We recall first the construction of the analytification of $X$. Let $U={\rm Spec}(A)$ be an open affine subset of $X$, then ${U^{\rm an}}$ is the set of multiplicative seminorms on $A$ extending the given absolute value $|\phantom{a}|$ on $K$. This set is endowed with the topology generated by the functions ${U^{\rm an}}\rightarrow{\mathbb{R}},p\mapsto p(a)$ with $a$ ranging over $A$. By glueing, we get a topological space ${X^{\rm an}}$ which is connected locally compact and Hausdorff. We can endow it with a sheaf of analytic functions leading to a Berkovich analytic space over $K$ which we call the analytification of $X$. For a morphism $\varphi:Y\rightarrow X$ of algebraic varieties over $K$, we get an analytic morphism $\varphi^{\rm an}:{Y^{\rm an}}\rightarrow{X^{\rm an}}$ induced by composing the multiplicative semiorms with $\varphi^{\sharp}$ on suitable affine open subsets. We refer to [Be90] for details, or to [BPS11], §1.2, for a neat description of the analytification.

We will define some local invariants in $x\in{X^{\rm an}}$. On an open affine neighbourhood $U={\rm Spec}(A)$, the point $x$ is given by a multiplicative seminorm $p$ on $A$ and we often write $|f(x)|:=p(f)$ for $f\in A$. Dividing out the prime ideal $I:=\{f\in A\mid p(f)=0\}$, we get a multiplicative norm on the integral domain $B:=A/I$ which extends to an absolute value $|\phantom{a}|_{x}$ on the quotient field of $B$. The completion of this field is denoted by ${\mathscr{H}}(x)$. It does not depend on the choice of $U$ and it may be also constructed analytically. The absolute value of ${\mathscr{H}}(x)$ is denoted by $|\phantom{a}|$ as it extends the given absolute value on $K$. Note that the completed residue field ${\mathscr{H}}(x)$ of $x$ remains the same if we replace the ambient variety $X$ by the Zariski closure of $x$ in $X$.

Let $s(x)$ be the transcendence degree of the residue field of ${\mathscr{H}}(x)$ over ${\tilde{K}}$. The quotient of the value group of ${\mathscr{H}}(x)$ by $\Gamma$ is a finitely generated abelian group and we denote its ${\mathbb{Q}}$-rank by $t(x)$. Finally, we set $d(x):=s(x)+t(x)$. By [Be90], Proposition 9.1.3, we have $\dim(X)=\dim(V)=\sup_{x\in V}d(x)$ for every open subset $V$ of ${X^{\rm an}}$.

Let $T:={\mathbb{G}}_{m}^{r}$ be a split multiplicative torus over $K$ with coordinates $z_{1},\dots,z_{r}$. Then we have the tropicalization map

It is immediate from the definitions that the map ${{\rm trop}}$ is continuous and proper. To get a coordinate free approach, we could use the character group $M$ and its dual $N$. Then ${\rm trop}$ is a map from ${T^{\rm an}}$ to $N_{\mathbb{R}}$. We refer to [Gu12] for details about tropical geometry.

For a closed subvariety $Y$ of $T$ of dimension $n$, the tropical variety associated to $X$ is defined by ${\rm Trop}(Y):={{\rm trop}}({Y^{\rm an}})$. The Bieri–Groves theorem says that ${\rm Trop}(Y)$ is a finite union of $n$-dimensional integral $\Gamma$-affine polyhedra in ${\mathbb{R}}^{r}$. It is shown in tropical geometry that ${\rm Trop}(Y)$ is an integral $\Gamma$-affine polyhedral complex. The polyhedral structure is only determined up to subdivision which does not matter for our constructions. We will see below that the tropical variety is endowed with a positive canonical weight $m$ satisfying the balancing condition from 3.7. We get a tropical cycle of pure dimension $n$ which we also denote by ${\rm Trop}(Y)$ forgetting the weight $m$ in the notation.

