Incompressibility of algebraic varieties

本文的主要目的是研究环量和射影齐次变量的数值和离散不变量，如本质维数和规范维数，以及它们在非闭域和相关结构上的代数群理论中的应用。从兼容的局部数据中构建全局信息/对象的想法，以及阻碍这样做的本质，对数学和物理来说是非常重要的（流形是通过“拼凑”局部数据构建全局对象的典型例子）。流形的代数版本（例如在数论的应用中需要，但也在遥远的领域，如遗传学中需要）被称为变种，或者更一般的方案。torsor理论提供了一种语言和工具来测量将局部数据拼凑在一起的障碍（比如构建一个变量或方案）。本质维度理论可以看作是一种“复杂性理论”。与此相关的是规范维度和可压缩变化的概念（一种工具，可以让我们知道某些类型的结构何时不能变得更简单）。在介绍了我的研究的核心部分之后，我现在将对我目前的一些研究和目标进行更详细的描述。如果一个代数变体的所有有理自同态都是显性的，那么它就是不可压缩的。关于变量不可压缩性的研究结果具有广泛的应用，特别是在计算代数结构的基本维数方面。申请人已经确定了几类品种的不可压缩性。在半单仿射代数群的作用下，大多数（但不是全部）是射影齐次的。这一建议的主要主旨是研究以下种类的不可压缩性（依赖于一个素数整数p）：(a)在特征p的可分离的p-原域的模环上的一般环量。在不同于p的特征中，结果已经建立。该证明利用了p元有限域中带系数的Chow群上的Steenrod运算。(b)超曲面Nrd = const，其中Nrd是p-初等中心简单代数的约简范数。该方法是基于对品种光滑紧化动机的研究。(c)与具有酉型对合的2-初等除法代数相关的酉格拉斯曼子。我们期望得到上一节关于辛对合所解释的结果的类似物。上述所有问题都可以用某些代数循环的合理性来表示。我的目标也是证明从函数域到基域的有理循环下降的结果（与R. Fino联合）。我们感兴趣的一个变种是由p次中心除法代数的约简范数给出的变种Nrd = const（结果在特征0中已经已知）。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。

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.