Meromorphic infinitesimal affine actions of the plane

2013/12/12 by Loïc Jean Dit Teyssier, Teyssier, Loïc Jean Dit
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS

paper · pdf · doi:10.48550/arxiv.1312.3452

31 pages

arxiv created 2013/12/12 · arxiv updated 2013/12/13

Abstract

We study complex Lie algebras spanned by pairs (Z,Y) of germs of a meromorphic vector field of the complex plane satisfying [Z,Y]=δY for some δ∈\ww C . This topic relates to Liouville-integrability of the differential equation induced by the foliation underlying Z . We give a direct geometric proof of a result by M. Berthier and F. Touzet characterizing germs of a foliation admitting a first-integral in a Liouvillian extension of the standard differential field. In so doing we study transverse and tangential rigidity properties when Z is holomorphic and its linear part is not nilpotent. A second part of the article is devoted to computing the Galois-Malgrange groupoid of meromorphic Z .

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