Derived categories in arithmetic and algebraic geometry

算术和代数几何的派生范畴

• 批准号: RGPIN-2022-03461
• 负责人: Honigs, Katrina
• 金额: $ 1.68万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758522
• 关键词: Derived; categories; arithmetic; algebraic; geometry

There are many unresolved questions about systems of polynomial equations, which are called varieties: How do we decide if they have solutions whose coordinates are all integers? How do we decide if two varieties have the same set of solutions? Directly computing answers to these questions is often impossible or prohibitively time-consuming, so mathematicians convert polynomials into other objects that are easier to analyze. One such object, called the derived category, has shown promising initial results in answering these questions, but it has only been studied in this regard in the last decade. More tools for analyzing the derived category and further knowledge of what it can detect are needed to take full advantage of this new technique. Polynomials may also be converted into cohomology theories. The l-adic étale cohomology theory was introduced by Grothendieck in 1960 in order to prove the Weil conjectures, and is considered indispensible. The derived category is a more refined measure since it can distinguish varieties that this theory cannot, but in general the relationship between cohomology and derived categories is unknown. To access the full utility of the derived category, it is important to uncover its relationship to this well-established theory. The most interesting varieties in regard to the questions above are those of Kodaira dimension 0 -- a tipping point between concave and convex. These include elliptic curves, which are well-known for theoretical applications like the proof of Fermat's last theorem as well as practical applications in cryptography. My long-term goal is to use and develop the derived category in order to classify varieties of Kodaira dimension 0. Short-term objectives: 1. Understand a portion of the l-adic étale cohomology of hyperkähler 4-folds of Kummer type, using a construction with connections to the derived category. 2. Develop tools for comparing derived categories of stacks over fields of positive characteristic. 3. Prove derived Torelli-type theorems, particularly for Enriques surfaces over fields of characteristic 2. 4. Explore whether the derived category detects the existence of points with integer coordinates on Calabi-Yau 3-folds. This program will provide new tools and insight to the study of derived categories, and more significantly, will bring the fruits of derived geometry to the larger community of algebraic geometers and number theorists. The derived category has already proven a useful setting for studying many major topics in algebraic geometry, including deformations, moduli, Torelli theorems, rational points, and mirror symmetry, which has applications in physics. This proposal will also support the training of highly qualified personnel in algebraic geometry, enhancing the mathematical community in Canada.

关于多项式方程组，有许多尚未解决的问题，这些问题被称为变种：我们如何确定它们是否有坐标都是整数的解？如何判断两个变量是否有相同的解集？直接计算这些问题的答案通常是不可能的，或者非常耗时，因此数学家将多项式转换为其他更容易分析的对象。 一个这样的对象，所谓的派生类别，已经显示出有希望的初步结果，在回答这些问题，但它只是在过去十年中在这方面进行了研究。为了充分利用这项新技术，需要更多的工具来分析派生类别，并进一步了解它可以检测到什么。多项式也可以转化为上同调理论。l-adic étale上同调理论是由Grothendieck在1960年为了证明Weil定理而引入的，并且被认为是必不可少的。导出范畴是一个更精确的度量，因为它可以区分这个理论不能区分的变种，但一般来说，上同调和导出范畴之间的关系是未知的。为了获得派生范畴的全部效用，重要的是要揭示它与这个已确立的理论的关系。关于上述问题，最有趣的变化是那些科代拉0维--一个介于凹和凸之间的临界点。其中包括椭圆曲线，它在理论上的应用，如费马最后定理的证明以及密码学的实际应用中是众所周知的。我的长期目标是使用和发展的派生类别，以分类品种的科代拉维度0。 短期目标：1。使用与导出范畴有联系的构造，理解库默型超kähler 4-折叠的l-adic étale上同调的一部分。2.开发工具，用于比较积极特征领域的堆栈派生类别。 3.证明导出的Torelli型定理，特别是特征为2的域上的Enriques曲面。4.探索导出的类别是否检测到卡-丘3-折叠上具有整数坐标的点的存在。该计划将为派生范畴的研究提供新的工具和见解，更重要的是，将把派生几何的成果带给更大的代数几何学家和数论家社区。导出范畴已经被证明是研究代数几何中许多主要课题的有用设置，包括变形、模、Torelli定理、有理点和镜像对称，这些在物理学中有应用。这项建议还将支持培训代数几何方面的高素质人员，加强加拿大的数学界。