Formal and Analytic normal forms of germs of holomorphic nondicritic foliations

Journal of Singularities
volume 9 (2014), 168-192

Received 11 April 2013. Received in revised form 2 December 2013.

DOI: 10.5427/jsing.2014.9m

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Abstract:

We consider the class V_n of germs of holomorphic vector fields in (C^2,0) with vanishing (n-1)-jet at the origin, n > 1. For generic germs v in V_2 we prove the existence of an analytical orbital normal form whose orbital formal normal form has the form v_{c,b} given in a previous paper of the current authors. Furthermore, fixing one representative vˆ of the analytic class of a germ v in V_2 having the y-axis invariant, the corresponding formal normal form vˆ_{c, bˆ} is analytic and unique (under strict orbital equivalence). Moreover for generic v in V_n, n>1, we give a preliminary orbital analytic normal form which is polynomial and of degree at most n in the y-variable.

Keywords:

nondicritic foliations, nondicritic vector fields, normal forms, formal equivalence, analytical equivalence

Mathematical Subject Classification:

34M35, 37F75

Author(s) information: