Let $X$ be an algebraic variety over $\mathbb{C}$ and $X^{{\text{\rm an}}}$ its associated complex analytic space. We recall the definition of differential forms on $X^{{\text{\rm an}}}$ introduced by Bloom and Herrera [BH69]. The space $X^{{\text{\rm an}}}$ can be covered by a family of open subsets $\{U_{i}\}_{i}$ such that each $U_{i}$ can be identified with a closed analytic subset of an open ball in $\mathbb{C}^{r}$ for some $r$. On each $U_{i}$, the differential forms are defined as the restriction to this subset of smooth complex-valued differential forms defined on an open neighbourhood of $U_{i}$ in $\mathbb{C}^{r}$. Two differential forms on $U_{i}$ are identified if they coincide on the non-singular locus of $U_{i}$. We denote by $\mathscr{A^{\ast}}(U_{i})$ the complex of differential forms of $U_{i}$, which is independent of the chosen embedding. In particular, if $U_{i}$ is non-singular, we recover the usual complex of differential forms. These complexes glue together to define a sheaf $\mathscr{A}^{\ast}_{X^{{\text{\rm an}}}}$. This sheaf is equipped with differential operators  d, $\operatorname{d^{c}}$, $\partial$, $\bar{\partial}$, an external product and inverse images with respect to analytic morphisms: these operations are defined locally on each $\mathscr{A^{\ast}}(U_{i})$ by extending the differential forms to a neighbourhood of $U_{i}$ in $\mathbb{C}^{r}$ as above and applying the corresponding operations for $\mathbb{C}^{r}$. We write $\mathcal{O}_{X^{{\text{\rm an}}}}$ and $C^{\infty}_{X^{{\text{\rm an}}}}=\mathscr{A}^{0}_{X^{{\text{\rm an}}}}$ for the sheaves of analytic functions and of smooth functions of $X^{{\text{\rm an}}}$, respectively.

Let $L$ be an algebraic line bundle on $X$ and $L^{{\text{\rm an}}}$ its analytification.

We remark that what we call “metric” in this text is called “continuous metric” in other contexts.

Let ${\overline{L}}=(L,\|\cdot\|)$ be a smooth metrized line bundle. Given a local section $s$ of $L^{{\text{\rm an}}}$ on an open subset $U$, the first Chern form of ${\overline{L}}$ is the $(1,1)$-form defined on $U$ as

$$\operatorname{c}_{1}({\overline{L}})=\partial\bar{\partial}\log\|s\|^{2}\in\mathscr{A}^{1,1}(U).$$

It does not depend on the choice of local section and can be extended to a global closed $(1,1)$-form. Observe that we are using the algebro-geometric convention, and so $\operatorname{c}_{1}({\overline{L}})$ determines a class in $H^{2}(X^{{\text{\rm an}}},2\pi i\,\mathbb{Z})$.

Many smooth metrics can be obtained as the inverse image of the Fubini-Study metric. Let $X$ be a variety over $\mathbb{C}$ and $L$ a line bundle on $X$, and assume that there is an integer $e\geq 1$ such that $L^{\otimes e}$ is generated by global sections. Choose a basis of the space of global sections $\Gamma(X,L^{\otimes e})$ and let $\varphi\colon X\to\mathbb{P}^{M}_{\mathbb{C}}$ be the induced morphism. Given a local section $s$ of $L$, let $s^{\prime}$ be a local section of ${\mathcal{O}}(1)$ such that $s^{\otimes e}=\varphi^{\ast}s^{\prime}$. Then, the smooth metric on $L^{{\text{\rm an}}}$ obtained from the Fubini-Study metric by inverse image is given by

$$\|s(p)\|=\|s^{\prime}(\varphi(p))\|^{1/e}_{{\operatorname{FS}}}$$

for any $p\in X^{{\text{\rm an}}}$ which is not a pole of $s$.

A family of smooth metrized line bundles ${\overline{L}}_{0},\dots,{\overline{L}}_{d-1}$ on $X$ and a $d$-dimensional cycle $Y$ of $X$ define a signed measure on $X^{{\text{\rm an}}}$ as follows. First suppose that $Y$ is a subvariety of $X$ and let $\delta_{Y}$ denote the current of integration along the analytic subvariety $Y^{{\text{\rm an}}}$, defined as $\delta_{Y}(\omega)=\frac{1}{(2\pi i)^{d}}\int_{Y^{{\text{\rm an}}}}\omega$ for $\omega\in\mathscr{A}^{2d}_{X^{{\text{\rm an}}}}$. Then the current

$$\operatorname{c}_{1}({\overline{L}}_{0})\wedge\cdots\wedge\operatorname{c}_{1}({\overline{L}}_{d-1})\wedge\delta_{Y}$$

is a signed measure on $X^{{\text{\rm an}}}$. This notion extends by linearity to $Y\in Z_{d}(X)$. If ${\overline{L}}_{i}$, $i=0,\dots,d-1$, are semipositive and $Y$ is effective, this signed measure is a measure.

In this section we recall Berkovich’s theory of analytic spaces. We will not present the most general theory developed in [Ber90] but we will content ourselves with the analytic spaces associated to algebraic varieties, that are simpler to define and enough for our purposes.

