Arakelov geometry, heights, equidistribution, and the Bogomolov conjecture

TL;DR

This paper introduces Arakelov geometry and its applications to heights, equidistribution, and the proof of the Bogomolov conjecture, emphasizing the interplay between classical and adelic frameworks.

Contribution

It reviews key concepts in Arakelov geometry, explains height properties, and discusses the proof of the Bogomolov conjecture using equidistribution techniques.

Findings

Explanation of arithmetic intersection numbers

Discussion of height properties in Arakelov geometry

Outline of Ullmo-Zhang's proof of the Bogomolov conjecture

Abstract

This is an introduction to the topics of the title, from the 2017 Grenoble Summer school on Arakelov geometry and arithmetic applications. We review Arithmetic intersection numbers, explain the definition of the height of a variety and its properties, both in the framework of classical Arakelov geometry and of Zhang's adelic formalism. We then discuss arithmetic ampleness and its application to the equidistribution theorem of Szpiro-Ullmo-Zhang-Yuan, both at complex and nonarchimedean places. We conclude with Ullmo-Zhang's proof of the Bogomolov conjecture over number fields.

Equations

$$\Gamma (\mathcal{X},{\mathcal{O}}_{\mathcal{X}}^{\times })\to {\mathcal{C}}^{\infty }(\mathcal{X}(\mathbf{C}),\mathbf{R})\to \widehat{\mathrm{Pic}}(\mathcal{X})\to \mathrm{Pic}(\mathcal{X})\to 0,$$

$$\widehat{\mathrm{deg}}\left({\hat{c}}_1(\overline{{\mathcal{L}}_1})\dots {\hat{c}}_1(\overline{{\mathcal{L}}_d})\mid \cdot \right)\text{nonscript}:plus1.0mu{\mathrm{Z}}_d(\mathcal{X})\to \mathbf{R},$$

$$\begin{gathered} m\widehat{\mathrm{deg}}\left({\hat{c}}_1(\overline{{\mathcal{L}}_1})\dots {\hat{c}}_1(\overline{{\mathcal{L}}_d})\mid Z\right) \\ =\widehat{\mathrm{deg}}\left({\hat{c}}_1(\overline{{\mathcal{L}}_1})\dots {\hat{c}}_1(\overline{{\mathcal{L}}_{d-1}})\mid \mathrm{div}(s)\right) \\ +{\int }_{Z(\mathbf{C})}\log {\Vert s\Vert }^{-1}{c}_1(\overline{{\mathcal{L}}_1})\dots c_1(\overline{{\mathcal{L}}_{d-1}}). \end{gathered}$$

$$\widehat{\mathrm{deg}}\left(z\right)=\log (\mathrm{Card}(\kappa (z))),$$

$$m\mathrm{deg}(c_1(L_1)\dots c_1(L_d)\mid Z)=\mathrm{deg}(c_1(L_1)\dots c_1(L_{d-1})\mid \mathrm{div}(s)).$$

$$\widehat{\mathrm{deg}}\left({\hat{c}}_1(\overline{{\mathcal{L}}_1})\dots {\hat{c}}_1(\overline{{\mathcal{L}}_d})\mid Z\right)=\mathrm{deg}\left(c_1({\mathcal{L}}_1{|}_{{\mathcal{X}}_{{\mathbf{F}}_p}})\dots c_1({\mathcal{L}}_d{|}_{{\mathcal{X}}_{{\mathbf{F}}_p}})\mid Z\right)\log (p).$$

$$\widehat{\mathrm{deg}}\left({\hat{c}}_1(\overline{\mathcal{L}})\mid Z\right)=\widehat{\mathrm{deg}}({\epsilon }_z^{\ast }\overline{\mathcal{L}})$$

$$\left({\hat{c}}_1(f^{\ast }\overline{{\mathcal{L}}_1})\dots {\hat{c}}_1(f^{\ast }\overline{{\mathcal{L}}_d})\mid Z\right)=0;$$

