Arkiv för Matematik

We study analytic integrable deformations of the germ of a holomorphic foliation given by $df=0$ at the origin $0\in {\mathbb{C}}^n,n\ge 3$. 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 $\le n\mathbf{-}3$ 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 $\mathrm{\omega }=df\mathbf{+}f\mathrm{\eta }$ 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 $\mathrm{\omega }=adf\mathbf{+}f\mathrm{\eta }$ admits a holomorphic first integral provided that: (i) $X_f:(f=0)$ is irreducible with an isolated singularity at the origin $0\in {\mathbb{C}}^n,n\ge 3$; (ii) the algebraic multiplicities of ω and f at the origin satisfy $\mathrm{\nu }(\mathrm{\omega })=\mathrm{\nu }(df)$. In the case of an isolated singularity for $(f=0)$ the writing $\mathrm{\omega }=adf\mathbf{+}f\mathrm{\eta }$ 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.