Let $K$ be a field complete with respect to a nontrivial non-Archimedean absolute value, as in the previous section. For simplicity, we will assume from now on that $K^{\circ}$ is a discrete valuation ring (DVR), and we will fix a generator $\varpi$ of its maximal ideal $K^{\circ\circ}$. This is the only case we will need in the sequel and it allows us to use a more elementary definition of measures and local heights. Nevertheless, the reader can consult [Gub03, Gub07] for the general case.

Let $X$ be an algebraic variety over $K$ and $L$ a line bundle on $X$. Let $X^{{\text{\rm an}}}$ and $L^{{\text{\rm an}}}$ be their respective analytifications.

Models of varieties and line bundles give rise to an important class of metrics. To introduce and study these metrics, we first consider the notion of model of varieties. Write $S=\operatorname{Spec}(K^{\circ})$. The scheme $S$ has two points: the special point $o$ and the generic point $\eta$. Given a scheme ${\mathcal{X}}$ over $S$, we set $\mathcal{X}_{o}=\mathcal{X}\times\operatorname{Spec}(k)$ and $\mathcal{X}_{\eta}=\mathcal{X}\times\operatorname{Spec}(K)$ for its special fibre and its generic fibre, respectively.

Given a model ${\mathcal{X}}$ of $X$, there is a reduction map defined on a closed subset of $X^{{\text{\rm an}}}$ with values in $\mathcal{X}_{o}$ [Ber90, §2.4]. This map can be described as follows. Let $\{\mathcal{U}_{i}\}_{i\in I}$ be a finite open affine cover of $\mathcal{X}$ by schemes over $S$ of finite type and, for each $i$, let ${\mathcal{A}}_{i}$ be a $K^{\circ}$-algebra such that ${\mathcal{U}}_{i}=\operatorname{Spec}({\mathcal{A}}_{i})$. Set $U_{i}=\mathcal{U}_{i}\cap X$ and let $C_{i}$ be the closed subset of $U_{i}^{{\text{\rm an}}}$ defined as

(2.12)

$$C_{i}=\{p\in U_{i}^{{\text{\rm an}}}\mid|a(p)|\leq 1,\forall a\in{\mathcal{A}}_{i}\}$$

For each $p\in C_{i}$, the prime ideal $\mathfrak{q}_{p}:=\{a\in{\mathcal{A}}_{i}\mid|a(p)|<1\}\subset{\mathcal{A}}_{i}$ contains $K^{\circ\circ}{\mathcal{A}}_{i}$ and so it determines a point ${\operatorname{red}}(p):=\mathfrak{q}_{p}/K^{\circ\circ}{\mathcal{A}}_{i}\in\mathcal{U}_{i,o}\subset\mathcal{X}_{o}$. Consider the closed subset $C=\bigcup_{i}C_{i}\subset X^{{\text{\rm an}}}$. The above maps glue together to define a map

(2.13)

$${\operatorname{red}}\colon C\longrightarrow\mathcal{X}_{o}.$$

This map is surjective and anti-continuous, in the sense that the preimages of the open subsets are closed [Ber90, §2.4]. For each irreducible component $V$ of ${\mathcal{X}}_{o}$, there is a unique point $\xi_{V}\in C$ such that

(2.14)

$${\operatorname{red}}(\xi_{V})=\eta_{V},$$

where $\eta_{V}$ denotes the generic point of $V$ [Ber90, Proposition 2.4.4]. The finite subset $\{\xi_{V}\}_{V}\subset X^{{\text{\rm an}}}$ is called the Shilov boundary of $X^{{\text{\rm an}}}$. Observe that it depends on the choice of ${\mathcal{X}}$.

If both $X$ and ${\mathcal{X}}$ are proper, then $C=X^{{\text{\rm an}}}$ and the reduction map is defined on the whole of $X^{{\text{\rm an}}}$. If both $X$ and ${\mathcal{X}}$ are normal, we can compute the Shilov boundary. Let $V$ be an irreducible component of $\mathcal{X}_{o}$ and choose a finite type affine open subset $\mathcal{U}=\operatorname{Spec}({\mathcal{A}})\subset{\mathcal{X}}$ containing $\eta_{V}$. Put ${A}={\mathcal{A}}\otimes_{K^{\circ}}K$ and $U={\mathcal{U}}\cap X$. Then the point $\xi_{V}\in U\subset X^{{\text{\rm an}}}$ is the multiplicative seminorm on $A$ given by

(2.15)

$$|a(\xi_{V})|=|\varpi|^{{\operatorname{ord}}_{V}(a)/{\operatorname{ord}}_{V}(\varpi)},$$

for each $a\in A$, where ${\operatorname{ord}}_{V}(f)$ is the order of $f$ at the generic point of $V$.

