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（称为簇）有许多未解决的问题：我们如何确定它们是否具有坐标均为整数的解？我们如何确定两个簇是否具有相同的解集？通常是不可能的或非常耗时，因此数学家将多项式转换为其他更容易分析的对象，称为派生类别，在回答这些问题方面已经显示出有希望的初步结果，但仅在此进行了研究。需要更多的工具来分析派生范畴，并进一步了解它可以检测到的内容，以充分利用这项新技术。l-adic étale 上同调理论也被引入。格洛腾迪克 (Grothendieck) 于 1960 年为了证明韦尔猜想而提出，并被认为是必不可少的。派生范畴是一种更精细的衡量标准，因为它可以区分该理论无法区分的变体，但一般来说可以区分它们之间的关系。上同调和派生范畴是未知的。为了充分发挥派生范畴的作用，重要的是要揭示它与这个完善的理论的关系。关于上述问题，最有趣的变化是 Kodaira 维度 0 的变化——a。其中包括椭圆曲线，它因费马大定理的证明等理论应用以及密码学的实际应用而闻名。我的长期目标是使用和开发派生类别。以便对 Kodaira 维数 0 的变体进行分类。 短期目标： 1. 使用与派生类别相关的构造，了解 Kummer 型 hyperkähler 4 倍的 l-adic étale 上同调的一部分 2. 开发工具。用于比较正特征域上的堆栈的派生类别 3. 证明派生的 Torelli 型定理，特别是特征 2 域上的 Enriques 曲面。 4. 探索派生类别是否成立。检测 Calabi-Yau 3 重上整数坐标点的存在 该程序将为派生类别的研究提供新的工具和见解，更重要的是，将为更大的代数几何学家群体带来派生几何的成果。派生范畴已经被证明是研究代数几何中许多主要主题的有用设置，包括变形、模、托雷利定理、有理点和镜像对称性，这些在物理学中都有应用。该提案还将支持代数几何领域高素质人才的培训，从而增强加拿大的数学界。