# Normal forms of invariant vector fields under a finite group action

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Publicacions Matemátiques, Vol 37 (1993), 75-82. NORMAL FORMS OF INVARIANT VECTOR FIELDS UNDER A FINITE GROUP ACTION Abstract FEDERICO SÁNCHEZ-BRINGAS Let I' be a finite subgroup of GL(n, (C) . This subgroup acts on the space of germs of holomorphic vector fields vanishing at the origin in Cn and on the group of germs of holomorphic diffeomorphisms of ((Cn, 0) . We prove a theorem of invariant conjugacy to a nor- mal form and linearization for the subspace of invariant germs of holomorphic vector fields and we give a description of this type of normal forms in dimension n = 2. Introduction The goal of this paper is to show that the classic theorems of Poincaré- Dulac [DU] and Siegel [SI] of conjugacy to a normal form and lineariza- tion of germs of holomorphic vector fields at 0 E Cn hold for the quotient space Cn/I', where I' is a finite subgroup of GL(n, C) . In this situation we consider the germs of holomorphic vector fields and the germs of con- jugating diffeomorphism of Cl invariant by the action of the subgroup . It is well known that Cn/I' has the structure of an algebraic variety and furthermore any variety which is the quotient of a finite group of local diffeomorphisms of Cn is of this form (in a specific system of coordi- nates) [CA], then we obtain here results for conjugacies to normal forms and linearizations of germs of holomorphic vector fields in this kind of algebraic varieties . In a different context, like bifurcation theory, sometimes conjugacy to a normal form of germs of holomorphic vector fields which preserves symmetries are needed, this results can also be applied . In the first section we prove the main theorem using the algebraic approach developed in [CH] . In the second section we analyse carefully the case C'/I` and we give a description of normal forms . Finally we wish to thank Xavier Gomez-Mont for líis helpful comments and remarks concerning this work . 76 F. SÁNCHEZ-BRINCAS 1 . Invariant conjugacy to a normal form and lineari

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