2.4. Approachable and integrable metrics, measures and local heights [02JC]

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2.4. Approachable and integrable metrics, measures and local heights

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 $L$ .

Definition 2.31.

Let ${\overline{L}}=(L,\|\cdot\|)$ be a metrized line bundle on $X$ . The metric $\|\cdot\|$ is approachable if there exists a sequence of semipositive smooth (in the Archimedean case) or semipositive algebraic (in the non-Archimedean case) metrics $(\|\cdot\|_{l})_{l\geq 0}$ on $L^{{\text{\rm an}}}$ such that

\lim_{l\to\infty}\operatorname{dist}(\|\cdot\|,\|\cdot\|_{l})=0.

If this is the case, we say that ${\overline{L}}$ is approachable. This metrized line bundle is integrable if there are approachable line bundles ${\overline{M}}$ , ${\overline{N}}$ such that ${\overline{L}}={\overline{M}}\otimes{\overline{N}}^{-1}$ .

The tensor product and the inverse image of approachable line bundles are also approachable. The tensor product, the dual and the inverse image of integrable line bundles are also integrable.

Example 2.32.

Let $X=\mathbb{P}^{n}$ be the projective space over $\mathbb{C}$ and $L=\mathcal{O}(1)$ . The canonical metric of $\mathcal{O}(1)^{{\text{\rm an}}}$ is the metric given, for $p=(p_{0}:\dots:p_{n})\in\mathbb{P}^{n}(\mathbb{C})$ , by

\|s(p)\|_{\operatorname{can}}=\frac{|\rho_{s}(p_{0},\dots,p_{n})|}{\max_{i}\{|p_{i}|\}},

for any rational section $s$ of $L$ defined at $p$ and the homogeneous rational function $\rho_{s}\in\mathbb{C}(x_{0},\dots,x_{n})$ associated to $s$ .

This is an approachable metric. Indeed, consider the $m$ -power map $[m]:\mathbb{P}^{n}\to\mathbb{P}^{n}$ defined as $[m](p_{0}:\dots:p_{n})=(p^{m}_{0}:\dots:p^{m}_{n})$ . The $m$ -th root of the inverse image by $[m]$ of the Fubini-Study metric of $\mathcal{O}(1)^{{\text{\rm an}}}$ is the semipositive smooth metric on $L^{{\text{\rm an}}}$ given by

\|s(p)\|_{m}=\frac{|s(p_{0},\dots,p_{n})|}{(\sum_{i}|p_{i}|^{2m})^{1/2m}}.

The family of metrics obtained varying $m$ converges uniformly to the canonical metric.

Proposition 2.33.

Let $Y$ be a $d$ -dimensional subvariety of $X$ and ${\overline{L}}_{i}=(L_{i},\|\cdot\|_{i})$ , $i=0,\dots,d-1$ , a collection of approachable metrized line bundles on $X$ . For each $i$ , let $(\|\cdot\|_{i,l})_{l\geq 0}$ be a sequence of semipositive smooth (in the Archimedean case) or algebraic (in the non-Archimedean case) metrics on $L_{i}^{{\text{\rm an}}}$ that converge to $\|\cdot\|_{i}$ . Then the measures $\operatorname{c}_{1}(L_{0},\|\cdot\|_{0,l})\land\dots\land\operatorname{c}_{1}(L_{d-1},\|\cdot\|_{d-1,l})\wedge\delta_{Y}$ converge weakly to a measure on $X^{\text{\rm an}}$ .

Proof.

The non-Archimedean case is proven in [Cha06, Proposition 2.7(b)] and in [Gub07, Proposition 3.12]. The Archimedean case can be proved similarly. ∎

Definition 2.34.

Let ${\overline{L}}_{i}=(L_{i},\|\cdot\|_{i})$ , $i=0,\dots,d-1$ , be a collection of approachable metrized line bundles on $X$ . For a $d$ -dimensional subvariety $Y\subset X$ , we denote by $\operatorname{c}_{1}({\overline{L}}_{0})\land\dots\land\operatorname{c}_{1}({\overline{L}}_{d-1})\wedge\delta_{Y}$ the limit measure in Proposition 2.33. For integrable bundles ${\overline{L}}_{i}$ and a $d$ -dimensional cycle $Y$ of $X$ , we can associate a signed measure $\operatorname{c}_{1}({\overline{L}}_{0})\land\dots\land\operatorname{c}_{1}({\overline{L}}_{d-1})\wedge\delta_{Y}$ on $X^{{\text{\rm an}}}$ by multilinearity.

This signed measure behaves well under field extensions.

Proposition 2.35.

