) admits a Liouvillean first integral in a neighbourhood of K.

The one-dimensional foliation \(\mathcal{H}\) is well defined on regular points as appearing in [loc. cit.]. Actually, in the non-singular case they can always find a path of infinite length, hence a contractive element in the holonomy pseudogroup based on a point \(p\). The leaf passing through \(p\) has to be algebraic.

However, the techniques involved in that article are not sufficient for the singular case and the author outlines three situations where the trajectories are affected by the singularities: there might be singular points in the divisor of zeros or poles, they might drastically reduce the domain of the holonomy and, as a specific case of the latter, ramified contractions may appear. Generalized Dulac transforms are introduced in the article and they play a crucial role in the argument provided by the author to overcome these difficulties.

The paper contains a very thorough introduction in Sections 1 and 2 and provides the necessary background in Section 3. The behaviour of \(\mathcal{H}\) around a Siegel singularity is studied in Section 4 where Dulac transforms are presented for the first time. In Section 5, the type of reduced singularities that might appear in a diffuse closed current are proven to be Siegel and irrational focus singularities, and the behaviour of \(\mathcal{H}\) is studied around the latter. Moreover, generalized Dulac transforms are introduced at the end of the section. In Section 6, a way of following trajectories through the singularities in order to find elements with contractive nature is presented. Once an infinite trajectory is assumed to exist in a minimal set contained in the support of a closed invariant positive current, in Section 7, using the contractive nature of the path, the author proves that either the minimal set is an algebraic curve or the trajectory is periodic passing through singularities of \(\mathcal{F}\). In Section 8, this last situation is studied and the proofs of the theorems are given.

Reviewer: Carlos Perez Garrandes (Madrid)