Rigidity Theorems for Generic Holomorphic Germs of Dicritic Foliations and Vector Fields in (C2, 0)

Abstract

We consider the class V d n+1 of dicritical germs of holomorphic vector fields in (C 2 , 0) with vanishing n-jet at the origin for n ≥ 1. We prove, under some genericity assumptions, that the formal equivalence of two generic germs implies their analytic equivalence. A similar result is also established for orbital equivalence. Moreover, we give formal, orbitally formal, and orbitally analytic classifications of generic germs in V d n+1 up to a change of coordinates with identity linear part.

Analytic Normal Forms of Germs of Holomorphic Dicritic Foliations

“…The same phenomenon of rigidity (formal equivalence implies analytic one) was obtained for generic dicritic degenerated germs (for the orbital and classical cases) in [ORV2]. There, rather simple formal normal forms for such germs were also constructed, and even their analytic orbital classification was obtained.…”

“…This classification was given in terms of some geometric invariants together with some formal invariants. However, the question on the analyticity of the formal normal forms constructed in [ORV2] stayed open. A similar situation appeared, at the time, for germs with nilpotent linear part.…”

“…In this work we use the highly inspiring method of Loray [Lo4] to give an answer to the analyticity of the formal normal forms constructed in [ORV2]: in the simplest case (n = 1, where n is the number of singular irreducible separatrices) we obtain the analyticity of the corresponding formal normal form as it was given in [ORV2]. For the case n > 1, the use of the method in [Lo2] allows only to transform the corresponding germ to a "quasipolynomial" normal form of a wider class than the formal normal form obtained in [ORV2]. Therefore, in this case the analyticity of the formal normal forms of [ORV2] still remains an open question.…”

Real-Formal Orbital Rigidity for Germs of Real Analytic Vector Fields on the Real Plane

“…Later, Ortiz, Rosales, and Voronin proved that the rigidity phenomenon also takes place for vector fields, that is, there is formal (non orbital) rigidity for generic germs of vector fields in V C n [21]. After that, they verified that formal rigidity and formal orbital rigidity takes place for generic dicritical germs of vector fields in V C n , obtaining in addition the minimal invariants for the strict analytic orbital classification 1 of such germs [22]. Recently, they gave the minimal invariants for the strict analytic orbital classification of generic nondicritic germs [24].…”

Topological Moduli Space for Germs of Holomorphic Foliations

“…any non dicritical foliation reduced after only one blow-up, its separatrices being smooth curves mutually transversal, or more generally any topologically quasi-homogeneous germ, see [12], -absolutely dicritical foliations of Cano-Corral [5], -dicritical foliations that are non singular after one blow-up, see [1] and [24].…”