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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635420
• 关键词: Incompressibility; algebraic; varieties

The main thrust of the proposal is to study numerical and discrete invariants of torsors andprojective homogeneous varieties, such as essential and canonical dimensions, and theirapplications 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 natureof the obstruction to doing so, are of great importance to mathematics and physics (manifoldsare the quintessential example of a global object constructed by "patching together" localdata). 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 patchinglocal data together (say to construct a variety or scheme). The theory of essential dimensioncould be thought as a sort of "complexity theory" of torsors. Linked to this is the concept ofcanonical dimension and of compressible variety (a tool that allows us to know when certaintype of constructions could not get simpler). Having introduced the central pieces of myresearch, I will proceed now to give a more detailed description of some of my current researchand objectives.An algebraic variety is called incompressible, if all its rational endomorphisms are dominant.Results on incompressibility of varieties have numerous applications, especially to computationof essential dimension of algebraic structures. Incompressibility for several classes of varietieshas been already established by the applicant. Most of the varieties (but not all of them) areprojective 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 ofvarieties (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 useof Steenrod operations on Chow groups with coefficients in the finite field of p elements. Suchoperations 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 ofunitary type. We expect to obtain an analogue of the results explained in the previous sectionconcerning 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 avariety to the base field (joint with R. Fino). One of the varieties we are interested in is thevariety Nrd = const given by the reduced norm of a degree p central division algebra (the result isalready known in characteristic 0).Finally, we plan to study operations in connective K-theory with possible application to the aboveproblems.

该提案的主旨是研究torsors和射影齐次簇的数值和离散不变量，如本质维数和规范维数，以及它们对非闭域上代数群理论和相关结构的应用。从兼容的局部数据中构建全局信息/对象的想法和这样做的障碍的性质，对数学和物理学来说非常重要（流形是通过“拼凑”本地数据构建全局对象的典型例子）。流形的代数形式（例如在数论的应用中需要，但在遗传学等遥远的领域也需要）被称为簇，或更一般的方案。torsors理论提供了语言和工具来衡量将局部数据拼凑在一起的障碍（比如构建簇或方案）。本质维数理论可以被看作是一种“复杂性理论”。与此相关的是规范维数和可压缩多样性的概念（这是一种工具，可以让我们知道什么时候某种类型的结构不能变得更简单）。在介绍了我的研究的中心部分之后，我将继续对我目前的一些研究和目标进行更详细的描述。一个代数簇被称为不可压缩的，如果它的所有有理自同态都是占优势的。关于簇的不可压缩性的结果有许多应用，特别是在计算代数结构的本质维数方面。申请人已经确定了几类品种的不可压缩性。大多数簇（但不是全部）在半单仿射代数群的作用下都是投射齐次的.本文的主要目的是研究以下几类簇（取决于素数p）的不可压缩性：（a）特征p的域的可分p-准素扩张的范环面上的一般扭体.在特征不同于p的情况下，结果已经成立.证明利用Steenrod操作的Chow群的系数在有限域的p个元素。这样的运算在特征为p的域上是不可用的。(b)超曲面Nrd = const，其中Nrd是p-准素中心单代数的约化范数，该方法基于对簇的光滑紧化动机的研究。(c)与具有酉型对合的2-素除代数相联系的酉格拉斯曼算子。我们期望得到与前一节中关于辛对合的结果类似的结果.所有上述问题都可以用某些代数圈的合理性来表示.我的目的也是证明关于有理圈从各种函数域到基域的下降的结果（与R. Fino）。我们感兴趣的变种之一是变种Nrd = const给出的约化模的一个度p中心除代数（结果是已知的特征0）。最后，我们计划研究的操作，可能的应用于上述问题的连接K-理论。