Deligne-Hodge-DeRham theory with coefficients

Abstract

Let $\mathcal{L}$ be a variation of Hodge structures on the complement $X^{\ast }$ of a normal crossing divisor (NCD) $Y$ in a smooth analytic variety $X$ and let $j:X^{\ast }=X-Y\to X$ denotes the open embedding. The purpose of this paper is to describe the weight filtration $W$ on a combinatorial logarithmic complex computing the (higher) direct image ${\mathbf{j}}_{\ast }\mathcal{L}$, underlying a mixed Hodge complex when $X$ is proper, proving in this way the results in the note [14] generalizing the constant coefficients case. When 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 $Y$ can be described by these logarithmic techniques and a comparison theorem shows that the filtration coincides with the weight defined by the logarithm of the monodromy which provides the link with various results on the subject.

33 pages

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