Non-abelian Hodge theory for positive characteristic and crystalline structures

这个项目的目的是将非阿贝尔Hodge理论推广到正特征或算术簇的代数簇。为此，我一直在研究一些使用更高范畴理论构建晶体拓朴的一般理论。高阶TOPO理论似乎是描述真实结晶TOPO的合适语言。许多研究者(C.Simson，B.Then，J.Lurie)在这十多年里构建了各种更高范畴理论(甚至更高的Topos理论)。直到去年，我主要使用辛普森的方法，但在阅读了J.Lurie最近的工作后，我意识到在我们的构造中使用Lurie的高等拓扑理论是合适的。这个项目将在不久的将来对算术几何和数学物理产生重要的影响。另一方面，要描述更高的晶体结构，似乎不能用比整数环上的结构公式来描述。N·杜罗夫最近的一项工作是用一个预期存在的元素F_1来解释这个场。如果能在F_1场上建立更高结晶度的TOOS理论，那么它应该是有用的。构建这样的理论是这个项目的下一个主题。

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.