What are applications of the theory of Berkovich analytic spaces? The analytification $X \mapsto X^{\mathrm{an}}$



applications of Berkovich spaces
I would first recommend the paper of Antoine Ducros ([Espaces analytiques $p$-adiques au sens de Berkovich](http://webusers.imj-prg.fr/%7Eantoine.ducros/asterisque.pdf), Séminaire Bourbaki, exposé 958, 2006) for a general survey of the theory, with applications.


Here is a list of applications which I find striking, starting from those mentioned by Ducros's survey.


* Étale cohomology. Berkovich developed a good theory of étale cohomology for his analytic spaces, which had applications in the Langlands program (for example, in the proof by Harris-Taylor of the local Langlands conjecture).
* Proof (by Berkovich) of a conjecture of Deligne that the vanishing/nearby cycles (for a scheme over a discrete valuation ring) only depend on the formal completion.
* Non-archimedean analogue of the classical potential theory on Riemann surfaces (Thuillier, Favre/Rivera-Letelier, Baker/Rumely).
* Non-archimedean equidistribution theorems in the framework of Arakelov geometry (myself, Favre/Rivera-Letelier, Baker/Rumely, Gubler, Yuan), with applications to the Bogomolov conjecture for abelian varieties of function fields (Gubler, Yamaki), algebraic dynamics of Manin-Mumford/Mordell-Lang type (Yuan/Zhang, Dujardin/Favre,...).
* Berkovich spaces of $\mathbf Z$ (Poineau) have applications to complicated rings of power series with integral coefficients introduced by Harbater and to their Galois theory. (In some sense, a geometrization of Harbater's formal patching.)
* Mirror symmetry (Kontsevich/Soibelman) via the study of non-archimedean degenerations of Calabi-Yau manifolds. Recent developments in birational geometry (Mustață/Nicaise, Nicaise/Xu, Temkin) and viz. a non-archimedean analogue of the Monge-Ampère equation (Boucksom/Favre/Jonsson, Yuan/Zhang, Liu Y.).
* Relation with tropical geometry (Baker/Payne/Rabinoff, my work with Ducros, Gubler/Rabinoff/Werner,...)
* Relations with non-archimedean Arakelov geometry (Gubler/Künnemann, Ducros and myself)


A notable feature of the Berkovich spaces is the presence of (sometimes canonical) closed subspaces endowed with canonical piecewise linear structures on which the analytic spaces retracts by deformations (Berkovich, Hrushovski/Loeser,...). Those subspaces (“skeleta”) carry a large amount of geometric information and are of tremendous use in the theory.


