Integrable deformations of local analytic fibrations with singularities

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, {\em 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 $ω= df + f η$ 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 $ω= adf + f η$ 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 $ω$ and $f$ at the origin satisfy $ν(ω) = ν(df)$. In the case of an isolated singularity for $(f=0)$ the writing $ω= adf + f η$ 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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