Algebraic Geometry Approaching Characteristic p

代数几何逼近特征p

• 批准号: 1501461
• 负责人: Bhargav Bhatt
• 金额: $ 28.48万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501461&HistoricalAwards=false
• 关键词: Algebraic; Geometry; Approaching; Characteristic

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).

该研究项目将有助于代数几何（多个变量中多项式方程的解决方案的研究）。代数几何形状是许多数学应用的基础，例如物理学，计算机科学以及最近的生物学，并且已经在古希腊人的工作中进行了研究，即使不是更早的话。但是，该主题在最近的过去经历了壮观的复兴，这在很大程度上要归功于亚历山大·格罗伦迪克（Alexander Grothendieck）及其合作者的工作。从这项工作中获得的主要见解之一是，对多项式方程式的“近似”解决方案（在模块化或“发条”算术的意义上）的研究对“真实”解决方案有很多启示。在该项目中，研究人员将扩展可用于研究“近似”解决方案的技术，并有助于弥合“近似”和“真实”解决方案之间的差距。该项目的目的是研究积极特征和P-ADIC环境中的代数几何形状。首先，使用Morrow和Scholze，PI打算给出与P-Adic歧管相关的Breuil-Kisin模块（或P-Adic Shtukas）的代数几何结构；这可以看作是霍奇结构与复杂歧管的关联的P-ADIC类似物，并且会对代数品种的共同体产生新的影响。其次，在Scholze的情况下，PI计划研究对完美方案的代数几何形状及其与H-Topology的关系。矢量束的寻求下降结果将用于构建确定性线束的某些束带络合物，这些束系在结构支架上不是线性的，这将提供一个新的代数循环来源。最后，借助Esnault和Kindler，PI将致力于Gieseker猜想的扩展（将基本组与特征P中的D模块相关）和Grothendieck猜想（将共同体分层的共同体与相干共同体学的完善相关联）。贯穿所有这些项目的中心主题是系统地使用“大”对象（例如，主层和H-拓扑，完美和衍生的方案）来研究“小”对象（例如，同居中的扭转或在草药上的线条捆绑包）。