The tropical weight $m$ on an $n$-dimensional polyhedron $\sigma$ of ${\rm Trop}(Y)$ is defined in the following way. By density of the value group $\Gamma$ in ${\mathbb{R}}$, there is $\omega\in\Gamma^{r}\cap{\rm relint}(\sigma)$. We choose $t\in{\mathbb{G}}_{m}^{r}(K)$ with ${\rm trop}(t)=\omega$. Then the closure of $t^{-1}Y$ in $({{\mathbb{G}}_{m}^{r}})_{{K^{\circ}}}$ is a flat variety over ${K^{\circ}}$ whose special fibre is called the initial degeneration ${\rm in}_{\omega}(Y)$ of $Y$ at $\omega$. Note that ${\rm in}_{\omega}(Y)$ is a closed subscheme of $({\mathbb{G}}_{m}^{r})_{{\tilde{K}}}$. Let $m_{W}$ be the multiplicity of the irreducible component $W$ of ${\rm in}_{\omega}(Y)$. Then the tropical weight $m_{\sigma}$ is defined by $m_{\sigma}:=\sum_{W}m_{W}$, where $W$ ranges over all irreducible components of ${\rm in}_{\omega}(Y)$. One can show that the definition is independent of the choices of $\omega$ and $t$. It is a non-trivial fact from tropical geometry that $({\rm Trop}(Y),m)$ is a tropical cycle (see [Gu12], §13, for details).

For an open subset $U$ of the algebraic variety $X$, a moment map is a morphism $\varphi:U\rightarrow T$ to a split multiplicative torus $T:={\mathbb{G}}_{m}^{r}$ over $K$. The tropicalization of $\varphi$ is

Obviously, this is a continuous map with respect to the topology on the analytification ${U^{\rm an}}$. Note that our moment maps are algebraic which differs from the moment maps in [CD12] which are defined analytically.

We say that the moment map $\varphi^{\prime}:U^{\prime}\rightarrow T^{\prime}$ of the open subset $U^{\prime}$ of $X$ refines the moment map $\varphi:U\rightarrow T$ if $U^{\prime}\subset U$ and if there is an affine homomorphism $\psi:T^{\prime}\rightarrow T$ of the multiplicative tori such that $\varphi=\psi\circ\varphi^{\prime}$ on $U^{\prime}$. Here, an affine homomorphism means a group homomorphism composed with a (multiplicative) translation on $T$. This group homomorphism induces a homomorphism $M\rightarrow M^{\prime}$ of character lattices. Its dual is the linear part of an integral affine map ${\rm Trop}(\psi):N^{\prime}_{\mathbb{R}}\rightarrow N_{\mathbb{R}}$ such that ${\varphi_{\rm trop}}={\rm Trop}(\psi)\circ\varphi_{\rm trop}^{\prime}$ on $(U^{\prime})^{\rm an}$.

If $\varphi_{i}:U_{i}\rightarrow T_{i}$ are finitely many moment maps of non-empty open subsets $U_{i}$ of $X$ with $i\in I$, then $U:=\cap_{i}U_{i}$ is a non-empty open subset of $X$ and $\varphi:U\rightarrow\prod_{i}T_{i},x\mapsto(\varphi_{i}(x))_{i\in I}$ is a moment map which refines every $\varphi_{i}$. Moreover, it follows easily from the universal property of the product that every moment map $\varphi^{\prime}:U^{\prime}\rightarrow T^{\prime}$ which refines every $\varphi_{i}$ refines also $\varphi$.

If $f:X_{1}\rightarrow X_{2}$ is a morphism of varieties over $K$, then we define the push-forward of $X_{1}$ with respect to $f$ as the cycle $f_{*}(X_{1}):=\deg(f)\overline{f(X_{1})}$, where the degree of $f$ is defined as $\deg(f):=[K(X_{1}):K(f(X_{1}))]$ if $f$ is generically finite and we set $\deg(f):=0$ if $[K(X_{1}):K(f(X_{1}))]=\infty$. By restriction, the push-forward can be defined in the same way on prime cycles of $X_{1}$ and extends by linearity to all cycles of $X_{1}$.

Now let $\varphi:U\rightarrow T={\mathbb{G}}_{m}^{r}$ be a moment map of the open subset $U$ of $X$. By 4.6,

is a tropical cycle on ${\mathbb{R}}^{r}$. If $\varphi$ is generically finite, then this tropical cycle is of pure dimension $\dim(X)$ and the support is equal to ${\varphi_{\rm trop}}({U^{\rm an}})$ (see Lemma 4.9).

