Independence of l and local terms

l 和局部项的独立性

• 批准号: 1303173
• 负责人: Martin Olsson
• 金额: $ 18万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1303173&HistoricalAwards=false
• 关键词: Independence; local; terms

This proposal focuses on questions that arise in the study of algebraic varieties and their cohomology, many of which have arithmetic applications. Earlier work of the PI and others has led to substantial recent developments in our understanding of operations on cohomology arising from correspondences, and also to a new suite of problems which the PI will investigate. The PI will study questions of independence of l for actions of correspondences on various cohomology theories (specifically etale, intersection, and crystalline cohomology), continue his work on localized chern classes, and study global questions using trace formulas. These lines of investigation are motivated by the theory of motives, which predicts independence of l results for certain operators on cohomology and suggests that their traces should have geometric significance.In addition, the PI will continue earlier work researching algebraic stacks, abelian varieties, log geometry, Fourier-Mukai transforms, and moduli spaces. The PI will continue to advise Ph.D. students working on projects related to the proposed research.The proposed work concerns basic geometric structures, such as cohomology, moduli spaces, and stacks, which lie at the core of algebraic geometry and its interactions with other fields, including number theory, representation theory, combinatorics, and practical applications. Roughly algebraic geometry is concerned with the study of geometric properties of algebraic varieties, which are solutions of systems of polynomial equations. Cohomology theories provide some of the most powerful techniques for the study of such varieties. For example, the Lefschetz trace formula can be used to estimate numbers of solutions of polynomials over finite fields, and more generally cohomology groups of algebraic varieties are one of the main sources of Galois representations, which are fundamental objects in modern number theory. The proposal addresses problems on cohomology theories arising in this context. Another fundamental approach to the study of algebraic varieties is through their classification and moduli spaces. The PI's work on moduli spaces and stacks will continue to advance our understanding of key moduli spaces and provide broadly applicable foundational tools.

该提案重点关注代数簇及其上同调研究中出现的问题，其中许多问题具有算术应用。 PI 和其他人的早期工作使我们对由对应关系产生的上同调运算的理解取得了重大进展，也导致了 PI 将研究的一系列新问题。 PI 将研究各种上同调理论（特别是 etale、交集和结晶上同调）的对应作用的 l 独立性问题，继续他在局部陈氏类上的工作，并使用迹公式研究全局问题。这些研究方向受到动机理论的推动，该理论预测了某些上同调算子的 l 结果的独立性，并表明它们的迹应该具有几何意义。此外，PI 将继续研究代数栈、阿贝尔簇、对数等早期工作。几何、傅里叶-穆凯变换和模空间。 PI 将继续为博士提供建议。学生正在从事与拟议研究相关的项目。拟议的工作涉及基本几何结构，例如上同调、模空间和堆栈，它们是代数几何的核心及其与其他领域的相互作用，包括数论、表示论、组合学和实际应用。粗略代数几何涉及代数簇的几何性质的研究，代数簇是多项式方程组的解。上同调理论为研究此类簇提供了一些最强大的技术。例如，Lefschetz 迹公式可用于估计有限域上多项式解的个数，更一般地，代数簇的上同调群是伽罗瓦表示的主要来源之一，而伽罗瓦表示是现代数论的基本对象。该提案解决了在此背景下出现的上同调理论的问题。研究代数簇的另一个基本方法是通过它们的分类和模空间。 PI 在模空间和堆栈方面的工作将继续增进我们对关键模空间的理解，并提供广泛适用的基础工具。