Irregularity of algebraic differential equations on varieties in positive characteristic

这个项目的建议是关于正特征域上光滑簇X上的代数微分方程组。这样的系统也被称为“分层捆绑”。复流形上对应的对象是具有平坦联系的向量丛，但在正特征中出现了新的现象。具体地说，层丛的边界行为与L局域系统的分支理论非常相似。本项目的总体目标是进一步发展这一类比。2012年，P.Deligne证明了固定秩不可约的L-进局部系统的一个有限定理，该系统的分支有界于无穷个固定因子。这项提议的具体目标是定义和研究“由无限支承的除数界定的不规则的分层丛”的概念。这个项目的一个成功成果将是能够表述关于分层丛的陈述，这类似于Deligne的定理。为了实现这一点，我首先打算假设X是一条曲线，并在形式上研究围绕无穷远点的分层丛的结构。在这里，我计划构造一个度量其奇点性质的不变量，它应该类似于扁平连接的不规则数和L-ADDY局域系统的天鹅导体。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。

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.