CHAT - Esnault, Codimension one in Algebraic and Arithmetic Geometry

Slides are available on the CHAT series website: https://chiyunhsu.github.io/activitie...

The CHAT series invite established professors to talk about either
(1) their math career in general
(2) their theorems or theories, but explained from a personal and historical perspective, like how they came up with the problem, what the Aha! moment was like, how the problem changes from its initial form to the published rigorous form.

The idea is that instead of talking about their latest theorems, the speakers would take a step back and talk about the trajectory of an idea, the path to the discovery of a theorem, the influence of ideas learned through a paper or a chance conversation with a colleague, and the hazards met and overcome along the way.

Date: Dec 4, 2023
Speaker: Hélène Esnault (FU Berlin/Harvard)
Title: Codimension one in Algebraic and Arithmetic Geometry
Abstract: The notions of weight in complex geometry and in \ell-adic theory in geometry over a finite field have been developed by Deligne and by the Grothendieck school. The analogy between the theories is foundational and led to predictions and theorems on both sides. On the complex Hodge theory side, not only do we have the weight filtration, but we also have the Hodge filtration.
The analogy on the \ell-adic side over a finite field hasn’t really been documented by Deligne.
Thinking of this gave the way to understand the Lang--Manin conjecture according to which smooth projective rationally connected varieties over a finite field possess a rational point.
http://page.mi.fu-berlin.de/esnault/p....

On the other hand, we know the formulation in complex geometry of the Hodge conjecture: on a smooth projective complex variety X, a sub-Hodge structure of H^{2j}(X) of Hodge type (j,j) should be supported on a codimension j cycle. The analog \ell-adic conjecture has been formulated by Tate, even over a number field. Grothendieck’s generalized Hodge conjecture is straightforwardly formulated: a sub-Hodge structure H of H^i(X) of Hodge type (i-1,1), (i-2,2), \ldots, (1,i-1) should be supported on a codimension 1 cycle. Equivalently it should die at the generic point of the variety.
This is difficult to formulate because Hodge structures are complicated to describe. But there is one instance for which we can bypass the Hodge formulation: H=H^i(X) and H^{0,i}=H^i(X, O)(=H^{i,0}=H^0(X, \Omega^i))=0. Then the conjecture descends to the field of definition of X and becomes purely algebraic. It is on the one hand related to the (quite bold) motivic conjectures predicting that H^i(X,O)=0 for all i\neq 0 should be equivalent to the triviality of the Chow group of 0-cycles over a large field (this brings us back to the proof of the Lang--Manin conjecture). On the other hand, as it is purely algebraic, one can try to think of it in the framework of today’s p-adic Hodge theory.