Local resolution of ideals subordinated to a foliation

Abstract

Let M be a complex- or real-analytic manifold, \(\theta \) be a singular distribution and \(\mathcal {I}\) a coherent ideal sheaf defined on M. We prove the existence of a local resolution of singularities of \(\mathcal {I}\) that preserves the class of singularities of \(\theta \), under the hypothesis that the considered class of singularities is invariant by \(\theta \)-admissible blowings-up. In particular, if \(\theta \) is monomial, we prove the existence of a local resolution of singularities of \(\mathcal {I}\) that preserves the monomiality of the singular distribution \(\theta \).

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Acknowledgments

I would like to thank Edward Bierstone for the useful suggestions and for reviewing the manuscript. The structure of this manuscript is strongly influenced by him. I would also like to express my gratitude to Daniel Panazzolo for the useful discussions concerning the problem and its applications. Finally, I would like to thank the anonymous reviewer for several very useful comments and, in particular, for suggesting a different title for the manuscript.

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Correspondence to André Belotto da Silva.

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Belotto da Silva, A. Local resolution of ideals subordinated to a foliation. RACSAM 110, 841–862 (2016). https://doi.org/10.1007/s13398-015-0264-0

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Keywords

  • Foliation
  • Resolution of singularities
  • Monomial

Mathematics Subject Classification

  • 32S45
  • 14E15