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Degeneration of algebraic varieties and log geometry

代数簇的退化和对数几何

基本信息

批准号：
14340010
负责人：
KATO Kazuya
金额：
$ 2.56万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-14340010/
关键词：
Hodge structure Clarifying space SL(2)-orbit Log geometry log abelian variety ramificaiton theory L-function Iwasawa theory アーベル多様体複素トーラス玉河数スキーム分岐退化導手 Rimemann-Roch l進層 nilpotent orbit

项目摘要

1. With S.Usui, I studied classifying spaces of polarized log Hodge structures. We published the paper on the space of SL_2-orbits, and completed the paper "Classifying spaces of degenerating polarized Hodge structures". The last paper has 197 pages in Tex typing, containing various new ideas in log Hodge theory.Using the new method of log mixed Hodge structures, C.Nakayama, T.Kajiwara and I improved our theory of analytic log abelian varieties and discovered the theory of log complex tori, and wrote joint papers. "Logarithmic abelian varieties, Analytic theory" and "Analytic log Picard varieties".C.Nakayama, L.Illusie, and I completed a paper "Analytic ket sites" on another study of degeneration of Hodge structures using the analytic Kummer etale site.2. T.Saito and I found a good ramification theory of schemes by using logarithmic intersection theory, and wrote a paper "Ramification theory for varieties over perfect fields" in which we gave a generalization of Grothendieck-Ogg-Shafarevivch formula to higher dimensions.3. The log geometry is related to the theory of L-functions via applications to p-adic Hodge theory. In this subject, I wrote a joint paper with T.Fukaya "A formulation of conjectures on p-adic zeta functions in non-commutative Iwasawa theory" and another joint paper with J.Coates, T.Fukaya, R.Sujatha, and O.Venjakob "The GL_2 main conjecture for elliptic curves without complex multiplication" on p-adic zeta functions in non-commutative Iwasawa theory. In these papers, we succeded to formulate the Iwasawa main conjectures in non-commutative Iwasawa theory.

1.我和S. J.一起研究了极化对数霍奇结构的分类空间。我们发表了关于SL_2-轨道空间的论文，并完成了“退化极化Hodge结构空间的分类”这篇论文。最后一篇论文有197页是Tex类型的，包含了log Hodge理论中的各种新思想，利用log混合Hodge结构的新方法，C.Nakayama，T.Kajiwara和我改进了我们的解析log Abel簇的理论，发现了log复环面理论，并写了联合论文。“对数阿贝尔簇，解析理论”和“解析对数皮卡德簇”C.中山，L.Illusie，我完成了一篇论文“解析ket网站”的另一个研究退化的霍奇结构使用的分析库默etale网站。我和T.Saito利用对数交理论找到了一个很好的格式的分歧理论，并写了一篇论文“完备域上簇的分歧理论”，其中我们给出了Grothendieck-Ogg-Shafarevich公式在高维上的推广.对数几何通过应用于p进霍奇理论而与L-函数理论相关。在这个主题中，我写了一篇与T.福谷的联合论文“非交换岩泽理论中p-adic zeta函数的公式”，以及另一篇与J.科茨，T.福谷，R.Sujatha和O.Venjakob的联合论文“非交换岩泽理论中p-adic zeta函数的GL_2主猜想”。在这些论文中，我们成功地制定了非交换岩泽理论的岩泽主要结构。