Hodge-DeRham theory with degenerating coefficients

Let ${\cal L}$ be a local system on the complement $X^{\star}$ of a normal crossing divisor (NCD) $Y$ in a smooth analytic variety $X$ and let $j: X^{\star} = X - Y \to X$ denotes the open embedding. The purpose of this paper is to describe a weight filtration $W$ on the direct image ${\bf j}_{\star}{\cal L}$ and in case a morphism $f: X \to D$ to a complex disc is given with $Y = f^{-1}(0)$, the weight filtration on the complex of nearby cocycles $\Psi_f ({\cal L})$ on $Y$. A comparison theorem shows that the filtration coincides with the weight defined by the logarithm of the monodromy and provides the link with various results on the subject.