Cohomology, Function Fields and Anabelian Geometry

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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.**