Incompressibility of algebraic varieties

该计划的主旨是研究torsors和 * 投射齐次簇的数值和离散不变量，如本质维数和标准维数，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）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 * 本地数据修补在一起的障碍（比如构建一个变体或方案）。本质维数理论可以被看作是一种“复杂性理论”。与此相关的是 * 标准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。大多数簇（但不是全部）在半单仿射代数群的作用下是 * 投射齐次的。该提案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-准素扩张的范数环面上的一般torsor。在不同于p的特征中，结果已经成立。证明利用 * 的Steenrod操作周群系数在有限域的p个元素。这种 * 操作在特征p的字段上不可用。** (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光滑紧化的品种。(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 ** 所解释的结果类似的结果。上述所有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理性循环从功能领域的 * 品种的基础领域（联合R。Fino）。我们感兴趣的变种之一是由p次中心除代数的约化范数给出的 * 变种Nrd = const（结果 * 在特征0中已知）。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.