Integrable deformations of local analytic fibrations with singularities

Dominique Cerveau Université de Rennes I Bruno Scárdua Universidade Federal do Rio de Janeiro

Complex Variables and Complex Analysis mathscidoc:1912.43003

Arkiv for Matematik, 56, (1), 33-44, 2018
We study analytic integrable deformations of the germ of a holomorphic foliation given by df=0 at the origin 0∈Cn,n≥3. We consider the case where f is a germ of an irreducible and reduced holomorphic function. Our central hypotheses is that, outside of a dimension ≤n−3 analytic subset Y⊂X, the analytic hypersurface Xf:(f=0) has only normal crossings singularities. We then prove that, as germs, such deformations also exhibit a holomorphic first integral, depending analytically on the parameter of the deformation. This applies to the study of integrable germs writing as ω=df+fη where f is quasi-homogeneous. Under the same hypotheses for Xf:(f=0) we prove that ω also admits a holomorphic first integral. Finally, we conclude that an integrable germ ω=adf+fη admits a holomorphic first integral provided that: (i) Xf:(f=0) is irreducible with an isolated singularity at the origin 0∈Cn,n≥3; (ii) the algebraic multiplicities of ω and f at the origin satisfy ν(ω)=ν(df). In the case of an isolated singularity for (f=0) the writing ω=adf+fη is always assured so that we conclude the existence of a holomorphic first integral. Some questions related to Relative Cohomology are naturally considered and not all of them answered.
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@inproceedings{dominique2018integrable,
  title={Integrable deformations of local analytic fibrations with singularities},
  author={Dominique Cerveau, and Bruno Scárdua},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191204092305460006559},
  booktitle={Arkiv for Matematik},
  volume={56},
  number={1},
  pages={33-44},
  year={2018},
}
Dominique Cerveau, and Bruno Scárdua. Integrable deformations of local analytic fibrations with singularities. 2018. Vol. 56. In Arkiv for Matematik. pp.33-44. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191204092305460006559.
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