Cohomology Theories for Algebraic Varieties

研究了最近由研究者和F.Morel构造的代数上同调理论，以及代数簇和方案的广义上同调理论，Morel-Voevodsky代数-同伦范畴中的有理等价关系，以及Asakura的算术Hodge上同调的一个改进。以Bloch的高阶Chow群为线索，建立了高次代数余边理论，考察了高次代数余边动机范畴，给出了代数高次椭圆亏格理论。此外，利用代数同伦范畴中的有理等价概念，研究了基元上同调到K-理论的上同调到谱序列到其他上同调理论的推广，并考虑了这个谱序列与Voevodsky的片谱序列之间的关系。最后，研究人员定义了Asakura的算术Hodge上同调的一个变体，并利用它构造了一个上同调理论的无穷序列，这些上同调理论猜想给出了一个越来越好的逼近动机上同调的方法。代数几何是利用代数和几何来研究方程的解的学科。例如，人们可以通过检查圆的方程或通过查看其几何性质来研究圆。代数拓扑学通过将代数不变量附加到空间来研究空间，这些不变量通常可以显式计算。研究者采用代数拓扑学中的方法和结构，然后对它们进行修改和提炼，使它们可以用来定义拓扑不变量的代数版本。这些新的代数不变量可用于研究方程解的微妙性质。

The investigator studies the theory of algebraic cobordism, recently constructed by the investigor and F. Morel, as well as generalized cohomology theories for algebraic varieties and schemes, the relation of rational equivalence in the Morel-Voevodsky algebraic-homotopy category, and a refinement of Asakura's arithmetic Hodge cohomology. The investigator attempts to construct a theory of higher algebraic cobordism, along the lines of Bloch's higher Chow groups, examines the category of cobordism motives, and tries to give a theory of algebraic higher elliptic genera. In addition, the investigator examines the generalization of the motivic cohomology to K-theoryspectral sequence to other cohomology theories on algebraic varieties, and considers the relationship of this spectral sequence to the slice spectral sequence of Voevodsky, using the notion of rational equivalence in the algebraic homotopy category. Finally the investigator defines a variation of Asakura's arithmetic Hodge cohomology and uses this to construct an infinitesequence of cohomology theories which conjecturely give a better and better approximation to motivic cohomology.The investigator's research involves a mixture of algebraic geometry and algebraic topology. Algebraic geometry is the study of solutions of equations using both algebra and geometry. For example, one can study a circle by examining its equation, or by looking at its geometric properties. Algebraic topology studies spaces by attaching algebraic invariants to them, invariants which can often be computed explicitly. The investigator takes methods and constructions in algebraic topology, and then modifies and refines them so that they can be used to define algebraic versions of the topological invariants. These new algebraic invariants are then applicable for studying subtle properties of solutions of equations.