$$\left({\hat{c}}_1(f^{\ast }\overline{{\mathcal{L}}_1})\dots {\hat{c}}_1(f^{\ast }\overline{{\mathcal{L}}_d})\mid Z\right)=\left({\hat{c}}_1(\overline{{\mathcal{L}}_1})\dots {\hat{c}}_1(\overline{{\mathcal{L}}_d})\mid f_{\ast }(Z)\right),$$

$$\widehat{\mathrm{deg}}\left({\hat{c}}_1(\overline{{\mathcal{L}}_1})\dots {\hat{c}}_1(\overline{{\mathcal{L}}_d})\mid Z\right)=\widehat{\mathrm{deg}}\left({\hat{c}}_1(f^{\ast }\overline{{\mathcal{L}}_1})\dots {\hat{c}}_1(f^{\ast }\overline{{\mathcal{L}}_d})\mid Z^{\prime }\right).$$

$${\mathrm{deg}}_{\mathcal{L}}(Z)=\mathrm{deg}(c_1(L{)}^d\mid Z)$$

$$\displaystyle h_{\overline{\mathscr{L}}}(Z)=\mathop{\widehat{\operatorname{deg}}}\left({\widehat{c}}_{1}(\overline{\mathscr{L}}^{d+1})\mid\mathscr{Z}\right)/\big{(}(d+1)\deg_{\mathscr{L}}(Z)\big{)}.$$

$$h_{f^{\ast }\overline{\mathcal{L}}}(Z)=h_{\overline{\mathcal{L}}}(f(Z)).$$

$${\hat{h}}_L([n](Z))=n^2{\hat{h}}_L(Z).$$

$${\hat{h}}_L([n](Z))=n{\hat{h}}_L(Z).$$

$$|h_{\overline{\mathcal{L}}}(Z)-h_{\overline{{\mathcal{L}}^{\prime }}}(Z)|⩽c$$

$$|\widehat{\mathrm{deg}}({\hat{c}}_1(\overline{\mathcal{L}}{)}^{d+1}\mid \mathcal{Z})-\widehat{\mathrm{deg}}({\hat{c}}_1(\overline{{\mathcal{L}}^{\prime }}{)}^{d+1}\mid {\mathcal{Z}}^{\prime })|⩽c\mathrm{deg}(c_1(L{)}^d\mid Z)$$

$$\sum _{i=0}^d\widehat{\mathrm{deg}}({\hat{c}}_1({\overline{\mathcal{L}}}^{\prime }\otimes {\overline{\mathcal{L}}}^{-1}){\hat{c}}_1({\overline{\mathcal{L}}}^{\prime }{)}^i{\hat{c}}_1(\overline{\mathcal{L}}{)}^{d-i}\mid \mathcal{Z}).$$

$$\mathrm{deg}(c_1({\mathcal{L}}^{\prime }{)}^i{c}_1(\mathcal{L}{)}^{d-i}\mid \mathrm{div}(s{|}_{\mathcal{Z}}))$$

$${\int }_{Z(\mathbf{C})}\log ({\Vert s\Vert }^{-1})c_1({\overline{\mathcal{L}}}^{\prime }{)}^i{c}_1(\overline{\mathcal{L}}{)}^{d-i}.$$

$${\Vert \log (\Vert s\Vert {)}^{-1}\Vert }_{\infty }{\int }_{Z(\mathbf{C})}{c}_1({\overline{\mathcal{L}}}^{\prime }{)}^i{c}_1(\overline{\mathcal{L}}{)}^{d-i}={\Vert \log (\Vert s\Vert {)}^{-1}\Vert }_{\infty }\mathrm{deg}(c_1(L{)}^d\mid Z).$$

$$-\sum _p{v}_p(n)[{\mathcal{Z}}_{{\mathbf{F}}_p}]⩽\mathrm{div}(s{|}_{\mathcal{Z}})⩽\sum _p{v}_p(n)[{\mathcal{Z}}_{{\mathbf{F}}_p}].$$

