Algebraic Geometry Approaching Characteristic p

这个研究项目将有助于代数几何（研究多变量多项式方程的解）。代数几何是数学的许多应用的基础--如物理学、计算机科学和最近的生物学--并且在古希腊人的工作中已经研究过，如果不是更早的话。然而，这一主题在最近经历了一次壮观的复兴，这主要归功于亚历山大格罗滕迪克和他的合作者的工作。从这项工作中获得的主要见解之一是，对多项式方程的“近似”解（在模块或“发条”算术的意义上）的研究揭示了“真实”解的相当多的光。在这个项目中，研究人员将扩大技术可用于研究“近似”的解决方案，并有助于弥合“近似”和“真正”的解决方案之间的差距差距。本计画的目标是研究正特征及p-adic设定下的代数几何。首先，与Morrow和Scholze一起，PI打算给出与p-adic流形相关联的Breuil-Kisin模（或p-adic shtukas）的代数几何构造;这可以被视为Hodge结构与复流形相关联的p-adic模拟，并将对代数簇的上同调产生新的影响。其次，与Scholze一起，PI计划研究完美方案上的代数几何，以及它与h-拓扑的关系;向量丛的下降结果将用于构造某些在结构层上不是线性的层的复形的行列式线丛，这将给出代数圈的新来源。最后，PI将与Esnault和Kindler一起研究Gieseker猜想（将基本群与特征p中的D-模联系起来）和Grothendieck猜想（将分层上同调与相干上同调的完善联系起来）的扩展。贯穿所有这些项目的中心主题是系统地使用“大”对象（如pro-etale和h-拓扑，完美和衍生计划）来研究“小”对象（如上同调的挠，或Grassmanians上的线丛）。

This research project will contribute to algebraic geometry (the study of solutions of polynomial equations in several variables). Algebraic geometry is fundamental to many applications of mathematics -- such as those to physics, computer science and, more recently, biology -- and was already studied in the work of the ancient Greeks, if not earlier. However, the subject experienced a spectacular revival in the recent past, thanks largely to the work of Alexander Grothendieck and his collaborators. One of the major insights gained from this work was that the study of "approximate" solutions (in the sense of modular or "clockwork" arithmetic) to polynomial equations sheds quite a bit of light on the "true" solutions. In this project, the investigator will expand the techniques available to study "approximate" solutions, and contribute to bridging the gap between "approximate" and "true" solutions. The goal of this project is to study algebraic geometry in the positive characteristic and p-adic settings. First, with Morrow and Scholze, the PI intends to give an algebro-geometric construction of Breuil-Kisin modules (or p-adic shtukas) associated to p-adic manifolds; this may be viewed as a p-adic analogue of the association of a Hodge structure to a complex manifold, and would have new consequences for the cohomology of algebraic varieties. Secondly, with Scholze, the PI plans to study algebraic geometry over perfect schemes, and its relation to the h-topology; the sought-for descent result for vector bundles would be used to construct determinant line bundles for certain complexes of sheaves that are not linear over the structure sheaf, which will give a new source of algebraic cycles. Finally, with Esnault and Kindler, the PI will work on extensions of Gieseker conjecture (relating fundamental groups to D-modules in characteristic p) and the Grothendieck conjecture (relating stratifying cohomology with the perfection of coherent cohomology). The central theme running through all these projects is the systematic use of "large" objects (such as the pro-etale and h- topologies, perfect and derived schemes) to study "small" objects (such as torsion in cohomology, or line bundles on Grassmanians).