nLab global Hodge theory

Idea

Global Hodge theory is the study of Hodge theoretical invariants in global analytic geometry and overconvergent global analytic geometry. The basic idea is to try to generalize/combine the results of classical (complex analytic) Hodge theory and of p-adic Hodge theory in a unique setting.

A first (non-archimedean) step in this direction was given by Bhatt in his paper on the derived approach to p-adic Hodge theory (originally due to Beilinson), where he proposed a Hodge theory for (say semi-stable) varieties over $\mathbb{Z}$, using Illusie’s derived de Rham cohomology? to define a convenient period ring?. From the point of view of global analytic geometry, this gives a Hodge theory for strict analytic spaces over $\mathbb{Z}$, equipped with its trivial norm. One may try to adapt these ideas to extend them to a Hodge theory for strict analytic spaces over $\mathbb{Z}$, equipped with its archimedean norm, but the use of a derived completion in the process prevents one from a straightforward generalization. Remark that the treatment of quasi-projective varieties should certainly involve a global mixed Hodge theory for pre-proper logarithmic analytic varieties.

The advantage of including the archimedean norm in global Hodge theory is that this seems to give a better setting for the study of completed arithmetic L-functions? and automorphic L-functions, whose functional equation may not be proved without using the archimedean information given by a convenient Gamma-factor. This extension is very easy to make if one works with rational coefficients, like in Beilinson’s original paper on p-adic Hodge theory (i.e., with the de Rham period ring). It remains a challenging problem to combine the non-archimedean and archimedean information on Hodge theory without forgetting to take care of torsion phenomena in cohomology, that are very important for the study of special values of L-functions. This may be doable if one follows the line of ideas used in Arakelov geometry, but formulated in a global analytic way. A very natural setting for this seems to be global strict analytic geometry over $\mathbb{Z}$ equipped with its archimedean norm.

Related concepts

• Berkovich space

• global analytic geometry

• overconvergent global analytic geometry

• arithmetic cryptography

References