We will show that every open affine subset $U$ of $X$ has a canonical moment map. We note that the abelian group $M_{U}:={\mathscr{O}}(U)^{\times}/K^{\times}$ is free of finite rank (see [Sa66], Lemme 1). Here, we use that $K$ is algebraically closed (or at least that $X$ is geometrically reduced). We choose representatives $\varphi_{1},\dots,\varphi_{r}$ in ${\mathscr{O}}(U)^{\times}$ of a basis. This leads to a moment map $\varphi_{U}:U\rightarrow T_{U}={\rm Spec}(K[M_{U}])$. By construction, $\varphi_{U}$ refines every other moment map on $U$. Note that this moment map $\varphi_{U}$ is canonical up to (multiplicative) translation by an element of $T_{U}(K)$.

Let $f:X^{\prime}\rightarrow X$ be a morphism of algebraic varieties over $K$ and let $U^{\prime}$ is an open subset of $X^{\prime}$ with $f(U^{\prime})\subset U$. Then $f^{\sharp}$ induces a homomorphism $M_{U}\rightarrow M_{U^{\prime}}$ of lattices. We get a canonical affine homomorphism $\psi_{U,U^{\prime}}:T_{U^{\prime}}\rightarrow T_{U}$ of the canonical tori with $\psi_{U,U^{\prime}}\circ\varphi_{U^{\prime}}=\varphi_{U}\circ f$. This will be applied very often in the case where $U^{\prime}$ is an open subset of $U$ in $X^{\prime}=X$ and $f={\rm id}$. Then we get a canonical affine homomorphism $\psi_{U,U^{\prime}}:T_{U^{\prime}}\rightarrow T_{U}$.

Recall that an open subset $U$ of $X$ is called very affine if $U$ has a closed embedding into a multiplicative torus. Clearly, the following conditions are equivalent for an open affine subset $U$ of $X$:

• (a)

  $U$ is very affine;

• (b)

  ${\mathscr{O}}(U)$ is generated as a $K$-algebra by ${\mathscr{O}}(U)^{\times}$;

• (c)

  the canonical moment map $\varphi_{U}$ from 4.12 is a closed embedding.

The intersection of two very affine open subsets is again very affine (see the proof of Proposition 4.16). Moreover, the very affine open subsets of $X$ form a basis for the Zariski topology. We conclude that all local considerations can be done using very affine open subsets.

On a very affine open subset, we will almost always use the canonical moment map $\varphi_{U}:U\rightarrow T_{U}$ which is a closed embedding by the above. To simplify the notation, we will set ${\rm Trop}(U)$ for the tropical variety of $U$ in $T_{U}$. It is a tropical cycle in $(N_{U})_{\mathbb{R}}$, where $N_{U}$ is the dual abelian group of $M_{U}$. The tropicalization map will be denoted by ${\rm trop}_{U}:=(\varphi_{U})_{\rm trop}:{U^{\rm an}}\rightarrow(N_{U})_{\mathbb{R}}$. Recall that $\varphi_{U}$ is only determined up to translation by an element of $T_{U}(K)$ and hence ${\rm trop}_{U}$ and ${\rm Trop}(U)$ are only canonical up to an affine translation. This ambiguity is no problem as our constructions will be compatible with affine translations.

A tropical chart $(V,\varphi_{U})$ on ${X^{\rm an}}$ consists of an open subset $V$ of ${X^{\rm an}}$ contained in ${U^{\rm an}}$ for a very affine open subset $U$ of $X$ with $V={\rm trop}_{U}^{-1}(\Omega)$ for some open subset $\Omega$ of ${\rm Trop}(U)$. Here the canonical moment map $\varphi_{U}:U\rightarrow T_{U}$ from 4.12 plays the role of (tropical) coordinates for $V$. By 4.13, $\varphi_{U}$ is an embedding. The condition $V={\rm trop}_{U}^{-1}(\Omega)$ means that $V$ behaves well with respect to the tropical coordinates. In particular, ${\rm trop}_{U}(V)=\Omega$ is an open subset of ${\rm Trop}(U)$.

We say that the tropical chart $(V^{\prime},\varphi_{U^{\prime}})$ is a tropical subchart of $(V,\varphi_{U})$ if $V^{\prime}\subset V$ and $U^{\prime}\subset U$. We note that the definition of tropical chart here is different from the tropical charts in [CD12], §3.1, which consist of an analytic morphism to a split torus and a finite union of polytopes containing the tropicalization.