Incompressibility of algebraic varieties

代数簇的不可压缩性

• 批准号: RGPIN-2014-05369
• 负责人: Karpenko, Nikita
• 金额: $ 2.48万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611598
• 关键词: Incompressibility; algebraic; varieties

The main thrust of the proposal is to study numerical and discrete invariants of torsors and projective homogeneous varieties, such as essential and canonical dimensions, and their applications to the theory of algebraic groups over non-closed fields and related structures. The idea of building global information/objects out of compatible local data and the nature of the obstruction to doing so, are of great importance to mathematics and physics (manifolds are the quintessential example of a global object constructed by "patching together" local data). The algebraic version of manifolds (needed for example in applications to number theory, but also in far removed areas such as genetics) are called varieties, or more generally schemes. The theory of torsors provides the language and tools to measure the obstructions to patching local data together (say to construct a variety or scheme). The theory of essential dimension could be thought as a sort of "complexity theory" of torsors. Linked to this is the concept of canonical dimension and of compressible variety (a tool that allows us to know when certain type of constructions could not get simpler). Having introduced the central pieces of my research, I will proceed now to give a more detailed description of some of my current research and objectives. An algebraic variety is called incompressible, if all its rational endomorphisms are dominant. Results on incompressibility of varieties have numerous applications, especially to computation of essential dimension of algebraic structures. Incompressibility for several classes of varieties has been already established by the applicant. Most of the varieties (but not all of them) are projective homogeneous under an action of a semisimple affine algebraic group. The main thrust of the proposal is the study of incompressibility for the following classes of varieties (depending on a prime integer p): (a) Generic torsors over norm tori of separable p-primary extensions of fields of characteristic p. In characteristic different from p, the result has been already established. The proof makes use of Steenrod operations on Chow groups with coefficients in the finite field of p elements. Such operations are not available over fields of characteristic p. (b) Hypersurfaces Nrd = const, where Nrd is the reduced norm of a p-primary central simple algebra. The approach is based on study of motives of smooth compactifications of the variety. (c) Unitary grassmannians associated to a 2-primary division algebra endowed with an involution of unitary type. We expect to obtain an analogue of the results explained in the previous section concerning symplectic involutions. All of the above questions can be expressed in terms of the rationality of certain algebraic cycles. My objective is also to prove results on descent of rational cycles from the function field of a variety to the base field (joint with R. Fino). One of the varieties we are interested in is the variety Nrd = const given by the reduced norm of a degree p central division algebra (the result is already known in characteristic 0). Finally, we plan to study operations in connective K-theory with possible application to the above problems.

该提案的主旨是研究扭转量和离散量的数值和离散不变量 投影同质簇，例如基本维数和规范维数，及其 在非闭域和相关结构上的代数群理论中的应用。 从兼容的本地数据和自然中构建全局信息/对象的想法 这样做的障碍，对于数学和物理学非常重要（流形 是通过“修补”本地对象构造的全局对象的典型示例 数据）。流形的代数版本（例如在数论应用中需要， 但在遥远的领域（如遗传学）也被称为品种，或更一般地说是计划。 扭转理论提供了测量修补障碍的语言和工具 本地数据在一起（比如构建一个品种或方案）。本质维度理论 可以被认为是一种扭转的“复杂性理论”。与此相关的是一个概念 规范维度和可压缩多样性（一种工具，让我们知道什么时候某些 结构类型再简单不过了）。介绍了我的核心部分之后 研究，我现在将继续对我当前的一些研究进行更详细的描述 和目标。 如果代数簇的所有有理自同态都是显性的，则该代数簇被称为不可压缩。 簇不可压缩性的结果有许多应用，特别是在计算方面 代数结构的基本维数。几类品种的不可压缩性 已经由申请人设立。大多数品种（但不是全部）都是 半单仿射代数群作用下的射影齐次。 该提案的主旨是研究以下类别的不可压缩性 品种（取决于素数 p）： (a) 特征 p 场的可分离 p 初级扩展的范数环面上的通用扭转量。 在与p不同的特性中，结果已经成立。证明有用 具有 p 元素有限域中的系数的 Chow 群上的 Steenrod 运算。这样的 操作不适用于特征 p 的字段。 (b) 超曲面 Nrd = const，其中 Nrd 是 p 初等中心简单代数的约化范数。 该方法基于对品种平滑紧凑化动机的研究。 (c) 与赋予了对合的 2 次除代数相关的一元格拉斯曼函数 单一型。我们期望获得与上一节中解释的结果类似的结果 关于辛对合。 所有上述问题都可以用某些代数环的合理性来表达。 我的目标也是证明有理循环从函数域下降的结果 品种到基地（与 R. Fino 联合）。我们感兴趣的品种之一是 变体 Nrd = 由 p 次中心除代数的约化范数给出的 const （结果为 特征 0) 中已知。 最后，我们计划研究连接 K 理论中的运算及其在上述方面的可能应用 问题。