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Generalized Flow-Box property for singular foliations

Abstract

We introduce a notion of generalized Flow-Box property valid for general singular distributions and sub-varieties (based on a dynamical interpretation). Just as in the usual Flow-Box Theorem, we characterize geometrical and algebraic conditions of (quasi) transversality in order for an analytic sub-variety X (not necessarily regular) to be a section of a line foliation. We also discuss the case of more general foliations. This study is originally motivated by a question of Jean-François Mattei (Invent Math 103(2):297–325, 1991, Theorem 3.2.1) about the existence of local slices for a (non-compact) Lie group action.

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Acknowledgements

We would like to thank Jean-François Mattei for bringing the problem to our attention and for the very useful discussions on the topic. Daniel Panazzolo was supported by ANR-16-CE40-0008. We would also like to thank the two anonymous referees for the many useful suggestions and the careful revision, in particular concerning Definition 4.10.

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Correspondence to André Belotto da Silva.

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Dedicado à Felipe Cano, em comemoração dos seus 60 anos.

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Belotto da Silva, A., Panazzolo, D. Generalized Flow-Box property for singular foliations. RACSAM 113, 3949–3965 (2019). https://doi.org/10.1007/s13398-018-0596-7

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Keywords

  • Flow-Box theorem
  • Vector-field
  • Foliations
  • Resolution of singularities

Mathematics Subject Classification

  • Primary 34C99
  • 34M35
  • 37F75
  • Secondary 14B05
  • 14E15
  • 34C08