Tropical charts (coordinates) and differential forms in non-archimedean geometry

Chambert-Loir and Ducros have introduced real differential forms and currents on Berkovich spaces.(See Gubler's survey for example). In that survey, a tropical chart $V$ is defined on an analytification $X^{\mathrm{an}}$, something like $\mathrm{val}^{-1}(\Delta)$ for a polyhedron $\Delta$. Here $\mathrm{val}: (K^\times)^n \to \mathbb R^n$ is defined by $(z_1,\dots,z_n) \mapsto (\mathrm{val}(z_1),\dots,\mathrm{val}(z_n))$

Question:

1. I am an beginner for this subject, I don't understand the reason to use tropical charts, what's the advantage? It seems not so good because they are contained in a torus $(K^\times)^n$ so they exclude zero. But in differential geometry local models are $\mathbb R^n$ which include zero.

2. Let's consider the rigid analytification $$\mathbb A_K^{n,rig} = \cup_{i=0}^\infty \mathrm{Sp} K \langle c^{-i}\zeta_1, \dots, c^{-i}\zeta_n\rangle$$ of the affine $n$-space $\mathbb A^n_K=\mathrm{Spec} K[\zeta_1,\dots,\zeta_n]$ (here $|c|>1$). In this case can we find some good tropical charts?

3. Other than the example in the question 2 above, can we explicitly give any other tropical charts? say on the Berkovich projective line?

• I don't undestand what do you mean in 2. by tropical charts on a rigid space, if it is not a Berkovich space. Do you mean the associated Berkovich space or do you want an analogous notion in a rigid space? – Xarles Jul 24 '18 at 15:54
• I think I mean the associated Berkovich space. By the way, do you mean tropical charts can only be defined on Berkovich spaces rather than rigid spaces? – Hang Jul 24 '18 at 16:14
• I am not sure how one can define "tropical charts" on rigid spaces, since the tropicalization maps are not in the appropriate category. – Xarles Jul 24 '18 at 16:45
• About your first question, I really don't understand what's your problem. You should think that the valuation($=-log(|\ |)$) is not defined on 0, so you cannot include it. – Xarles Jul 24 '18 at 17:00
• You choose two points $a,b\ne 0, \infty$, and take the open set $U=\mathbb P^{1 an}\setminus \{a,b\}$. You can map $U$ to $\mathbb Gm^{an}$ by the analytic (in fact algebraic) bijective morphism $f(z)=(z-a)/(z-b)$. Now, tropicalization map $v:\mathbb Gm^{an}\to \mathbb R$ defines $U$ as a tropical chart. – Xarles Jul 26 '18 at 16:53

I will try to explain what it is a tropical chart on an algebraic variety over a non-archimedean field $K$ (complete with respect to a non-archimedean absolute value, algebraically closed by simplicity).

First of all, for the algebraic tori (of the form $T:=\mathbb G_m^n$, for $n\ge 1$), we have a continuous map $Trop\colon T^{an}\to \mathbb R^n$, where $T^{an}$ is the Berkovich analytic space associated to $T$, usually called the tropicalization. Then in Gubler's notation, the preimage $V$ for $Trop$ of any open subset $\Omega$ of $\mathbb R^n$ is a tropical chart of $T$.

Now, for any algebraic variety $X$, consider affine open subsets $U$ such that there is an closed immersion to an algebraic torus $T$ (they are called very affine). For this open affine subsets there is in fact a canonical closed immersion to a torus $\varphi_U\colon U \to T_U$.

Note that any algebraic variety can be covered by very affine open sets, refining the usual covering by affine open sets.

Now, a tropical chart $(V,\varphi_U)$ is an open subset $V \subset X^{an}$ contained in a very affine subset $U$ such that $V=\varphi_U^{-1}(Trop^{-1}(\Omega))$ for an open subset $\Omega \subset \mathbb R^n$. Using that the very affine open sets are a basis for the Zarisky topology, one can show that the tropical charts are a basis for the analytic topology of $X^{an}$ (see the Proposition 4.16).

So any algebraic variety has a covering by tropical charts, hence any Zarisky open set of an algebraic variety.

Compare with the usual analytic charts, where a chart can be seen as $(V,\varphi_U)$, with $V\subset X^{an}$ and $U$ an open subset with a map $\varphi_U\colon U\to \mathbb C^n$ such that $V=\varphi_U^{-1}(\Omega)$, for $\Omega$ an open subset of $\mathbb C^n$.

The fact that here we have $\mathbb G_m^{n,an}$ instead of $\mathbb A^{n,an}$ (which should be the natural analogue of $\mathbb C^n$) is a quite interesting question; it happens in several place in the non-arquimedean world (e.g. Tate elliptic curves).

• By an algebraic variety, I follow "Serre's convention": mathoverflow.net/a/7798/24442. – Xarles Jul 28 '18 at 14:11
• Thanks a lot for a great answer. I take some time to digest. But I still feel confused about one point: people usually consider $Trop^{-1}(\Delta)$ for a rational polyhedral $\Delta\subset \mathbb R^n$. (See Einsiedler-Kapranov-Lind's paper for example). In this case, it is simple and is given by an affinoid algebra. But here a tropical chart is defined for an arbitrary $\Omega\subset \mathbb R^n$. When $\Omega$ is not a rational polyhedral, what can we say? I even don't know if $Trop^{-1}(\Omega)$ is still a Berkorvich space. – Hang Aug 8 '18 at 22:07