12
$\begingroup$

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

$\endgroup$
3
  • 5
    $\begingroup$ It looks like the end of your sentence is missing. Also, may I ask why Berkovich spaces (as opposed to some other choice, such as adic spaces for instance, or $p$-adic analytic geometry in general)? $\endgroup$ Mar 16, 2015 at 20:49
  • $\begingroup$ Complex geometry can be seen as some sort of degeneration of Non-Archimedean geometry and from Non-Archimedean geometry we can get Tropical geometry and from Tropical geometry we can get Non-Archimedean geometry by approximation method . The Strominger-Yau-Zaslow conjecture gives a geometric description of the relation between mirror pairs of Calabi-Yau varieties. One of the applications of Berkovich space is the study of Strominger-Yau-Zaslow conjecture for Berkovich spaces to get better picture on Tropical methods in Arithmetric geometry due to Kontsevich and Soibelman $\endgroup$
    – user21574
    Jun 4, 2017 at 17:29
  • $\begingroup$ In fact Strominger-Yau-Zaslow conjecture for non-Archimedean geometry is just Berkovich’s deformation retraction. See paper of Gross, Siebert, Hacking and Keel. Moreover, A non-archimedean analogue of Kähler geometry was proposed by Kontsevich and Tschinkel. Boucksom, Chambert-Loir, Favre, Gubler, Jonsson and Zhang $\endgroup$
    – user21574
    Jun 4, 2017 at 17:38

4 Answers 4

18
$\begingroup$

I would first recommend the paper of Antoine Ducros (Espaces analytiques $p$-adiques au sens de Berkovich, 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.

$\endgroup$
2
  • $\begingroup$ I guess you mean Gubler/Rabinoff/Werner in your "relation with tropical geometry". $\endgroup$ Mar 16, 2015 at 20:52
  • $\begingroup$ @JérômePoineau: Oops. Thanks. I edit at once. $\endgroup$
    – ACL
    Mar 16, 2015 at 21:26
7
$\begingroup$

Let me add a few more applications to what has already been mentioned.

  • Relation with Bruhat-Tits buildings (Berkovich, then Rémy/Thuillier/Werner). If $G$ is a reductive group over a non-archimedean valued field, then the Bruhat-Tits building $\mathscr{B}(G)$ of $G$ embeds into the analytification $G^{an}$ of $G$. If you choose a parabolic subgroup $P$, you have a map to $(G/P)^{an}$, which is the analytification of a proper variety, hence a compact space. This can help you compactify the building, describe the strata of the compactification, etc.

  • Complex dynamics. Favre and Jonsson used Berkovich spaces (actually some instance of it that they call "valuative tree") in order to study the dynamics of a polynomial endomorphism of $\mathbb{C}^2$ near a superattracting fixed point or at infinity.

  • Resolution of singularities (Thuillier). Let $X$ be an algebraic variety over a perfect field $k$ with an isolated singular point $x$. Let $f \colon Y \to X$ be a resolution of it such that $f^{-1}(x)$ is a normal crossing divisor $E$. Then the homotopy type of the incidence complex of $E$ is independent of the choice of the resolution. (Remark here that $k$ is any perfect field and that the Berkovich spaces that come up in the proof are over the field $k$ endowed with the trivial valuation.)

  • $p$-adic differential equations. André used Berkovich spaces in order to prove a conjecture of Dwork on the logarithmic growth of the solutions of $p$-adic differential equations (he actually needs Berkovich spaces only when the base field is not locally compact). In a different direction, Baldassarri (then Pulita and myself, and also Kedlaya) took up the study of $p$-adic differential equations on Berkovich curves. For instance, in my work with Pulita, we manage to give conditions for the finiteness of the de Rham cohomology of a curve with coefficients in some module with a connection. (Here, my point is that even if you can easily state the results in any theory you like, rigid geometry for instance, you will have a hard time proving them without Berkovich spaces.)

$\endgroup$
4
$\begingroup$

Jan Kiwi (Duke Journal) used Berkovich spaces to give the first proof of my conjecture that any sequence of quadratic rational maps which divergence in moduli space (of Mobius conjugacy classes) has at most two "rescaling limits".

$\endgroup$
2
$\begingroup$

I have found the following:

  • Walter Gubler applied it to the Bogomolov conjecture.
  • Jérôme Poineau applied it to the inverse Galois problem: http://arxiv.org/abs/0809.2880
  • Michael Stoll uses them in http://arxiv.org/abs/1307.1773 "Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell-Weil rank" for uniform Mordell.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.