Cohomological study of arithmetic varieties

正如我在项目建议书中所写的那样，我研究了l进轴和分支基过滤的黎曼-洛克公式。黎曼-罗氏公式的计算结果比预期的要好得多。在与其中一位研究者Kazuya Kato的联合研究中，我提出并证明了一个任意维度的Grothendieck-Ogg-Shafarevich公式，该公式根据l进束沿边界的分支计算欧拉数。该公式在60年代已被证明适用于曲线，但仍有待推广到高维。首先，我们将Swan类定义为边界上支持的0循环类，作为l进轴的分枝不变量。通过建立开变的Lefschetz迹公式，证明了其度计算欧拉数。这个证明写在一篇论文中，被《数学年鉴》接受发表。在与Kato的一项联合研究中，我们还获得了局部场上光滑品种上l进束的导体公式。我们利用了在研究布洛赫导体公式中引入的k理论的局域交理论和对数Lefschetz迹公式的开变的推广。我们现在正在准备一篇关于证明的论文。对于正特征变化上的l进轴，定义了特征类。我在与Ahmed Abbes的联合研究中研究了它他是这个研究项目的外国合作者。首先，我们与Swan班建立了关系。我们还将细化定义为在闭合光纤上支持的上同源类。进一步，在秩一的情况下，我们证明了它与之前Kato定义的0循环类相同。在证明过程中，我们还证明了在等特征情况下，对任意局部域所定义的分支群滤除可以推导出先前由加藤在阿贝尔化商上所定义的滤除。

As I wrote in the project proposal, I studied Riemann-Roch formulas for l-adic sheaves and filtration by ramification groups. The results obtained on Riemann-Roch formulas are much better than expected.I formulated and proved a Grothendieck-Ogg-Shafarevich formula in an arbitrary dimension, that computes the Euler number of an l-adic sheaf in terms of its ramification along the boundary, in a joint research with one of the investigator, Kazuya Kato. The formula was proved in 60's for curves but remained to be generalized to higher dimension. First, we define the Swan class as a 0-cycle class supported on the boundary as a ramification invariant of an l-adic sheaf. By establishing a Lefschetz trace formula for open varieties, we prove that its degree computes the Euler number. The proof is written in a paper accepted for publication at the Annales of Mathematics.We also obtained a conductor formula for a l-adic sheaf on a smooth variety over a local field, in a joint research with Kato. We use the K-theoretic localized intersection theory and a generalization to the open varieties of log Lefschetz trace formula, that we introduced in the study of the conductor formula of Bloch. We are now preparing a paper on the proof.For an l-adic sheaf on a variety in positive characteristic, the characteristic class is defined. I studied it in a joint research with Ahmed Abbes who is a foreign collaborator of this research project. First, we established a relation with the Swan class. We also defined a refinement as a cohomology class supported on the closed fiber. Further, in the rank one case, we proved that it is the same as the 0-cycle class defined by Kato previously. In the course of proof, we have also established that the filtration by ramification groups defined for an arbitrary local field induces the filtration previously defined by Kato on the abelianized quotient in the equal characteristic case.