Incompressibility of algebraic varieties

该方案的主要目的是研究扭量和射影齐次簇的数值不变量和离散不变量，例如本质维度和标准维度，以及它们在非闭域上的代数群理论和相关结构中的应用。从兼容的本地数据中构建全局信息/对象的想法及其障碍的性质，对数学和物理都非常重要(流形是通过将本地数据“拼凑在一起”而构建的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，但在遗传学等遥远的领域中也需要)被称为变种，或者更一般的方案。扭量理论提供了语言和工具来衡量将本地数据拼接在一起的障碍(比如构建一种多样性或方案)。本质维度理论可以看作是一种扭力的“复杂性理论”。与此相关的是规范维度和可压缩多样性的概念(一种工具，允许我们知道何时某些类型的结构不能变得更简单)。在介绍了我的研究的核心部分之后，我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占优的。关于簇的不可压缩性的结果有着广泛的应用，特别是在计算代数结构的本质维度方面。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，大多数簇(但不是所有簇)是射影齐次的。这一建议的主要目的是研究下列种类的变种(依赖于素数p)的不可压缩性：(A)特征p的域的可分p-初等扩张的范数环面上的一般扭子。在与p不同的特征中，结果已经建立。该证明利用了系数在p元有限域上的Chow群上的Steenrod运算。这种运算在特征为p的域上是不可用的。(B)超曲面NRd=Const，其中NRd是p-素数中心单代数的约化范数。这种方法是基于对品种平滑压实的动机的研究。(C)与赋有酉型对合的2-初等除法代数有关的酉草曼。我们期望得到与上一节关于辛对合解的结果类似的结果。所有上述问题都可以用某些代数圈的合理性来表示。我的目的也是证明有理循环从簇的函数域下降到基域(与R.Fino联合)的结果。我们感兴趣的变种之一是由p次中心除代数的约化范数给出的变种NRD=const(其结果在特征0中已知)。最后，我们计划研究连通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 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.