$$\mathrm{deg}(c_1({\mathcal{L}}^{\prime }{)}^i{c}_1(\mathcal{L}{)}^{d-i}\mid \mathrm{div}(s{|}_{\mathcal{Z}}{)}_p)$$

$$⩽v_p(n)\mathrm{deg}(c_1({\mathcal{L}}^{\prime }{)}^i{c}_1(\mathcal{L}{)}^{d-i}\mid [{\mathcal{Z}}_{{\mathbf{F}}_p}])$$

$$=v_p(n)\mathrm{deg}(c_1(L{)}^d\mid Z),$$

$$\Gamma (X,{\mathcal{O}}_X^{\times })\to {\mathcal{C}}_{\mathrm{c}}(X_{\mathrm{ad}},\mathbf{R})\to \overline{\mathrm{Pic}}(X_{\mathrm{ad}})\to \mathrm{Pic}(X)\to 0.$$

$${\delta }_v(\Vert \cdot \Vert ,{\Vert \cdot \Vert }^{\prime })={\Vert {\phi }_v\Vert }_{\infty }=\underset{x\in X_v^{\mathrm{an}}}{\sup }\left|\log \frac{{\Vert \cdot \Vert }^{\prime }}{\Vert \cdot \Vert }(x)\right|.$$

$$\delta (\Vert \cdot \Vert ,{\Vert \cdot \Vert }^{\prime })=\sum _{v\in S}{\delta }_v({\Vert \cdot \Vert }^{\prime },\Vert \cdot \Vert ).$$

$$\Vert s_f\Vert (x)=\frac{|f(x_0,\dots ,x_N)|}{{\left(|x_0{|}^2+\cdots +|x_N{|}^2\right)}^{1/2}}.$$

$$\Big{|}\mathop{\widehat{\operatorname{deg}}}\left({\widehat{c}}_{1}(\overline{\mathscr{L}^{\prime}_{0}})\dots\widehat{c}_{1}(\overline{\mathscr{L}^{\prime}_{d}})\mid Z\right)-\mathop{\widehat{\operatorname{deg}}}\left({\widehat{c}}_{1}(\overline{\mathscr{L}_{0}})\dots\widehat{c}_{1}(\overline{\mathscr{L}_{d}})\mid Z\right)\Big{|}\\ \leqslant\sum_{j=0}^{d}\delta(\overline{L_{j}},\overline{L^{\prime}_{j}})\deg\left(c_{1}(L_{0})\dots\widehat{c_{1}(L_{j})}\dots c_{1}(L_{d})\mid Z\right),$$

$$\begin{gathered} \widehat{\mathrm{deg}}\left({\hat{c}}_1(\overline{{\mathcal{L}}_0^{\prime }})\dots {\hat{c}}_1(\overline{{\mathcal{L}}_d^{\prime }})\mid Z\right)-\widehat{\mathrm{deg}}\left({\hat{c}}_1(\overline{{\mathcal{L}}_0})\dots {\hat{c}}_1(\overline{{\mathcal{L}}_d})\mid Z\right) \\ =\sum _{j=0}^d\widehat{\mathrm{deg}}\left({\hat{c}}_1(\overline{{\mathcal{L}}_0^{\prime }})\dots {\hat{c}}_1(\overline{{\mathcal{L}}_{j-1}^{\prime }})({\hat{c}}_1(\overline{{\mathcal{L}}_j^{\prime }})-{\hat{c}}_1(\overline{{\mathcal{L}}_j})){\hat{c}}_1(\overline{{\mathcal{L}}_{j+1}})\dots {\hat{c}}_1(\overline{{\mathcal{L}}_d})\mid Z\right) \end{gathered}$$

$$\widehat{\mathrm{deg}}\left({\hat{c}}_1(\overline{{\mathcal{L}}_0^{\prime }})\dots {\hat{c}}_1(\overline{{\mathcal{L}}_{j-1}^{\prime }})({\hat{c}}_1(\overline{{\mathcal{L}}_j^{\prime }})-{\hat{c}}_1(\overline{{\mathcal{L}}_j})){\hat{c}}_1(\overline{{\mathcal{L}}_{j+1}})\dots {\hat{c}}_1(\overline{{\mathcal{L}}_d})\mid Z\right)$$

