  Let $X$ be a compact Riemann surface, $\Sigma$ a finite set of points and $M
= X\setminus \Sigma$. We study the $L^2$ cohomology of a polarized complex
variation of Hodge structure on a Galois covering of the Riemann surface of
finite type $M$. In this article we treat the case when the covering comes from
a branched covering of $X$, and where $M$ is endowed with a metric asymptotic
to a Poincar\'e metric. We prove that after tensorisation with the algebra of
affiliated operators, the $L^2$ cohomology admits a pure Hodge structure.

Algebraic Geometry