Non-abelian Hodge theory for positive characteristic and crystalline structures

本课题的目的是将非阿贝尔霍奇理论推广到正特征的代数变量或算术变量。为此，我一直在研究使用高等范畴理论构建晶体拓扑的一般理论。高等拓扑理论似乎是描述真实晶体拓扑的合适语言。近一二十年来，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Simpson的方法，但在阅读了J. Lurie最近的工作后，我意识到Lurie的高等拓扑理论适用于我们的结构。本课题在不久的将来将对算术几何和数学物理具有重要意义。另一方面，要描述更高级的晶体结构，似乎在有理整数环上的结构的公式是不充分的。N. Durov最近的一项工作用一个元素F_1来构造场，这个元素F_1是预期存在的。如果能在F_1场上建立更高的晶体拓扑理论，它应该是有用的。构建这样的理论是本项目的下一个主题。

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.