Full text

1. 1 Introduction

2. 2 Arithmetic intersection numbers

3. 2.1

4. 2.3

5. 2.4

6. 3 The height of a variety

7. 3.1

8. 3.2

9. 4 Adelic metrics

10. 4.1

11. 4.2

12. 4.3

13. 4.5

14. 4.7

15. 4.10

16. 5 Arithmetic ampleness

17. 6 Measures

18. 7 Volumes

19. 7.1

20. 7.3

21. 8 Equidistribution

22. 8.3

23. 9 The Bogomolov conjecture

24. 9.1

25. 9.2

26. 9.7

27. 9.8

28. 9.10

1 Introduction

Fermat’s method of infinite descent studies the solutions to diophantine equations by constructing, from a given solution of a diophantine equation, a smaller solution, and ultimately deriving a contradiction. In order to formalize the intuitive notion of “size” of an algebraic solution of a diophantine equation, Northcott (1950) and Weil (1951) have introduced the notion of height of an algebraic point of an algebraic variety defined over a number field and established their basic functorial properties, using the decomposition theorem of Weil (1929). The height machine is now an important tool in modern diophantine geometry.

The advent of arithmetic intersection theory with Arakelov (1974) and, above all, its extension in any dimension by Gillet & Soulé (1990) (“Arakelov geometry”) has led Faltings (1991) to extend the concept further by introducing the height of a subvariety, defined in pure analogy with its degree, replacing classical intersection theory with arithmetic intersection theory. This point of view has been developed in great depth by Bost \bbletal (1994) and Zhang (1995a).

Although I shall not use it in these notes, I also mention the alternative viewpoint of Philippon (1991) who defines the height of a subvariety as the height of the coefficients-vector of its “Chow form”.

The viewpoint of adelic metrics introduced in Zhang (1995b) is strengthened by the introduction of Berkovich spaces in this context, based on Gubler (1998), and leading to the definition by Chambert-Loir (2006) of measures at all places analogous to product of Chern forms at the archimedean place.

We describe these notions in the first sections of the text. We then present the equidistribution theorem of Szpiro \bbletal (1997) and its extension by Yuan (2008). Finally, we use these ideas to explain the proof of Bogomolov’s conjecture, following Ullmo (1998) and Zhang (1998).

I thank the editors of this volume for their invitation to lecture on this topic in Grenoble, and the referees for their patient comments that helped me improve this survey.

I started my graduate education under the guidance of Lucien Szpiro. He introduced me to heights, abelian varieties and Arakelov geometry almost 30 years ago. As a sad coincidence, he passed away a few weeks before I was to make the final modifications to this text, at a time when a viral pandemy locked down half of humanity. I dedicate this survey to his memory.

2 Arithmetic intersection numbers

2.1

Let $\mathscr{X}$ be a proper flat scheme over $\mathbf{Z}$. For every integer $d\geqslant 0$, let $\mathrm{Z}_{d}(\mathscr{X})$ be the group of $d$-cycles on $\mathscr{X}$: it is the free abelian group generated by integral closed subschemes of dimension $d$.

\remaname* 2.2**.*

Let $f\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathscr{X}\to\operatorname{Spec}(\mathbf{Z})$ be the structural morphism. By assumption, $f$ is proper so that the image of an integral closed subscheme $Z$ of $\mathscr{X}$ is again an integral closed subscheme of $\operatorname{Spec}(\mathbf{Z})$. There are thus two cases:

Either $f(Z)=\operatorname{Spec}(\mathbf{Z})$, in which case we say that $Z$ is horizontal; 2. 2.

