Connectedness of the Fibers of a Liouvillian Function

Abstract

Let $D$ be a normal crossing divisor in a complex analytic manifold of dimension $n$, and let $\Omega$ be a closed logarithmic one-form, with poles on $D$. Under appropriate hypothesis, we prove the connectedness of the fibers for a primitive of $\Omega$ in “good” neighborhoods of $D$. We deduce the connectedness of the fibers of Liouvillian functions of type $f=f_1^{{\lambda }_1}\cdots f_p^{{\lambda }_p}$ at the origin of ${\mathbf{C}}^n$, under two conditions: the first extends the usual notion that “$f$ is not a power”. The second excludes certain meromorphic functions.