Arkiv för Matematik

Volume 56 (2018)

Number 1

Integrable deformations of local analytic fibrations with singularities

Pages: 33 – 44

DOI: http://dx.doi.org/10.4310/ARKIV.2018.v56.n1.a3

Authors

Dominique Cerveau (IRMAR, Université de Rennes I, Rennes, France)

Bruno Scárdua (Instituto de Matemática, Universidade Federal do Rio de Janeiro, Brazil)

Abstract

We study analytic integrable deformations of the germ of a holomorphic foliation given by $df=0$ at the origin $0 \in \mathbb{C}^n , n \geq 3$. 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 $\leq n-3$ analytic subset $Y \subset X$, the analytic hypersurface $X_f : (f=0)$ 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 $\omega = df + f \eta$ where $f$ is quasi-homogeneous. Under the same hypotheses for $X_f : (f=0)$ we prove that ω also admits a holomorphic first integral. Finally, we conclude that an integrable germ $\omega = adf + f \eta$ admits a holomorphic first integral provided that: (i) $X_f : (f=0)$ is irreducible with an isolated singularity at the origin $0 \in \mathbb{C}_n , n \geq 3$; (ii) the algebraic multiplicities of $\omega$ and $f$ at the origin satisfy $\nu (\omega) = \nu (df)$. In the case of an isolated singularity for $(f=0)$ the writing $\omega = adf + f \eta$ 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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Received 18 May 2016