The question is inspired by Carnahan's comment in the following MO question

What is the intuition for $\mathbb{Q}^{ab}$ having cohomological dimension $1$?

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* ?

Thank you.