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Mixed Hodge realization of triangulated mixed motives and comparison of triangulated categories of motives


In the 1960s Grothendieck made conjectures as to the existence of a universal framework for cohomology of algebraic varieties called the category of mixed motives. During the 1980's the conjectural formalism was enriched by conjectures of Deligne, Beilinson, Bloch and others. In the 1990's V. Voevodsky, M. Levine and M. Hanamura have given several constructions of triangulated categories that have many of the expected properties of the derived category of mixed motives. As part of Grothendieck and Beilinson 's conjectural formalism comparison of those theories with a particular derived Bloch-Ogus theory should be made via realization functors.

The aim of this proposal is to construct the Hodge realization functor for Voevodsky's triangulated category of mixed motives over a smooth scheme of finite type over C and to construct equivalences between the triangulated categories of Voevodsky, Levine and Hanamura.

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