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.
Global analytic geometry
combines non-archimedean and archimedean analytic geometry;
contains algebraic geometry as a sub-theory;
treats all places on equal footing, contrary to scheme theory. This implies that it is closer in spirit to the automorphic philosophy and Langlands program, and more historically to Tate’s proof of the functional equation.
The main aim of the theory is to define, using global analytic tools, a good Hodge theory for arithmetic varieties.
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.
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).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.
A short introduction for large audience is in
For more see
Frédéric Paugam,_Overconvergent global analytic geometry (preprint)
Frédéric Paugam, Global analytic geometry (arXiv:0803.0148)
Frédéric Paugam, Global analytic geometry and the functional equation, lecture notes (pdf)
Jérôme Poineau, La droite de Berkovich sur Z, Asterisque 334 (2010).
(on fundamental properties of the affine line)
Jérôme Poineau, Espaces de Berkovich sur Z: etude locale arXiv:1202.0799
(on the coherence of the sheaf of analytic functions on higher dimensional affine spaces)
Oren Ben-Bassat, Kobi Kremnizer, section 7 of Non-Archimedean analytic geometry as relative algebraic geometry (arXiv:1312.0338)