The question is inspired by Carnahan's comment in the following MO question
Precisely, Borel proved (not Grothendieck, I think, who proved the arithmetic version) that, when S is a punctured disk, and H is a variation of polarized Hodge structures on S, then the monodromy operator is quasi-unipotent. Are there any counter-examples when S is a complex annulus? If yes, any examples of geometric origin? Namely f: X ---> S is a holomorphic family of projective manifolds ... .
To get started, what is the universal covering (as a complex manifold) of an annulus a < |z| < b ?