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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=588371
• 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torsors的数值和离散不变量， 投射齐次变体，如本质维数和规范维数，以及它们的 应用于非闭域上的代数群理论和相关结构。 从兼容的本地数据中构建全局信息/对象的想法以及 的障碍，这样做，是非常重要的数学和物理（流形 是通过“拼凑”局部对象来构造全局对象的典型示例。 数据）。流形的代数形式（例如在数论应用中需要， 但也在遥远的领域，如遗传学）被称为品种，或更普遍的计划。 Torsors理论为度量修补障碍提供了语言和工具 本地数据在一起（比如说构建一个品种或方案）。本质维数理论 可以被认为是一种关于torsors的“复杂性理论”。与此相关的是 规范维数和可压缩的品种（一个工具，让我们知道当某些 建筑物不能变得更简单）。在介绍了我的 研究，我现在将继续更详细地描述我目前的一些研究 和目标。 一个代数簇称为不可压缩的，如果它的所有有理自同态都是支配的。 关于簇的不可压缩性的结果有许多应用，特别是在计算中 代数结构的本质维数几类簇的不可压缩性 申请人已经成立。大多数品种（但不是全部）是 在半单仿射代数群的作用下的射影齐性。 该提案的主要目的是研究以下几类不可压缩性： 变种（取决于素数p）： (a)特征p的域的可分p-准素扩张的范环面上的通有扭体。 在不同于p的特征中，结果已经成立。证明使用 在有限p元域上系数的Chow群上的Steenrod运算。等 在特征p的字段上不能进行操作。 (b)超曲面Nrd = const，其中Nrd是p-准素中心单代数的约化范数。 该方法是基于研究的动机顺利紧化的品种。 (c)与赋对合的2-素除代数相联系的酉Grassmannian 单一类型我们期望得到与上一节所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圈的合理性来表示。 我的目标也是证明结果下降的有理循环从功能领域的一个 品种的基本领域（联合与R。Fino）。我们感兴趣的品种之一是 variety Nrd = const，由p次中心除代数的约化范数给出（结果是 已知特征0）。 最后，我们计划研究连接K理论中的运算，并可能应用于上述问题 问题