# nLab global analytic geometry

### Context

#### Analytic geometry

analytic geometry (complex, rigid, global)

## Basic concepts

analytic function

analytification

GAGA

# Contents

## Idea

Global analytic geometry is a developing subject that gives an alternative/complementary approach to scheme theory in arithmetic geometry and analytic number theory. The starting point of this theory is in Vladimir Berkovich’s book about spectral theory and non-archimedean analytic geometry. It was then developped further by Jérôme Poineau.

Many interesting results on polynomial equations can be proved using the mysterious interactions between algebraic, complex analytic and p-adic analytic geometry. The aim of global analytic geometry is to construct a category of spaces which contains these three geometries.

## Motivation

Global analytic geometry

The main aim of the theory is to define, using global analytic tools, a good Hodge theory for arithmetic varieties.

## Possible set of constraints

for a relaxed approach to global Hodge theory: it is not an easy task to find a good set of constraints on such a global Hodge theory?, but they are useful to understand better the motivations underlying the construction of global analytic spaces.

1. having a good theory of linear coefficients on global analytic spaces, with the Grothendieck six operations (this should be done by the use of the sub-analytic? topology in characteristic $0$, and by a probably quite hard to develop model theoretical description of definable? sets for the G-topology on strict and non-strict overconvergent analytic spaces). It seems that global analytic motivic spectral coefficients (given by imposing homotopy invariance with respect to the unit disc are not so well adapted to the study of torsion phenomena in the characteristic p situation).
2. having a good theory of higher and derived global analytic spaces, with a well-behaved notion of de Rham type cohomology theory and a Chern character. The constraints on such a theory would be:
• get back (or be isomorphic to) the usual algebraic de Rham Chern character when one works with usual schemes.
• get back the p-adic analytic de Rham Chern character (on Ayoub’s motivic cohomology) of dagger spaces when one works with dagger p-adic spaces.
• get back the usual de Rham Chern character when one works over $\C$.
1. the functional equation of zeta functions of arbitrary arithmetic varieties;
2. an analytic langlands program, giving a correspondence between general (non-algebraic) automorphic representations and global analytic motives.

Argument in favor of its use are:

• the fact that archimedean factors are deeply related to (real and complex analytic) Hodge theory;

• the fact that all proofs of parts of local Langlands program use deeply non-archimedean analytic spaces that are out of the scope of classical algebraic scheme theory.

## References

A short introduction for large audience is in

• Jérôme Poineau, Global analytic geometry, pages 20-23 in EMS newsletter September 2007 (pdf)

For more see

Revised on October 23, 2014 05:14:58 by Anonymous Coward (78.194.68.97)