Hodge-DeRham theory with degenerating coefficients

Abstract

Let $\mathcal{L}$ 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 ${\mathbf{j}}_{\star }\mathcal{L}$ 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 ${\Psi }_f(\mathcal{L})$ 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.