Arkiv för Matematik

Integrable deformations of local analytic fibrations with singularities

Dominique Cerveau and Bruno Scárdua

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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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Ark. Mat., Volume 56, Number 1 (2018), 33-44.

Received: 18 May 2016
First available in Project Euclid: 19 June 2019

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Cerveau, Dominique; Scárdua, Bruno. Integrable deformations of local analytic fibrations with singularities. Ark. Mat. 56 (2018), no. 1, 33--44. doi:10.4310/ARKIV.2018.v56.n1.a3.

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