Mixed Hodge Complexes on Algebraic Varieties

The notion of cohomological mixed Hodge complex was introduced by Deligne [4] as a tool to construct the mixed Hodge structure on the cohomology of complex algebraic varieties. This is defined by using logarithmic forms and simplicial resolutions of varieties. It is called cohomological, because its conditions are imposed only after taking the global section functor. Forgetting the rational (or integral) structure and also the weight filtration, a refinement of this notion was obtained by Du Bois [6] (following Deligne’s idea). For a complex algebraic variety X and a closed subvariety D of X , he introduced a filtered complex (Ω̃•X〈D〉, F ) on X , whose differential is given by differential operators of order at most one. It is well defined in a certain triangulated category, and gives the Hodge filtration of Deligne’s mixed Hodge structure on the cohomology of X\D by taking the hypercohomology if X is proper. On the other hand, the notion of mixed Hodge Module is introduced in [14], [15]. This gives also a mixed Hodge structure of the cohomology of a complex algebraic variety (without using a simplicial resolution). It is generally considered that the theory of mixed Hodge Modules is a generalization of Deligne’s mixed Hodge theory. There were, however, some gaps between the two theories. One is that the theory of mixed Hodge Modules does not work on simplicial schemes, because the direct images and the pull-backs are defined only in the derived categories, and not in the level of complexes. In particular, it was not clear whether the two mixed structures obtained coincide in general (except when the variety is embeddable into a smooth variety). The difficulty comes from the fact that the double complex construction associated with a cosimplicial complex on a simplicial scheme does not work in the derived category, because d is not zero, but only homotopic to zero. Note that this difficulty is not solved in this paper, nor seems to be solved soon. In fact, the problem is avoided in this paper, because we do not have to construct a complex of mixed Hodge Modules on the simplicial scheme in the proof of (0.2). Another problem is the difference in the systems of weight. In Deligne’s theory [4], the weight filtrationW on a mixed Hodge complex K is defined so that the weight of HGrk K is i + k. This weight filtration W is called the standard weight. On the other hand, the weight of HiGr ′ k M is k for a complex of mixed Hodge structures M (or a mixed Hodge complex in the sense of [1, 3.2]) with the weight filtration W . This weight filtration W ′ is called the absolute weight. For a complex of mixed Hodge structures, the passage from W ′ to W is done by taking the convolution of W ′ and σ (the filtration “bête” in [4]). But