With the previous notation, let $K^{\prime}$ be a finite extension of $K$ . Set $(X^{\prime},Y^{\prime})=(X,Y)\times\operatorname{Spec}(K^{\prime})$ and let $\varphi\colon{X^{\prime}}^{{\text{\rm an}}}\to X^{{\text{\rm an}}}$ be the induced map. Let $\varphi^{\ast}{\overline{L}}_{i}$ , $i=0,\dots,d-1$ , be the line bundles with algebraic metrics on $X^{\prime}$ obtained by base change. Then

\varphi_{\ast}\left(\operatorname{c}_{1}(\varphi^{\ast}{\overline{L}}_{0})\land\dots\land\operatorname{c}_{1}(\varphi^{\ast}{\overline{L}}_{d-1})\land\delta_{{Y^{\prime}}}\right)=\operatorname{c}_{1}({\overline{L}}_{0})\land\dots\land\operatorname{c}_{1}({\overline{L}}_{d-1})\land\delta_{Y}.

Proof.

This follows from [Gub07, Remark 3.10]. ∎

We also have the following functorial property.

Proposition 2.36.

Let $\varphi\colon X^{\prime}\to X$ be a morphism of proper varieties over $K$ , $Y^{\prime}$ a $d$ -dimensional cycle of $X^{\prime}$ , and ${\overline{L}}_{i}=(L_{i},\|\cdot\|_{i})$ , $i=0,\dots,d-1$ , a collection of integrable metrized line bundles on $X$ . Then

\varphi_{\ast}\left(\operatorname{c}_{1}(\varphi^{\ast}{\overline{L}}_{0})\land\dots\land\operatorname{c}_{1}(\varphi^{\ast}{\overline{L}}_{d-1})\land\delta_{{Y^{\prime}}}\right)=\operatorname{c}_{1}({\overline{L}}_{0})\land\dots\land\operatorname{c}_{1}({\overline{L}}_{d-1})\land\delta_{\varphi_{*}Y}.

Proof.

In the non-Archimedean, this follows from [Gub07, Corollary 3.9(2)]. In the Archimedean case, this follows from the functoriality of Chern classes, the projection formula, and the continuity of direct image of measures. ∎

These signed measures allow us to integrate continuous functions on $X^{{\text{\rm an}}}$ . Indeed, it is also possible to integrate certain functions with logarithmic singularities that play an important role in the definition of local heights.

Proposition 2.37.

Let $Y$ be a $d$ -dimensional cycle of $X$ , ${\overline{L}}_{i}=(L_{i},\|\cdot\|_{i})$ , $i=0,\dots,d$ , a collection of integrable metrized line bundles, and $s_{d}$ a rational section of $L_{d}$ such that $\operatorname{div}(s_{d})$ intersects $Y$ properly. Then $\log\|s_{d}\|$ is integrable with respect to the measure $\operatorname{c}_{1}({\overline{L}}_{0})\land\dots\land\operatorname{c}_{1}({\overline{L}}_{d-1})\wedge\delta_{Y}$ .

Proof.

This is proved in [CT09, Theorem 4.1] for completions of number fields. The argument can be easily extended to cover the general case. ∎

Definition 2.38.

Let $Y$ be a $d$ -dimensional cycle of $X$ and $L_{i}$ a line bundle on $X$ and $s_{i}$ a rational section of $L_{i}$ , $i=0,\dots,d$ . We say that $s_{0},\dots,s_{d}$ meet properly $Y$ if, for all $I\subset\{0,\dots,d\}$ ,

\dim\left(Y\cap\bigcap_{i\in I}|\operatorname{div}s_{i}|\right)=d-\#I.

Definition 2.39.

The local height on $X$ is the function that, to each $d$ -dimensional cycle $Y$ and each family of integrable metrized line bundles with sections $({\overline{L}}_{i},s_{i})$ , $i=0,\dots,d$ , such that the sections meet $Y$ properly, associates a real number $\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y;s_{0},\dots,s_{d})$ determined inductively by the properties:

(1)

$\operatorname{h}(\emptyset)=0$ ;

(2)

if $Y$ is a cycle of dimension $d\geq 0$ , then

\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y;s_{0},\dots,s_{d})=\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d-1}}(Y\cdot\operatorname{div}s_{d};s_{0},\dots,s_{d-1})\\ -\int_{X^{{\text{\rm an}}}}\log\|s_{d}\|\operatorname{c}_{1}(\overline{L}_{0})\land\dots\wedge\operatorname{c}_{1}(\overline{L}_{d-1})\land\delta_{Y}.

In particular, for $p\in X(K)\setminus|\operatorname{div}(s_{0})|$ ,

(2.40)

\operatorname{h}_{{\overline{L}}_{0}}(p;s_{0})=-\log\|s_{0}(p)\|.

Remark 2.41.

Definition 2.39 works better when the variety $X$ is projective. In this case, for every cycle $Y$ there exist sections that meet $Y$ properly, thanks to the moving lemma. This does not necessarily occur for arbitrary proper varieties. Nevertheless, we will be able to define the global height (Definition 2.56) of any cycle of a proper variety by using Chow’s lemma. Similarly we will be able to define the toric local height (Definition 6.1) of any cycle of a proper toric variety.

Remark 2.42.