Or $f(Z)=\{(p)\}$ for some prime number $p$, in which case we say that $Z$ is vertical.

2.3

The set $\mathscr{X}(\mathbf{C})$ of complex points of $\mathscr{X}$ has a natural structure of a complex analytic space, smooth if and only if $\mathscr{X}_{\mathbf{Q}}$ is regular. This gives rise to the notions of continuous, resp. smooth, resp. holomorphic function on $\mathscr{X}(\mathbf{C})$: by definition, this is a function which, for every local embedding of an open subset $U$ of $\mathscr{X}(\mathbf{C})$ into $\mathbf{C}^{n}$, extends to a continuous, resp. smooth, resp. holomorphic function around the image of $U$.

Let $\mathscr{L}$ be a line bundle on $\mathscr{X}$. A hermitian metric on $\mathscr{L}$ is the datum, for every open subset $U$ of $\mathscr{X}(\mathbf{C})$ and every section $s\in\Gamma(U,\mathscr{L})$ of a continuous function $\left\|s\right\|\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0muU\to\mathbf{R}_{+}$, subject to the following conditions:

For every open subset $V$ of $U$, one has $\left\|s|_{V}\right\|=\left\|s\right\||_{V}$; 2. 2.

For every holomorphic function $f\in\mathscr{O}_{\mathscr{X}}(U)$, one has $\left\|fs\right\|=\mathopen{|}f\mathclose{|}\left\|s\right\|$; 3. 3.

If $s$ does not vanish on $U$, then the function $\left\|s\right\|$ is strictly positive and smooth.

A hermitian line bundle $\overline{\mathscr{L}}$ on $\mathscr{X}$ is a line bundle $\mathscr{L}$ endowed with a hermitian metric.

With respect to the tensor product of underlying line bundles and the tensor product of hermitian metrics, the set of isomorphism classes of hermitian line bundles on $\mathscr{X}$ is an abelian group, denoted by ${\widehat{\operatorname{Pic}}}(\mathscr{X})$. This group fits within an exact sequence of abelian groups:

[TABLE]

where the first map is $f\mapsto\log\mathopen{|}f\mathclose{|}$, the second associates with $\varphi\in\mathscr{C}^{\infty}(\mathscr{X}(\mathbf{C}),\mathbf{R})$ the trivial line bundle $\mathscr{O}_{\mathscr{X}}$ endowed with the hermitian metric for which $\log\left\|1\right\|^{-1}=\varphi$, and the last one forgets the metric.

2.4

The starting point of our lectures will be the following theorem that asserts existence and uniqueness of “arithmetic intersection degrees” of cycles associated with hermitian line bundles. It fits naturally within the arithmetic intersection theory of Gillet & Soulé (1990), we refer to the foundational article by Bost \bbletal (1994) for such an approach; see also Faltings (1992) for a direct construction, as well as to the notes of Soulé in this volume.

\theoname** 2.5****.**

Let $n=\dim(\mathscr{X})$ and let $\overline{\mathscr{L}_{1}},\dots,\overline{\mathscr{L}_{n}}$ be hermitian line bundles on $\mathscr{X}$. There exists a unique family of linear maps:

[TABLE]

for $d\in\{0,\dots,n\}$ satisfying the following properties:

For every integer $d\in\{1,\dots,n\}$, every integral closed subscheme $Z$ of $\mathscr{X}$ such that $\dim(Z)=d$, every integer $m\neq 0$ and every regular meromorphic111that is, defined over a dense open subscheme of $Z$* section $s$ of $\mathscr{L}_{d}^{m}|_{Z}$, one has*

[TABLE] 2. 2.

For every closed point $z$ of $\mathscr{X}$, viewed as a integral closed subscheme of dimension $d=0$, one has

[TABLE]

where $\kappa(z)$ is the residue field of $z$.222It is indeed a finite field.

