Connectedness of the Fibers of a Liouvillian Function

Abstract

Let D be a normal crossing divisor in a complex analytic manifold of dimension «, and let Q be a closed logarithmic one-form, with poles on D. Under appropriate hypothesis, we prove the connectedness of the fibers for a primitive of Q in good neighborhoods of D. We deduce the connectedness of the fibers of Liouvillian functions of type f=fil ••• //p at the origin of C, under two conditions: the first extends the usual notion that /is not a power. The second excludes certain meromorphic functions.