Applications of the theory of mixed motifs

1.对于具有奇点的簇，Barthel，Schrielet，Fiesler，Gabber和Kaup的一个定理证明了Borel-Moore同调代数圈的圈类可以提升为交上同调代数圈类。我们根据分解定理给出了这个定理的另一种证明。此外，我们制定了一个motivic类似的这个定理，并证明了它成立下的“标准”的代数圈（由于格罗滕迪克，布洛赫贝林森Murre，和贝林森Soule）。给出了交Chow群的定义，它是交上同调的一个模似.我们在一篇论文中对这一理论作了详细的叙述.我们表明动机动机分解定理（动机类似的分解定理）举行的莱夫谢茨铅笔与表面的总空间。这同样适用于任何维度的Lefschetz铅笔的一些假设。我们写了一篇论文的建设的triagulted类别的混合motivic层的基础品种。这将我们先前建立的理论推广到一个领域。

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.