Incompressibility of algebraic varieties

该提案的主旨是研究torsors和投射齐次簇的数值和离散不变量，如基本和规范维度，以及它们在非封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/对象的想法以及这样做的障碍的性质对数学和物理学非常重要（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）被称为变种，或更一般的方案。Torsors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（比如构建一个变体或方案）。本质维数理论可以看作是一种关于torors的“复杂性理论”。与此相关的是规范维数和可压缩簇的概念（一种工具，允许我们知道某些类型的构造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。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。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。大多数的品种（但不是所有的）是投射齐次下的一个行动的半单仿射代数群。该建议的主要目的是研究以下类别的品种（取决于一个素数p）的不可压缩性：（a）一般torsors在范环面的可分p-主要扩展的领域的特点p.在特点不同的p，结果已经建立。证明利用Steenrod操作周群的系数在有限域的p个元素。这样的操作在特征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。该方法是基于研究的动机顺利紧化的品种。(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。我的目标也是证明结果下降的有理循环从功能领域的品种的基础领域（联合R。Fino）。我们感兴趣的变种之一是变种Nrd = const由一个度p的中心除代数的约化范数给出（结果在特征0中已经知道）。最后，我们计划研究可能应用于上述问题的连接K-理论的操作。

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.