Cohomology, Function Fields and Anabelian Geometry

代数和算术几何研究多项式和/或丢番图方程的解，而伽罗瓦理论研究这些方程的对称性。通过考虑这些算术/几何对象一直到两个民族的等价性，人们经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出生不变量的性质，因此研究代数簇的出生不变量有着悠久的传统，例如阿尔巴尼亚簇(和/或它的Tate模)，未分枝的Brauer群等。几个重要的公开问题，如关于因子的Tate猜想，也是出生不变的。另一方面，阿纳贝尔几何学从伽罗瓦理论的角度研究算术和/或几何。此外，近几年来，二元非交换几何出现了一次重大的复兴，主要是在几乎交换的背景下，它使用了可以上同调研究的基本群的幂零截断。建议的研究将应用Anabelian几何的工具，并开发新的工具，通过上同调的镜头研究函数域的算术/几何及其几何模型。除了Galois上同调之外，建议研究的一个核心参与者是函数域的“类属上同调”。这取决于上同调理论的选择，但不同的比较同构导致类属上同调的实现之间相应的比较同构。虽然一般上同调在许多方面类似于Galois上同调，但它经常继承额外的结构，如混合Hodge结构或绝对Galois群的作用，这使这些群具有更丰富和更精细的结构。现在已知(见[40])，高维函数域的同构类型完全由它的具有有理系数的普通上同调环决定，并被赋予1次典型的混合Hodge结构。因此，一旦人们将一些额外的基元数据附加到H^1，尽管是以高度间接的方式，所有的混生不变量都被编码在泛上同调环中。此外，一般上同调群通常带有与给定函数域的Galois上同调的典范比较态射，当上同调完成后，它变成同构。在这一点上，类属上同调可以被视为伽罗瓦理论语境和理据语境之间的桥梁，伽罗瓦理论语境适用于再交换技术，而理据语境则以几何意义的方式对(双态)不变量进行编码。这项拟议研究的总体目标是以双向的方式利用这座桥梁：Anabelian技术将被用来研究有动机的物体，而Motivic结构将被用来在Anabelian几何中获得新的见解。

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.**