Algebraic Geometry Approaching Characteristic p

这项研究项目将对代数几何(研究多变量多项式方程的解)做出贡献。代数几何是许多数学应用的基础--例如在物理、计算机科学和最近的生物学中的应用--古希腊人的工作中已经研究过代数几何，如果不是更早的话。然而，这个主题在最近经历了壮观的复兴，这在很大程度上要归功于亚历山大·格罗森迪克和他的合作者的工作。从这项工作中获得的一个主要见解是，研究多项式方程的“近似”解(在模算法或“发条”算法的意义上)可以揭示出相当多的“真实”解。在这个项目中，研究人员将扩展可用于研究“近似”解的技术，并为弥合“近似”和“真实”解之间的差距做出贡献。这个项目的目标是研究正特征和p-进环境下的代数几何。首先，与Morrow和Scholze一起，PI打算给出与p-进流形相关的Breuil-Kisin模(或p-进shtukas)的代数几何构造；这可以被看作是Hodge结构到复流形的结合的p-进类似，并且将对代数簇的上同调有新的结果。其次，PI计划与Scholze一起研究完美方案上的代数几何及其与h-拓扑的关系；向量丛的下降结果将被用来构造某些在结构层上不是线性的层的复形的行列式线丛，这将给出一个新的代数圈的来源。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(将分层上同调与凝聚上同调的完备性联系起来)的推广。贯穿所有这些项目的中心主题是系统地使用“大”对象(如原拓扑和h-拓扑、完美和衍生方案)来研究“小”对象(如上同调中的扭转或Grassmanian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).