Applications of the theory of mixed motifs

1. Barthel, Brasselet, Fiesler， Gabber和Kaup的一个定理证明了一个代数环在Borel-Moore同调中的循环类可以提升为一个交上同调中的循环类。我们根据分解定理给出了这个定理的另一种证明。我们进一步构造了这个定理的一个动机类比，并证明了它在代数循环的“标准”猜想下是成立的（由于Grothendieck， Bloch-Beilinson-Murre和Beilinson-Soule）。给出了交点周群的定义，它是交点上同调的一个动力类比。我们在一篇论文中详细阐述了这一理论。我们证明了以曲面为总空间的Lefschetz铅笔的动机分解定理（分解定理的动机类比）是成立的。对于任何维度的Lefschetz铅笔，在某些假设下也是如此。我们写了一篇论文，在一个基本品种的混合动力轮轴的三角形类别的结构。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.