Cohomology, Function Fields and Anabelian Geometry

上同调、函数域和阿贝尔几何

• 批准号: RGPIN-2019-04762
• 负责人: Topaz, Adam
• 金额: $ 1.75万
• 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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=737842
• 关键词: Cohomology; Function; Fields; Anabelian; Geometry

Algebraic and arithmetic geometry study solutions to polynomial and/or diophantine equations, while Galois theory studies the symmetries of such equations. By considering these arithmetic/geometric objects up-to birational equivalence, one can often obtain new information about the relationships between them. In fact, many important questions in arithmetic/algebraic geometry are of birational nature, and so there is a longstanding tradition of studying birational invariants of algebraic varieties, such as the Albanese variety (and/or its Tate module), the unramified Brauer group, etc. Several important open problems, such as the Tate conjecture for divisors, are also known to be birationally invariant. On the other hand, anabelian geometry studies arithmetic and/or geometry from the point of view of Galois theory. Moreover, birational anabelian geometry has seen a major resurgence in recent years, primarily in the almost-abelian context, which uses nilpotent truncations of fundamental groups that can be studied cohomologically. The proposed research will apply tools from anabelian geometry, and develop new ones, in studying the arithmetic/geometry of function fields and their geometric models, through the lens of cohomology. In addition to Galois cohomology, a central player in the proposed research is the "generic cohomology" of function fields. This depends on a choice of a cohomology theory, but the various comparison isomorphisms induce corresponding comparison isomorphisms between the realizations of generic cohomology. Although generic cohomology resembles Galois cohomology in many ways, it often inherits additional structure, such as a mixed Hodge structure or an action by an absolute Galois group, which gives these groups a richer and more refined structure. It is now known (see [40]) that the isomorphy type of a higher dimensional function field is completely determined by its generic cohomology ring, with rational coefficients, endowed with the canonical mixed Hodge structure in degree 1. Thus, all birational invariants are encoded in the generic cohomology ring, once one attaches some additional motivic data to H^1, albeit in a highly indirect way. Moreover, the generic cohomology groups often come equipped with a canonical comparison morphism to the \ell-adic Galois cohomology of the given function field, which becomes an isomorphism after \ell-adic completion. In this regard, generic cohomology can be seen as a bridge between the Galois-theoretical context, which is amenable to anabelian techniques, and the motivic context, where (birational) invariants are encoded in geometrically meaningful ways. The general goal of the proposed research is to utilize this bridge in a bidirectional way: anabelian techniques will be used to investigate motivic objects, while motivic structures will be used to gain new insight in anabelian geometry.

代数和算术几何研究多项式和/或丢芬图方程的解，而伽罗瓦理论研究这些方程的对称性。通过将这些算术/几何对象考虑到两种等价，人们通常可以获得关于它们之间关系的新信息。事实上，算术/代数几何中的许多重要问题都具有双分性，因此研究代数变量的双分不变量有着悠久的传统，例如Albanese变量（和/或其Tate模），未分枝的Brauer群等。一些重要的开放问题，如关于除数的Tate猜想，也被认为是双不变的。另一方面，安娜贝尔几何从伽罗瓦理论的角度研究算术和/或几何。此外，近年来，双对数阿贝尔几何有了很大的复兴，主要是在几乎阿贝尔的背景下，它使用幂零截断的基本群，可以研究上同调。本研究将从上同调的角度出发，应用并开发新的工具来研究函数场的算术/几何及其几何模型。除了伽罗瓦上同调之外，本研究的核心是函数域的“一般上同调”。这取决于上同构理论的选择，但各种比较同构在一般上同构的实现之间会产生相应的比较同构。虽然一般上同调在许多方面类似于伽罗瓦上同调，但它通常继承了额外的结构，例如混合Hodge结构或绝对伽罗瓦群的作用，这使这些群具有更丰富和更精细的结构。现在我们知道（见[40]），高维函数场的同构类型完全由它的泛上同调环决定，它具有理性系数，具有1度的正则混合Hodge结构。因此，一旦将一些额外的动机数据附加到H^1上，尽管以一种高度间接的方式，所有的双相不变量都编码在泛上同调环中。此外，一般上同调群通常具有给定函数域的\进伽罗瓦上同调的正则比较仿射，在\进补全后成为同构。在这方面，一般上同调可以被看作是伽罗瓦理论背景和动机背景之间的桥梁，伽罗瓦理论背景适用于再推理技术，而动机背景则以几何上有意义的方式编码（两族）不变量。所提出的研究的总体目标是以双向的方式利用这种桥梁：可溯性技术将用于研究动机对象，而动机结构将用于在可溯性几何中获得新的见解。