Applications of the theory of mixed motifs

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1. For a variety with singularities, a theorem of Barthel, Brasselet, Fiesler, Gabber and Kaup asserts that the cycle class of an algebraic cycle in Borel-Moore homoloty can be lifted to a class in intersection cohomology. We gave an alternative proof of this theorem based on the decomposition theorem. Further we formulated a motivic analogue of this theorem, and proved it holds true under the "standard" conjectures on algebraic cycles (due to Grothendieck, Bloch-Beilinson-Murre, and Beilinson-Soule).2. We gave a definition of intersection Chow group, which is a motivic analogue of intersection cohomology. We gave a detailed account of this theory in a paper.3. We showed the motivic motivic decomposition theorem (motivic analogue of the decomposition theorem) holds for a Lefschetz pencil with a surface as the total space. The same holds under some hypotheses for a Lefschetz pencil of any dimension.4. We wrote a paper on the construction of the triagulted category of mixed motivic sheaves over a base variety. This generalizes our previously established theory over a field.