Cohomological study of arithmetic varieties

正如我在项目建议书中所写的，我研究了L的黎曼-罗克公式--二进轮和按分支群过滤。在Riemann-Roch公式上得到的结果比预期的要好得多。在与研究员Kazuya Kato的联合研究中，我建立并证明了任意维上的Grothendieck-Ogg-Shafarevich公式，该公式根据L-进制束在边界上的分支来计算其欧拉数。该公式在60年代的S中得到了证明，但仍有待推广到更高维。首先，我们将SWAN类定义为边界上支持的0-圈类，作为L-进制叶的分支不变量。通过建立开簇的Lefschetz迹公式，证明了它的次数是计算欧拉数的。这一证明写在一篇发表在《数学年鉴》上的论文中。我们还与加藤共同研究，得到了局部域上光滑簇上的L-进数层的一个导数公式。利用K-理论的局域交理论，推广了Bloch导数公式中引入的对数Lefschetz迹公式的开簇。我们正在准备一篇关于证明的论文.对于一个L-二进制的正特征簇，定义了它的特征类.我在与艾哈迈德·阿贝斯的一项联合研究中进行了研究，艾哈迈德·阿贝斯是这项研究项目的外国合作者。首先，我们和天鹅班建立了关系。我们还将精化定义为闭纤维上支撑的上同调类。进一步，在一阶情形下，我们证明了它与Kato以前定义的0-圈类相同。在证明过程中，我们还证明了在特征相等的情况下，定义在任意局部域上的分支群的滤子诱导出了加藤先前定义的关于商数的滤子。

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.