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

Variation Of Mixed Hodge Structures

Variation of mixed Hodge structures(VMHS), introduced by P. Deligne, is a linear structure reflecting the geometry on cohomology of the fibers of an algebraic family, generalizing variation of Hodge structures for smooth proper families, introduced by P. Griffiths. Hence, it is a strong tool to study the variation of the geometric structure of fibers of a morphism. We describe here the degenerating properties of a VMHS of geometric origin and the existence of a relative monodromy filtration, as well the definition and properties of abstract admissible VMHS.Comment: 46 page