Invariant hypersurfaces and nodal components for codimension one singular foliations
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Invariant hypersurfaces and nodal components for codimension one singular foliations Felipe Cano1
· Jean François Mattei2 · Marianna Ravara-Vago3
Received: 9 October 2019 / Accepted: 5 August 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract It is known that there is at least an invariant analytic curve passing through each of the components in the complement of nodal singularities, after reduction of singularities of a germ of singular foliation in (C2 , 0). Here, we state and prove a generalization of this property to any ambient dimension. Keywords Singular holomorphic foliations · Invariant hypersurfaces · Simplicial complexes · Desingularization Mathematics Subject Classification 32S65 · 14M25 · 14E15
1 Introduction This paper deals with the presence of invariant hypersurfaces in each component of the “space of leaves” of germs holomorphic codimension one foliations. More precisely, we provide a generalization to any ambient dimension of the two dimensional refined version of Camacho–Sad’s theorem [8], stated an proved by Ortiz et al. [25]. The main result in this paper is the following one: Theorem 1 Consider a nodal reduction of singularities π : (M, E, F ) → ((Cn , 0), ∅, F0 )
Partially supported by MTM-2016-77642-C2-1-P. Spain Partially supported by CNPq-406034/2016-8. Brasil.
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Felipe Cano [email protected] Jean François Mattei [email protected] Marianna Ravara-Vago [email protected]
1
Universidad de Valladolid. Fac. Ciencias, Paseo Belén, 7, 47011 Valladolid, Spain
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Université Paul Sabatier, Toulouse, France
3
Univ. Federal de Santa Catarina, Florianopólis, Brasil 0123456789().: V,-vol
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of a GH-foliation F0 on (Cn , 0). Let |S | be the support of the nodal separator set of (M, E, F ). For any connected component C of E\|S |, there is an invariant hypersurface H0 of F0 such that H ∩ C = ∅, where H ⊂ M is the strict transform of H0 by π. Moreover, we have that H ∩ E ⊂ C. Concerning the existence of invariant hypersufaces, let us recall that it was a Thom’s question if any singular holomorphic foliation on (C2 , 0) has at least one invariant branch. A positive answer has been obtained by Camacho and Sad in [8]. In higher ambient dimension, Jouanolou gave in [18] examples of codimension one dicritical holomorphic foliations without invariant hypersurface. For any ambient dimension and non dicritical foliations, the existence of invariant hypersurface is proved in [3,4]. The space of leaves of foliations on (C2 , 0) is naturally separated by the so-called “nodal points”. This property has been remarked in [20]. After a “nodal” reduction of singularities, we can put all the nodal points as “corners” of the exceptional divisor. When we remove them, the exceptional divisor is decomposed into several connected components: we get exactly “s + 1” pieces if we have “s” nodal points. By a result of Ortiz–Rosales–Voronin [25], each of these pieces intersects the strict transform of at least one invariant branch of the foliation. This statem
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