Let $K$ be either $\mathbb{R}$, $\mathbb{C}$ or a complete field with respect to an absolute value associated to a nontrivial discrete valuation. When $K=\mathbb{R}$ we will use the technique of Remark 2.5 and in the non-Archimedean case we will use the notations of §2.3. Let $\mathbb{T}$ be an $n$-dimensional split torus over $K$ and let $N$ and $M=N^{\vee}$ be the corresponding lattices. Let $\Sigma$ be a fan in $N_{\mathbb{R}}$. For each cone $\sigma\in\Sigma$, we will denote by $X_{\sigma}^{{\text{\rm an}}}$ the complex analytic space $X_{\sigma}(\mathbb{C})$ in Archimedean case or the Berkovich analytic space associated to the scheme $X_{\sigma,K}$ in the non-Archimedean case. These analytic spaces glue together in an analytic space $X_{\Sigma}^{{\text{\rm an}}}$.

Given any cone $\sigma\in\Sigma$, we write

On $X_{\sigma}(\mathbb{R}_{\geq 0})$, we put the coarsest topology such that, for each $m\in M_{\sigma}$, the map $X_{\sigma}(\mathbb{R}_{\geq 0})\to\mathbb{R}_{\geq 0}$ given by $\gamma\mapsto\gamma(m)$ is continuous. Observe that if $\tau$ is a face of $\sigma$, then there is a dense open immersion $X_{\tau}(\mathbb{R}_{\geq 0})\hookrightarrow X_{\sigma}(\mathbb{R}_{\geq 0})$. Hence the topological spaces $X_{\sigma}(\mathbb{R}_{\geq 0})$ glue together to define a topological space $X_{\Sigma}(\mathbb{R}_{\geq 0})$. This is the variety with corners associated to $X_{\Sigma}$. Analogously to the algebraic case, one can prove that this topological space is Hausdorff and that the spaces $X_{\sigma}(\mathbb{R}_{\geq 0})$ can be identified with open subspaces of $X_{\Sigma}(\mathbb{R}_{\geq 0})$ satisfying

For each $\sigma\in\Sigma$ there is a continuous map $\rho_{\sigma}\colon X^{{\text{\rm an}}}_{\sigma}\to X_{\sigma}(\mathbb{R}_{\geq 0})$. This map is given, in the Archimedean case, by

While, in the non-Archimedean case, since a point $p\in X_{\sigma}^{{\text{\rm an}}}$ corresponds to a multiplicative seminorm on $K[M_{\sigma}]$ and a point in $X_{\sigma}(\mathbb{R}_{\geq 0})$ corresponds to a semigroup homomorphism from $M_{\sigma}$ to $(\mathbb{R}_{\geq 0},\times)$, we can define $\rho_{\sigma}(p)$ as the semigroup homomorphism that, to an element $m\in M_{\sigma}$, corresponds $|\chi^{m}(p)|$. These maps glue together to define a continuous map $\rho_{\Sigma}:X_{\Sigma}^{{\text{\rm an}}}\to X_{\Sigma}(\mathbb{R}_{\geq 0})$.

When $\Sigma$ is complete, the analytic space $X_{\Sigma}^{{\text{\rm an}}}$ is compact, and the map $\rho_{\Sigma}$ is proper. By Lemma 5.1, for each cone $\sigma\in\Sigma$, the map $\rho_{\sigma}$ is proper. Since every rational cone belongs to a complete fan, the map $\rho_{\Sigma}$ is proper even if $\Sigma$ is not complete. Of particular interest is the case when $\sigma=\{0\}$. Then $\mathbb{T}^{{\text{\rm an}}}:=X_{0}^{{\text{\rm an}}}$ is an Abelian analytic group, that is, an Abelian group object in the category of analytic spaces. In particular, for any field extension $K^{\prime}$ of $K$, the set $X_{0}^{{\text{\rm an}}}(K^{\prime})$ is an Abelian group. Also $\mathbb{T}(\mathbb{R}_{\geq 0}):=X_{0}(\mathbb{R}_{\geq 0})\simeq(\mathbb{R}_{\geq 0})^{n}$ is a topological Abelian group. Moreover, $\mathbb{T}^{{\text{\rm an}}}$ acts on $X^{{\text{\rm an}}}_{\Sigma}$, $\mathbb{T}(\mathbb{R}_{\geq 0})$ acts on $X_{\Sigma}(\mathbb{R}_{\geq 0})$ and the map $\rho_{\Sigma}$ is equivariant with respect to these actions. The kernel of the map $\rho_{0}$ is a closed subgroup, that we call the compact torus of $\mathbb{T}^{{\text{\rm an}}}$ and we denote by $\mathbb{S}^{{\text{\rm an}}}$. In the Archimedean case it is isomorphic to $(S^{1})^{n}$, while in the non-Archimedean case it is the compact torus of Example 2.8. In fact, the fibres of the map $\rho_{\Sigma}$ are orbits under the action of $\mathbb{S}^{{\text{\rm an}}}$. Therefore the space $X_{\sigma}(\mathbb{R}_{\geq 0})$ is the quotient of $X_{\sigma}^{{\text{\rm an}}}$ by the action of the closed subgroup $\mathbb{S}^{{\text{\rm an}}}$. We warn the reader that the compact topological space underlying $\mathbb{S}^{{\text{\rm an}}}$ is not an abstract group (see [Ber90, Chapter 5]).