Next we recall the definition of models of line bundles. Let $L$ be a line bundle on $X$.

We assume that the variety $X$ is proper for the rest of this section. To a proper model of a line bundle we can associate a metric.

Different models may give rise to the same metric.

The inverse image of an algebraic metric is algebraic.

Next we give a second description of an algebraic metric. As before, let $X$ be a proper variety over $K$ and $L$ a line bundle on $X$, and $\|\cdot\|_{\mathcal{X},\mathcal{L},e}$ an algebraic metric on $L^{{\text{\rm an}}}$. Let $p\in X^{{\text{\rm an}}}$ and put $H=\mathscr{H}(p)$, which is a complete extension of $K$. Let $H^{\circ}$ be its valuation ring, and $o$ and $\eta$ the special and the generic point of $\operatorname{Spec}(H^{\circ})$, respectively. The point $p$ induces a morphism of schemes $\operatorname{Spec}(H)\to X$. By the valuative criterion of properness, there is a unique extension

(2.20)

$$\widetilde{p}\colon\operatorname{Spec}(H^{\circ})\longrightarrow\mathcal{X}.$$

It satisfies $\widetilde{p}(\eta)=\pi(p)$, where $\pi\colon X^{{\text{\rm an}}}\rightarrow X$ is the natural map introduced at the beginning of §2.2, and $\widetilde{p}(o)={\operatorname{red}}(p)$.

We give a third description of an algebraic metric in terms of intersection theory that makes evident the relationship with higher dimensional Arakelov theory. Let $({\mathcal{X}},{\mathcal{L}},e)$ be a proper model of $(X,L)$ and $\iota\colon{\mathcal{Y}}\to{\mathcal{X}}$ a closed algebraic curve. Let ${\widetilde{{\mathcal{Y}}}}$ be the normalization of ${\mathcal{Y}}$ and ${\widetilde{\iota}}\colon{\widetilde{{\mathcal{Y}}}}\to{\mathcal{X}}$ and $\rho\colon{\widetilde{{\mathcal{Y}}}}\to\operatorname{Spec}(K^{\circ})$ the induced morphisms. Let $s$ be a rational section of ${\mathcal{L}}$ such that $\operatorname{div}(s)$ intersects properly ${\mathcal{Y}}$. Then the intersection number $(\iota\cdot\operatorname{div}(s))$ is defined as

$$(\iota\cdot\operatorname{div}(s))=\deg(\rho_{\ast}(\operatorname{div}({\widetilde{\iota}}^{\ast}s))).$$

Many other algebraic metrics can be obtained from Example 2.25, by considering maps of varieties to projective spaces. Let $X$ be a proper variety over $K$ equipped with a line bundle $L$ such that $L^{\otimes e}$ is generated by global sections for an integer $e\geq 1$. A set of global sections in $\Gamma(X,L^{\otimes e})$ that generates $L^{\otimes e}$ induces a morphism $\varphi\colon X\to\mathbb{P}_{K}^{n}$ and, by inverse image, a metric $\varphi^{*}\|\cdot\|_{{\operatorname{can}}}$ on $L$. If $X$ admits a a proper model, Proposition 2.19 shows that this metric is algebraic.

Now we recall the notion of semipositivity for algebraic metrics. A curve $C$ in ${\mathcal{X}}$ is vertical if it is contained in ${\mathcal{X}}_{o}$.

With the hypothesis in Proposition 2.19, the inverse image of a semipositive algebraic metric is also a semipositive algebraic metric.

Finally, we recall the definition of the signed measures associated with algebraic metrics.

This signed measure only depends on the metrics and not on the particular choice of models [Cha06, Proposition 2.7]. Observe that ${\operatorname{ord}}_{V}(\varpi)$ is the multiplicity of the component $V$ in ${\widetilde{{\mathcal{Y}}}}_{o}$ and that the total mass of this measure equals $\deg_{L_{0},\dots,L_{d-1}}(Y)$. If ${\overline{L}}_{i}$ is semipositive for all $i$ and $Y$ is effective, this signed measure is a measure.