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Irregularity of algebraic differential equations on varieties in positive characteristic

正特征品种代数微分方程的不正则性

基本信息

批准号：
274476424
负责人：
Dr. Lars Kindler
金额：
--
依托单位：
Institut für Mathematik
依托单位国家：
德国
项目类别：
Research Fellowships
财政年份：
2015
资助国家：
德国
起止时间：
2014-12-31 至 2015-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/274476424?language=en
关键词：
Irregularity algebraic differential equations varieties

项目摘要

This project proposal is concerned with systems of algebraic differential equations on smooth varieties X over a field of positive characteristic. Such systems are also called "stratified bundles''. The corresponding objects on complex manifolds are vector bundles with flat connection, but in positive characteristic new phenomena appear. In particular, the boundary behavior of a stratified bundle bears close resemblance to the ramification theory of l-adic local systems.The overall goal of this project is to further develop this analogy. In 2012, P. Deligne proved a finiteness theorem for irreducible l-adic local systems of fixed rank, with ramification bounded by a fixed divisor supported at infinity. The concrete goal of this proposal is to define and study the notion of a "stratified bundle with irregularity bounded by a divisor supported at infinity''. One successful outcome of the project would be to be able to formulate a statement about stratified bundles, which is analogous to Deligne's theorem.To achieve this, I first intend to assume that X is a curve and to study the structure of a stratified bundle formally locally around a point at infinity. Here I plan to construct an invariant measuring the quality of its singularity, which should be analogous to the irregularity number of a flat connection and the Swan conductor of an l-adic local system. If X has higher dimension, I plan to define the notion of bounded irregularity by probing X with curves.

本计画的建议是关于正特征域上光滑簇X上的代数微分方程组。此类系统也称为“分层束”。复流形上对应的对象是具有平坦联络的向量丛，但在正特征上出现了新的现象。特别是，分层丛的边界行为与l-adic局部系统的分歧理论有着密切的相似之处。本项目的总体目标是进一步发展这种类比。2012年，P. Deligne证明了一个固定秩的不可约l-adic局部系统的有限性定理，其中分支由一个支持在无穷远的固定因子所限制。这个建议的具体目标是定义和研究“一个分层丛的概念，不规则的边界上的一个除数支持在无穷大”。这个项目的一个成功的结果是能够制定一个关于分层丛的声明，这是类似于德利涅定理。为了实现这一点，我首先打算假设X是一条曲线，并研究一个分层丛的结构，正式当地围绕一个点在无穷远。在这里，我计划构建一个不变量测量其奇异性的质量，这应该是类似于不规则数的平面连接和天鹅导体的l-adic本地系统。如果X有更高的维度，我计划通过用曲线探测X来定义有界不规则性的概念。