Integrable deformations of local analytic fibrations with singularities

TL;DR

This paper investigates conditions under which analytic deformations of holomorphic foliations with singularities admit holomorphic first integrals, especially focusing on cases with normal crossings and isolated singularities.

Contribution

It proves that under certain geometric and singularity conditions, integrable deformations of holomorphic foliations also possess holomorphic first integrals, extending previous results to more general singular settings.

Findings

Deformations with normal crossings singularities admit holomorphic first integrals.

Holomorphic first integrals exist for certain quasi-homogeneous cases.

Existence of first integrals is guaranteed when the singularity is isolated and multiplicities match.

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, {\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 $\omega = df + f \eta$ where $f$ is quasi-homogeneous. Under the same hypotheses for $X_f : (f=0)$ we prove that $\omega$ 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.