Non-abelian Hodge theory for positive characteristic and crystalline structures

正特征和晶体结构的非阿贝尔霍奇理论

• 批准号: 17540015
• 负责人: YOKOGAWA Koji
• 金额: $ 2.26万
• 依托单位: Ochanomizu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540015/
• 关键词: Algebra; Topology; Non-abelian Hodge theory; Homotopy Mathematics; Crystalline; Positive characteristic; Algebraic Geometry; Vector bundle

The purpose of this project is to extend non-abelian Hodge theory to algebraic varieties for positive characteristic or arithmetic varieties. For this purpose I have been studying general theories for some construction of crystalline topos using higher category theory. It seems that higher topos theory is a suitable language to describe real crystalline topoi. Various higher category theories (even higher topos theories) have been constructed in these ten twenty years by many researchers (C. Simson, B. Then, J. Lurie). Until last year I used mainly Simpson's methods, but after reading recent work of J. Lurie I realized that it is suitable to use Lurie's higher topos theory for our constructions. This project will be important for arithmetic geometry and mathematical physics in the near future. On the other hand, to describe higher crystalline structure, it seems that formulations of the structure over the ring of ratioinal integers are not adequate. A recent work of N. Durov construes the field with one element F_1 which has been expected to exist. If higher crystalline topos theory could be constructed over the field F_1, it should be useful. A construction of such theory is the next theme for this project.

该项目的目的是将非交换霍奇理论扩展到代数簇的正特征或算术簇。为此，我一直在研究使用更高范畴论构建晶体拓扑的一般理论。看来高等拓扑结构理论是描述真实晶体拓扑结构的合适语言。在这十二十年里，许多研究者（C. Simson、B. then、J. Lurie）构建了各种更高范畴理论（甚至更高拓扑理论）。直到去年我主要使用辛普森的方法，但在阅读了 J. Lurie 最近的著作之后，我意识到使用 Lurie 的更高拓扑理论来进行构建是合适的。该项目对于不久的将来的算术几何和数学物理将具有重要意义。另一方面，为了描述更高的晶体结构，有理整数环上的结构公式似乎还不够。 N. Durov 最近的一项工作用一个预计存在的元素 F_1 解释了该场。如果可以在 F_1 场上构建更高结晶拓扑理论，它应该是有用的。构建这样的理论是该项目的下一个主题。