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  • Author: Bruno Scárdua x
  • General Mathematics x
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Abstract

We study the classification of polynomial vector fields in two complex variables under the hypotheses that the singularities are isolated and the flow is complete. Normal forms are obtained for the case the generic orbit is diffeomorphic to ℂ. For the case the generic orbit is diffeomorphic to ℂ ∖ {0} and there is an affine singularity we classify the linear part of the vector field and prove the existence of entire linearization or first integral.

Abstract

We study groups of germs of complex diffeomorphisms having a property called irreducibility. The notion is motivated by the similar property of the fundamental group of the complement of an irreducible hypersurface in the complex projective space. Natural examples of such groups of germ maps are given by holonomy groups and monodromy groups of integrable systems (foliations) under certain conditions. We prove some finiteness results for these groups extending previous results in [D. Cerveau and F. Loray, Un théorème de Frobenius singulier via l’arithmétique élémentaire, J. Number Theory 68 1998, 2, 217–228]. Applications are given to the framework of germs of holomorphic foliations. We prove the existence of first integrals under certain irreducibility or more general conditions on the tangent cone of the foliation after a punctual blow-up.