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.

关于多项式方程式系统有许多未解决的问题，这些问题称为品种​​：我们如何决定它们是否具有坐标都是整数的解决方案？我们如何决定两个品种是否具有相同的解决方案？直接计算这些问题的答案通常是不可能的或禁止的时间耗时，因此数学家将多项式转换为更容易分析的对象。一个称为派生类别的这样的对象在回答这些问题时显示了有望的初始结果，但是在过去的十年中，它才在这方面进行了研究。更多用于分析派生类别的工具，并进一步了解它可以检测到的东西，以充分利用这项新技术。多项式也可以转化为同胞理论。 L-ADIC典型的典型共同体学理论是由Grothendieck于1960年引入的，以证明Weil的猜想，并且被认为是不可或缺的。派生类别是一个更精致的测量，因为它可以区分该理论不能的品种，但总的来说，同时学与派生类别之间的关系尚不清楚。要访问派生类别的全部效用，重要的是要揭示其与这一公认理论的关系。关于上述问题的最有趣的品种是Kodaira Dimension 0 - 凹入和凸之间的转折点。包括椭圆形曲线，这些曲线以理论应用而闻名，例如Fermat的最后一个定理以及密码学中的实际应用。我的长期目标是使用和开发派生类别，以对Kodaira维度0的变化进行分类。短期目标：1。了解Hyperkähler4倍Kummer类型的L-Adicétale共同体的一部分，使用与派生类别的连接的结构。 2。开发用于比较积极特征领域的堆栈类别的工具。 3。证明了衍生的Torelli型定理，特别是对于特征性领域的Enriques表面2。4。探索派生类别是否检测到Calabi-yau上有整数坐标的点是否存在3倍。该计划将为衍生类别的研究提供新的工具和见解，更重要的是，将使派生的几何形状的成果带入更大的代数几何学和数字理论家的社区。派生类别已经证明是研究代数几何学的许多主要主题的有用设置，包括变形，模量，Torelli定理，理性点和镜像对称性，该主题在物理学中应用。该建议还将支持对代数几何学高素质人员的培训，从而增强了加拿大的数学社区。