Deligne-Hodge-DeRham theory with coefficients
Abstract
Description
Let ${\cal L}$ be a variation of Hodge structures on the complement $X^{*}$ of a normal crossing divisor (NCD) $ Y$ in a smooth analytic variety $X$ and let $ j: X^{*} = 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 ${\bf j}_{*}{\cal 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 $Ψ_f ({\cal L})$ 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
33 pages