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Vector fields, separatrices and Kato surfaces. (Champs de vecteurs, séparatrices et surfaces de Kato.) (English. French summary) Zbl 1335.32025
In [Ann. Math. (2) 115, 579–595 (1982; Zbl 0503.32007)], C. Camacho and P. Sad affirmatively solved a question dating back to Poincaré, proving that every germ of holomorphic foliation near the origin in \(\mathbb C^2\) having an isolated singularity at \(O\), admits a separatrix at \(O\). Such a result is however false in case of a holomorphic foliations on singular surfaces, even if the foliation is given by a vector field, although in such a case the counter-examples are very particular. More precisely, the only examples of compact normal, irreducible complex singular surfaces admitting holomorphic foliations defined by vector fields with no separatrices through isolated singularities are obtained as quotients of intermediate Kato surfaces.
In the paper under review, the author proves that compactness of the surface can be replaced by completeness of the vector field. More precisely, let \(S\) be a connected, normal, irreducible, complex two-dimensional singular surface with a singularity at \(p\in S\). Let \(X\) be a complete holomorphic vector field on \(S\). If the foliation induced by \(X\) has no separatrix through \(p\), then the minimal resolution of \(S\) at \(p\) is a Kato surface. As a consequence, \(S\) is compact.
In particular, every complete holomorphic vector field on a non-compact, normal, irreducible, complex two-dimensional singular surface has separatrices at every isolated singularity. In the case of Stein spaces, the author obtains a more precise result: let \(S\) be a normal, irreducible, complex two-dimensional Stein space and let \(X\) be a complete holomorphic vector field on \(S\) with an isolated singularity at \(p\in S\). Assume that \(p\) is an isolated equilibrium point of \(X\). Then, either there are one or two separatrices of \(X\) at \(p\) and \(S\) at \(p\) is a cyclic quotient singularity, or there exist infinitely many separatrices of \(X\) at \(p\) and \(X\) induces an action of \(\mathbb C^\ast\).
The paper also contains a relatively self-contained proof, not relying on the Enriques-Castelnuovo classification of surfaces, of part of the Dloussky-Oeljeklaus-Toma classification of holomorphic vector fields on compact complex surfaces.

32S65 Singularities of holomorphic vector fields and foliations
32C20 Normal analytic spaces
34M45 Ordinary differential equations on complex manifolds
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