Mixed motivic sheaves, category theory, and study of cycle complexes

拟DG范畴是主要研究者给出的DG范畴的推广，是混合动机层理论中的一个概念。发展了拟DG范畴的基本理论。主要的特征是拟DG范畴的基本公理，值在拟DG范畴中的C-图的概念，函数复形的可加性，C-图的拟DG范畴的构造，以及C-图的拟DG范畴的同伦范畴具有三角范畴结构的证明。当构成概念的一部分的函数复形（依赖于n个对象）在每个变量中是可加的时，证明了对于C-图的函数复形也具有相同的性质。特别地，我们给出了Chow上同调与同调重合的条件，给出了拟投射簇的模群和高阶Chow群的爆破公式。

Quasi DG category, a generalization of DG category given by the principal researcher, is a notion that appears in the theory of mixed motivic sheaves. We developed the basic theory of quasi DG categories. The mail features are basic axioms of a quasi DG category, the notion of C-diagrams with values in a quasi DG category, additivity of the function complexes, the construction of the quasi DG category of C-diagrams, and the proof that the homotopy category of the quasi DG category of C-diagrams has the structure of a triangulated category.In particular, when the function complex (depending on n objects) that constitutes part of the notion, is additive in each variable, the same property was proven to hold for the function complex for C-diagrams.We described the Chow cohomology of an algebraic surface using its resolution of singularities. In particular, we gave a condition for Chow cohomology and homology to coincide.We showed the blow-up formula for the motives and higher Chow groups of a quasi-projective variety.