Chambert-Loir and Ducros have introduced real differential forms and currents on Berkovich spaces.(See Gubler's [survey](https://arxiv.org/abs/1303.7364) 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?



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


