Formal and Analytic normal forms of germs of holomorphic nondicritic foliations

L. Ortiz-Bobadilla, E. Rosales-González, and S. M. Voronin

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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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.


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

Mathematical Subject Classification:

34M35, 37F75

Author(s) information:

L. Ortiz-Bobadilla E. Rosales-González S. M. Voronin
Instituto de Matemáticas Instituto de Matemáticas Department of Mathematics
Universidad Nacional Autónoma de México Universidad Nacional Autónoma de México Chelyabinsk State University
Area de la investigación científica Area de la investigación científica Kashirin's str. 129
Ciudad Universitaria Ciudad Universitaria 454021, Chelyabinsk, Russia
C.P. 04510, México City, México C.P. 04510, México City, México email:
email: email: