Algebraic Cycles and Higher Abel-Jacobi map

上同调是算术几何和代数几何中最重要的研究对象之一。例如，设K是数域，O_K是它的整数环。则K的理想类群和O_K中的单位群是方案Spec(O_K)的动机上同调.算术几何中的一个重要猜想是算术方案的动机上同调是有限的.这是上述例子的有限性结果的自然推广，这是经典数论中的一个基本事实。到目前为止，除了Spec(O_K)或有限域上的曲线外，关于这个问题的结果很少。在我们的研究中，我们证明了一个新的关于动机上同调的有限结果。为了说明一个结果，设X是Spec(O_K)上的正则射影平坦(算术情形)或有限域F上的射影光滑簇(几何情形)。第一个重要的观察结果是，X的某个动机上同调的有限性源于Kato关于整数q〓1的KH_q(X)的消失的猜想.这里KH_q(X)是附在X上的某个算术不变量.在X=Spec(O_K)的情形下的Kato猜想等价于数论中关于K的Brauer群的一个基本事实，这意味着K上中心单代数的哈斯原理.我们在奇点归结的假设下证明了几何情形下的Kato猜想.更准确地说，我们得到了如下结果：定理设X是有限域上的射影光滑簇。设γ〓1为整数。假设嵌入在F上光滑簇中的维维子簇〓_K的奇点分解，则对1〓q〓γ+2，KH_q(X)=0.当γ=2时，我们还成功地证明了上述意义下奇点的分解.从而无条件地得到了1〓q〓4的KH_q(X)=0，从而得到了X的动机上同调的一个新的有限性结果.

Motivic cohomology is one of the most significant objects to study in arithmetic and algebraic geometry. For example, let K be a number field and O_K be its ring of integers. Then the ideal class group of K and the group of units in O_K are motivic cohomology of the scheme Spec(O_K).An important conjecture in arithmetic geometry is finiteness of motivic cohomology of arithmetic schemes. This is a natural generalization of the finiteness result for the above examples, which is a fundamental fact in classical number theory. There have been very few results on the problem so far except the case of Spec(O_K) or a curve over a finite field.In our research we have proved a new finiteness result for motivic cohomology. To state a result, let X be either regular projective flat over Spec(O_K) (arithmetic case) or a projective smooth variety over a finite field F (geometric case). The first crucial observation is that the finiteness of a certain motivic cohomology of X follows from a conjecture of Kato on the vanishing of KH_q(X) for integers q〓1. Here KH_q(X) is a certain arithmetic invariant attached to X. The Kato conjecture in case X=Spec(O_K) is equivalent to a fundamental fact in number theory concerning the Brauer group of K, which implies the Hasse principle for central simple algebras over K.We have shown the Kato conjecture in geometric case under the assumption of resolution of singu-larities. To be more precise we have obtain the following:Theorem Let X be a projective smooth variety over a finite field. Let γ〓1 be an integer. Assume resolution of singularities for subvarieties of dimension〓_K embedded in a smooth variety over F. Then KH_q(X)=0 for 1〓q〓γ+2.We have also succeeded to show the resolution of singularities in the above sense in case γ=2. Thus we get KH_q(X)=0 for 1〓q〓4 unconditionally and it gives rise to a new finiteness result for motivic cohomology of X.