When $X$ is regular and the metrics are smooth (in the Archimedean case) or algebraic (in the non-Archimedean case), the local heights of Definition 2.39 agree with the local heights that can be derived using the Gillet-Soulé arithmetic intersection product. In particular, in the Archimedean case, this local height agrees with the Archimedean contribution of the Arakelov global height introduced by Bost, Gillet and Soulé in [BGS94]. In the non-Archimedean case, the local height can be interpreted in terms of an intersection product. Assume that $Y$ is prime and choose models $(\mathcal{X}_{i},\mathcal{L}_{i},e_{i})$ of $(X,L_{i})$ that realize the algebraic metrics of ${\overline{L}}_{i}$ . Without loss of generality, we may assume that all the models $\mathcal{X}_{i}$ agree with a common model $\mathcal{X}$ . The sections $s^{\otimes e_{i}}_{i}$ can be seen as rational sections of $\mathcal{L}_{i}$ over $\mathcal{X}$ . With the notations in Definition 2.28, the equation (2.15) implies that

\log\|s_{d}(\xi_{V})\|=\frac{\log|\varpi|{\operatorname{ord}}_{V}(s_{d}^{\otimes e_{d}})}{e_{d}{\operatorname{ord}}_{v}(\varpi)}.

Therefore, in this case the equation in Definition 2.39(2) can be written as

(2.43)

\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y;s_{0},\dots,s_{d})=\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d-1}}(Y\cdot\operatorname{div}(s_{d});s_{0},\dots,s_{d-1})\\ -\frac{\log|\varpi|}{e_{0}\dots e_{d}}\sum_{V\in{\widetilde{\mathcal{Y}}}_{0}^{(0)}}{\operatorname{ord}}_{V}(s_{d}^{\otimes e_{d}})\deg_{\mathcal{L}_{0},\dots,\mathcal{L}_{d-1}}(V).

Remark 2.44.

It is a fundamental observation by Zhang [Zha95b] that the non-Archimedean contribution of the Arakelov global height of a variety can be expressed in terms of a family of metrics. In particular, this global height only depends on the metrics and not on a particular choice of models, exhibiting the analogy between the Archimedean and non-Archimedean settings. The local heights were extended by Gubler [Gub02, Gub03] to non-necessarily discrete valuations and he also weakened the hypothesis of proper intersection.

Remark 2.45.

The local heights of Definition 2.39 agree with the local heights introduced by Gubler, see [Gub03, Proposition 3.5] for the Archimedean case and [Gub03, Remark 9.4] for the non-Archimedean case. In the Archimedean case, the local height in [Gub03] is defined in terms of a refined star product of Green currents based on [Bur94]. The hypothesis needed in Gubler’s definition of local heights are weaker than the ones we use. We have chosen the current definition because it is more elementary and suffices for our purposes.

Theorem 2.46.

The local height function satisfies the following properties.

(1)

It is symmetric and multilinear with respect to $\otimes$ in the pairs $({\overline{L}}_{i},s_{i})$ , $i=0,\dots,d$ , provided that all terms are defined.

(2)

Let $\varphi\colon X^{\prime}\to X$ be a morphism of proper varieties over $K$ , $Y$ a $d$ -dimensional cycle of $X^{\prime}$ , and $({\overline{L}}_{i},s_{i})$ an integrable metrized line bundle on $X$ and a section, $i=0,\dots,d$ . Then

\operatorname{h}_{\varphi^{\ast}{\overline{L}}_{0},\dots,\varphi^{\ast}{\overline{L}}_{d}}(Y;\varphi^{\ast}s_{0},\dots,\varphi^{\ast}s_{d})=\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(\varphi_{\ast}Y;s_{0},\dots,s_{d}),

provided that both terms are defined.

(3)

Let $Z$ be the zero-cycle $Y\cdot\operatorname{div}(s_{0})\cdots\operatorname{div}(s_{d-1})$ and $f$ a rational function such that the section $fs_{d}$ meets $Z$ properly. Then

\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y;s_{0},\dots,s_{d})-\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y;s_{0},\dots,fs_{d})=\log|f(Z)|,

where, if $Z=\sum_{l}m_{l}p_{l}$ , then $f(Z)=\prod_{l}f(p_{l})^{m_{l}}$ .

(4)

Let ${\overline{L^{\prime}}}_{d}=(L_{d},\|\cdot\|^{\prime})$ be another choice of metric. Then

\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y;s_{0},\dots,s_{d})-\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}^{\prime}_{d}}(Y;s_{0},\dots,s_{d})=\\ -\int_{Y}\log(\|s_{d}(p)\|/\|s_{d}(p)\|^{\prime})\operatorname{c}_{1}(\overline{L}_{0})\land\dots\wedge\operatorname{c}_{1}(\overline{L}_{d-1})\land\delta_{Y}

is independent of the choice of sections.

Proof.

In the Archimedean case, statement (1) is [Gub03, Proposition 3.4], statement (2) is [Gub03, Proposition 3.6]. In the non-Archimedean case, statement (1) and (2) are [Gub03, Remark 9.3]. The other two statements follow easily from the definition. ∎