Let $K$ be a field complete with respect to a nontrivial non-Archimedean absolute value $|\cdot|$. Such fields will be called non-Archimedean fields. Let $K^{\circ}=\{\alpha\in K\mid|\alpha|\leq 1\}$ be the valuation ring, $K^{\circ\circ}=\{\alpha\in K\mid|\alpha|<1\}$ the maximal ideal and $k=K^{\circ}/K^{\circ\circ}$ the residue field.

Let $X$ be a scheme of finite type over $K$. Following [Ber90, §1 and Remark 3.4.2], we can associate an analytic space $X^{{\text{\rm an}}}$ to the scheme $X$ as follows. First assume that $X=\operatorname{Spec}(A)$, where $A$ is a finitely generated $K$-algebra. Then, the points of $X^{{\text{\rm an}}}$ are the multiplicative seminorms of $A$ that extend the absolute value of $K$, see [Ber90, §1.1]. Every element $a$ of $A$ defines a function $|a(\cdot)|\colon X^{{\text{\rm an}}}\to\mathbb{R}_{\geq 0}$ given by evaluation of the seminorm. The topology of $X^{{\text{\rm an}}}$ is the coarsest topology that makes the functions $|a(\cdot)|$ continuous for all $a\in A$.

To each point $p\in X^{{\text{\rm an}}}$ we attach a prime ideal

$$\mathfrak{p}_{p}=\{a\in A\mid|a(p)|=0\}.$$

This induces a map $\pi\colon X^{{\text{\rm an}}}\to X$ defined as $\pi(p)=\mathfrak{p}_{p}$. The point $p$ is a multiplicative seminorm on $A$ and so it induces a non-Archimedean absolute value on the field of fractions of $A/\mathfrak{p}_{p}$. We denote by $\mathscr{H}(p)$ the completion of this field with respect to that absolute value.

Let $U$ be an open subset of $X^{{\text{\rm an}}}$. An analytic function on $U$ is a function

$$f\colon U\longrightarrow\coprod_{p\in{U}}\mathscr{H}(p)$$

such that, for each $p\in U$, $f(p)\in\mathscr{H}(p)$ and there is an open neigborhood $U^{\prime}\subset U$ of $p$ with the property that, for all $\varepsilon>0$, there are elements $a,b\in A$ with $b\not\in\mathfrak{p}_{q}$ and $|f(q)-a(q)/b(q)|<\varepsilon$ for all $q\in U^{\prime}$. The analytic functions form a sheaf, denoted $\mathcal{O}_{X^{{\text{\rm an}}}}$, and $(X^{{\text{\rm an}}},\mathcal{O}_{X^{{\text{\rm an}}}})$ is a locally ringed space [Ber90, §1.5 and Remark 3.4.2]. In particular, every element $a\in A$ determines an analytic function on $X^{{\text{\rm an}}}$, also denoted $a$. The function $|a(\cdot)|$ can then be obtained by composing $a$ with the absolute value map

$$|\cdot|\colon\coprod_{p\in X^{{\text{\rm an}}}}\mathscr{H}(p)\longrightarrow\mathbb{R}_{\geq 0},$$

which justifies its notation.

Now, if $X$ is a scheme of finite type over $K$, the analytic space $X^{{\text{\rm an}}}$ is defined by gluing together the affine analytic spaces obtained from an affine open cover of $X$. If we want to stress the base field we will denote $X^{{\text{\rm an}}}$ by $X_{K}^{{\text{\rm an}}}$.

Let $K^{\prime}$ be a complete extension of $K$ and $X^{\text{\rm an}}_{K^{\prime}}$ the analytic space associated to the scheme $X_{K^{\prime}}$. There is a natural map $X_{K^{\prime}}^{{\text{\rm an}}}\to X_{K}^{{\text{\rm an}}}$ defined locally by restricting seminorms.

The basic properties of $X^{{\text{\rm an}}}$ are summarized in the following theorem.

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.

Let $K$ be either $\mathbb{R}$ or $\mathbb{C}$ (the Archimedean case) as in §2.1, or a complete field with respect to a nontrivial non-Archimedean absolute value (the non-Archimedean case) as in §2.3. Let $X$ be a proper variety over $K$. Its analytification $X^{{\text{\rm an}}}$ will be a complex analytic space in the Archimedean case (equipped with an anti-linear involution when $K=\mathbb{R}$), or an analytic space in the sense of Berkovich, in the non-Archimedean case. A metrized line bundle on $X$ is a pair ${\overline{L}}=(L,\|\cdot\|)$, where $L$ is a line bundle on $X$ and $\|\cdot\|$ is a metric on $L^{{\text{\rm an}}}$. Recall that the operations on line bundles of tensor product, dual and inverse image under a morphism extend to metrized line bundles.

Given two metrics $\|\cdot\|$ and $\|\cdot\|^{\prime}$ on $L^{{\text{\rm an}}}$, their quotient defines a continuous function $X^{{\text{\rm an}}}\to\mathbb{R}_{>0}$ given by $\|s(p)\|/\|s(p)\|^{\prime}$ for any local section $s$ of $L$ not vanishing at $p$. The distance between $\|\cdot\|$ and $\|\cdot\|^{\prime}$ is defined as the supremum of the absolute value of the logarithm of this function. In other words,

$$\operatorname{dist}(\|\cdot\|,\|\cdot\|^{\prime})=\sup_{p\in X^{\text{\rm an}}\setminus\operatorname{div}(s)}|\log(\|s(p)\|/\|s(p)\|^{\prime})|,$$

for any non-zero rational section $s$ of $L$.