The maps $\rho_{\sigma}$, $\sigma\in\Sigma$, have canonical sections that we denote $\theta_{\sigma}$. These sections glue together to give a section $\theta_{\Sigma}$ of $\rho_{\Sigma}$. In the Archimedean case $\theta_{\sigma}$ is induced by the semigroup inclusion $\mathbb{R}_{\geq 0}\subset\mathbb{C}$. In the non-Archimedean case $\theta_{\sigma}$ is defined by the following result.

Let now

and denote by ${\operatorname{\mathbf{e}}}_{K}\colon\mathbb{R}\to\mathbb{R}_{>0}$ the map $u\mapsto\exp(-\lambda_{K}u)$. This map induces an homeomorphism $N_{\mathbb{R}}\rightarrow X_{0}(\mathbb{R}_{>0})$ that we also denote by ${\operatorname{\mathbf{e}}}_{K}$.

In the non-Archimedean case, the map ${\operatorname{val}}\colon\mathbb{T}(K)\to N$ of Definition 4.71, can be extended to a map $\mathbb{T}^{{\text{\rm an}}}\to N_{\mathbb{R}}$ that we denote ${\operatorname{val}}_{K}$ or, when $K$ is clear from the context by ${\operatorname{val}}$. For each $p\in X^{{\text{\rm an}}}_{0}$ we denote by ${\operatorname{val}}_{K}(p)\in\operatorname{Hom}_{\operatorname{sg}}(M,\mathbb{R})=N_{\mathbb{R}}$ the morphism

In the Archimedean case we will denote by ${\operatorname{val}}_{\mathbb{C}}$ or simply by ${\operatorname{val}}$ the map defined by the same equation. Then, the diagram

is commutative.

The map ${\operatorname{\mathbf{e}}}_{K}$ allows us to see $X_{\Sigma}(\mathbb{R}_{\geq 0})$ as a partial compactification on $N_{\mathbb{R}}$. Following [AMRT75, Chapter I, §1] we can give another description of the topology of $X_{\Sigma}(\mathbb{R}_{\geq 0})$. For $\sigma\in\Sigma$, we denote

We choose a positive definite bilinear pairing in $N_{\mathbb{R}}$. Hence we can identify the quotient spaces $N(\tau)_{\mathbb{R}}$ with subspaces of $N_{\mathbb{R}}$, that, for simplicity, we will denote also by $N(\tau)_{\mathbb{R}}$. For a point $u\in N(\tau)_{\mathbb{R}}$, let $U\subset N(\tau)_{\mathbb{R}}$ be a neighbourhood of $u$. For each $\tau^{\prime}$ face of $\tau$, $\tau$ induces a cone $\pi_{\tau^{\prime}}(\tau)$ contained in $N(\tau^{\prime})_{\mathbb{R}}$. If $p\in\tau$ its image $\pi_{\tau^{\prime}}(p)$ in $N(\tau^{\prime})_{\mathbb{R}}$, is contained in $\pi_{\tau^{\prime}}(\tau)$. We write

Moving $U$ and $p$ we obtain a basis of neighbourhoods of $u$ in $N_{\sigma}$. This defines a topology on $N_{\sigma}$ such that the map ${\operatorname{\mathbf{e}}}_{K}\colon N_{\mathbb{R}}\to X_{\sigma}(\mathbb{R}_{\geq 0})$ extends to a homeomorphism $N_{\sigma}\to X_{\sigma}(\mathbb{R}_{\geq 0})$.

We write

and put in $N_{\Sigma}$ the topology that makes $\{N_{\sigma}\}_{\sigma\in\Sigma}$ an open cover. Then the map ${\operatorname{\mathbf{e}}}_{K}$ extends to a homeomorphism between $N_{\Sigma}$ and $X_{\Sigma}(\mathbb{R}_{\geq 0})$ and the map ${\operatorname{val}}_{K}$ extends to a proper continuous map $X^{{\text{\rm an}}}_{\Sigma}\to N_{\Sigma}$ such that the diagram

is commutative.

We end this section stating the functorial properties of the space $X_{\Sigma}(\mathbb{R}_{\geq 0})$. The proofs are left to the reader. Let $N$ and $\Sigma$ be as before and $\sigma\in\Sigma$. Recall that the associated closed subvariety $V(\sigma)$ is canonically isomorphic to the toric variety $X_{\Sigma(\sigma)}$.

Let $N_{1}$ and $N_{2}$ be lattices and let $\Sigma_{1}$ and $\Sigma_{2}$ be complete fans in $N_{1,\mathbb{R}}$ and $N_{2,\mathbb{R}}$ respectively. Let $H\colon N_{1}\to N_{2}$ be a linear map such that, for each cone $\sigma_{1}\in\Sigma_{1}$, there is a cone $\sigma_{2}\in\Sigma_{2}$ with $H(\sigma_{1})\subset\sigma_{2}$. Let $p\in X_{\Sigma_{2},0}(K)$ and let $A\colon N_{1,\mathbb{R}}\to N_{2,\mathbb{R}}$ be the affine map $A=H+{\operatorname{val}}(p)$.