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Constructing equivalences with some extensions to the divisor and topological invariance of projective holonomy

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Abstract

Given topologically equivalent germs of holomorphic foliations F and F, under some hypothesis, we construct topological equivalences extending to some regions of the divisor after resolution of singularities. As an application we study the topological invariance of the projective holonomy representation.

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... In the case of Generic General Type foliations, Theorem 3 is a consequence of the work of Marín and Mattei [4] -Generic General Type foliations are generalized curves with an additional generic dynamical property which guarantees that the conjugation h is transversely holomorphic -. In fact, in [4] the authors prove much more: if F is of Generic General Type and F is any foliation topologically equivalent to F, then there exists a topological equivalence between F and F extending to the exceptional divisor after the resolutions of F and F. On the other hand, if F is a generalized curve not necessarily of Generic General Type, in [5] is proved that always exists a topological equivalence between F and F extending after resolution to a neighborhood of each linearizable or resonant singularity which is not a corner. In particular, this topological equivalence extends to each nodal singularity which is not a corner. ...
... Moreover, Theorem 2 guarantees that p andp are in "isomorphic positions" in their corresponding exceptional divisors. From this point the construction of a topological equivalence extending to p follows some ideas already used in [5]. ...
... Proof of Theorem 3. It is a direct consequence of Theorem 23. Proof of Theorem 4. By [5] there exists a topological equivalence h between F and F which, after resolution, extends as a homeomorphism to a neighborhood of each linearizable or resonant singularity which is not a corner. We denote by E and E the exceptional divisors in the resolutions of F and F, respectively. ...
Preprint
We study a special kind of local invariant sets of singular holomorphic foliations called nodal separators. We define notions of equisingularity and topological equivalence for nodal separators as intrinsic objects and, in analogy with the celebrated theorem of Zariski for analytic curves, we prove the equivalence of these notions. We give some applications in the study of topological equivalences of holomorphic foliations. In particular, we show that the nodal singularities and its eigenvalues in the resolution of a generalized curve are topological invariants.
... In the case of Generic General Type foliations, Theorem 3 is a consequence of the work of Marín and Mattei [4] -Generic General Type foliations are generalized curves with an additional generic dynamical property which guarantees that the conjugation h is transversely holomorphic -. In fact, in [4] the authors prove much more: if F is of Generic General Type and F is any foliation topologically equivalent to F, then there exists a topological equivalence between F and F extending to the exceptional divisor after the resolutions of F and F. On the other hand, if F is a generalized curve not necessarily of Generic General Type, in [5] is proved that always exists a topological equivalence between F and F extending after resolution to a neighborhood of each linearizable or resonant singularity which is not a corner. In particular, this topological equivalence extends to each nodal singularity which is not a corner. ...
... Moreover, Theorem 2 guarantees that p andp are in "isomorphic positions" in their corresponding exceptional divisors. From this point the construction of a topological equivalence extending to p follows some ideas already used in [5]. ...
... Proof of Theorem 3. It is a direct consequence of Theorem 23. Proof of Theorem 4. By [5] there exists a topological equivalence h between F and F which, after resolution, extends as a homeomorphism to a neighborhood of each linearizable or resonant singularity which is not a corner. We denote by E and E the exceptional divisors in the resolutions of F and F, respectively. ...
Article
Full-text available
We study a special kind of local invariant sets of singular holomorphic foliations called nodal separators. We define notions of equisingularity and topological equivalence for nodal separators as intrinsic objects and, in analogy with the celebrated theorem of Zariski for analytic curves, we prove the equivalence of these notions. We give some applications in the study of topological equivalences of holomorphic foliations. In particular, we show that the nodal singularities and its eigenvalues in the resolution of a generalized curve are topological invariants.
... Also in [4] the authors give a positive answer for a generic class of foliations F and assuming that h is a topologically trivial deformation. Stronger results in relation to this subject are obtained in [8], [10], [9], and [17]. We must remark the work of Marín and Mattei ( [9]), who prove the topological invariance of the projective holonomy for a generic class of generalized curves, although the problem is still unsettled if we allow saddle node singularities after resolution. ...
... Thus, Proposition 12 and Lemma 13 discard any homological obstruction to perform the constructions in the rest of the paper. Proposition 12 is a special version of a kind of results previously obtained in [9] (Theorem 6.2.1) and [17] (section 5). ...
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In this paper we study bilipschitz equivalences of germs of holomorphic foliations in (C2,0)(\mathbb{C}^2,0). We prove that the algebraic multiplicity of a singularity is invariant by such equivalences. Moreover, for a large class of singularities, we show that the projective holonomy representation is also a bilipschitz invariant.
... Among them, let us mention, without trying to be exhaustive, works of D. Marín and J.-F. Mattei [18], and of R. Rosas [30]. ...
Preprint
In this article we study the analytic classification of certain types of quasi-homogeneous cuspidal holomorphic foliations in (\CC^3,{\bf 0}) via the essential holonomy defined over one of the components of the exceptional divisor that appears in the reduction of the singularities of the foliation.
... is proven by R. Rosas under weak hypothesis [38,Proposition 13], see also [22,Theorem 1.12]. Index Formula (23) implies that: ...
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The object of this survey is to give an overview on the topology of singularities of holomorphic foliation germs on (C2,0)(\mathbb C^2,0).
... a) λ := CS(F ♯ , D, s) is an irrational real number. If λ is positive, s is a nodal singular point, and (26) was obtained by R. Rosas in [30,Proposition 13]. ...
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This work deals with the topological classification of germs of singular foliations on (C2,0)(\mathbb C^{2},0). Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and the projective holonomy representations and we compute the moduli space of topological classes in terms of the cohomology of a new algebraic object that we call group-graph. This moduli space may be an infinite dimensional functional space but under generic conditions we prove that it has finite dimension and we describe its algebraic and topological structures.
... Among them, let us mention, without trying to be exhaustive, works of D. Marín and J.-F. Mattei [18], and of R. Rosas [30]. ...
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This work deals with the topological classification of germs of singular foliations on (C2,0)(\mathbb C^{2},0). Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and the projective holonomy representations and we compute the moduli space of topological classes in terms of the cohomology of a new algebraic object that we call group-graph. This moduli space may be an infinite dimensional functional space but under generic conditions we prove that it has finite dimension and we describe its algebraic and topological structures.
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