Theory of motives and algebraic cycles

1. 变量的动机：设D(k)为域k上的混合动机的范畴。我们从拟射影变量范畴中得到一个函子到D(k)。建筑采用立方体超分辨率方法。动机分解定理：构造和证明拓扑分解定理（Beilinson， Bernstein和Deligne）的动机类比是一个有趣的问题。对于希尔伯特模品种上的阿贝尔品种的泛族，我们证明了期望动力分解的存在性，并由此推导出纤维品种的Grothendieck-Murre猜想。链上的同源对应：考虑同源对应和它们的组成是众所周知的。我们在链的层次上考虑过。即给出具有上同调的阿贝尔群的复形，并以复形映射的形式生成复合映射。将该构造应用于由混合动机产生上同调实现函子。混合动机：我们在一个拟射影变量上勾画了混合动机的三角范畴的构造。

1. Motives of varieties: Let D(k) be the category of mixed motives over a field k. We produced a functor from the category of quasi-projective varieties into D(k). The construction uses the method of cubical hyperresolution.2. Motivic decomposition theorem: It is of interest to formulate and prove the motivic analogue of the topological decomposition theorem (of Beilinson, Bernstein and Deligne). In the case of the universal family of abelian varieties over the Hilbert modular variety, we showed the existence of the expected motivic decomposition, and deduced from it the Grothendieck-Murre conjecture for the fiber variety.3. Homology correspondence at chain level: It is well-known to consider homological correspondences and their compositions. We considered this at the chain level. Namely we gave a complex of abelian groups which gives cohomology, and produced the composition map as a map of complexes. This construction is applied to produce the cohomology realization functor from mixed motives.4. Mixed motivic sheaves: We sketched the construction of the triangulated category of mixed motives over a quasi-projective variety.