Moreover, these maps are multilinear and symmetric in the hermitian line bundles $\overline{\mathscr{L}_{1}},\dots,\overline{\mathscr{L}_{n}}$ and only depend on their isomorphism classes in ${\widehat{\operatorname{Pic}}}(\mathscr{X})$.

\remaname* 2.6**.*

This theorem should be put in correspondence with the analogous geometric result for classical intersection numbers. Let $F$ be a field and let $X$ be a proper scheme over $F$, let $n=\dim(X)$ and let $L_{1},\dots,L_{n}$ be line bundles over $X$. The degree $\deg(c_{1}(L_{1})\dots c_{1}(L_{d})\mid Z)$ of a $d$-cycle $Z$ in $X$ is characterized by the relations:

It is linear in $Z$; 2. 2.

If $d=0$ and $Z$ is a closed point $z$ whose residue field $\kappa(Z)$ is a finite extension of $F$, then $\deg(Z)=[\kappa(Z):F]$; 3. 3.

If $d\geqslant 1$ and $Z$ is an integral closed subscheme of $X$ of dimension $d$, $m$ a non-zero integer, $s$ a regular meromorphic section of $L_{d}^{m}|_{Z}$, then

[TABLE]

The additional integral that appears in the arithmetic degree takes into account the fact that $\operatorname{Spec}(\mathbf{Z})$ does not behave as a proper variety.

\exemname* 2.7**.*

Assume that $Z$ is vertical and lies over a maximal ideal $(p)$ of $\operatorname{Spec}(\mathbf{Z})$. Then $Z$ is a proper scheme over $\mathbf{F}_{p}$ and it follows from the inductive definition and the analogous formula in classical intersection theory that

[TABLE]

\exemname* 2.8**.*

Assume that $d=1$ and that $Z$ is horizontal, so that $Z$ is the Zariski-closure in $\mathscr{X}$ of a closed point $z\in\mathscr{X}_{\mathbf{Q}}$. Let $F=\kappa(z)$ and let $\mathfrak{o}_{F}$ be its ring of integers; by properness of $\mathscr{X}$, the canonical morphism $\operatorname{Spec}(F)\to\mathscr{X}$ with image $z$ extends to a morphism $\varepsilon_{z}\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\operatorname{Spec}(\mathfrak{o}_{F})\to\mathscr{X}$, whose image is $Z$. The formula

[TABLE]

expresses the arithmetic intersection number as the arithmetic degree of the hermitian line bundle $\varepsilon_{z}^{*}\overline{\mathscr{L}}$ over $\operatorname{Spec}(\mathfrak{o}_{F})$.

\propname** 2.9****.**

Let $f\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathscr{X}^{\prime}\to\mathscr{X}$ be a generically finite morphism of proper flat schemes over $\mathbf{Z}$, let $Z$ be an integral closed subscheme of $\mathscr{X}^{\prime}$ and let $d=\dim(Z)$.

If $\dim(f(Z))<d$, then

[TABLE] 2. 2.

Otherwise, $\dim(f(Z))=d$ and

[TABLE]

where $f_{*}(Z)=[\kappa(Z):\kappa(f(Z))]f(Z)$ is a $d$-cycle on $\mathscr{X}$.

This is the “projection formula” for height. It can be proved by induction using the inductive definition of the heights and the change of variables formula.

\remaname* 2.10**.*

Let $n=\dim(\mathscr{X})$ and assume that $\mathscr{X}$ is regular. As the notation suggests rightly, the arithmetic intersection theory of Gillet & Soulé (1990) allows another definition of the real number $\mathop{\widehat{\operatorname{deg}}}\left({\widehat{c}}_{1}(\overline{\mathscr{L}_{1}})\dots\widehat{c}_{1}(\overline{\mathscr{L}_{n}})\mid\mathscr{X}\right)$ as the arithmetic degree of the [math]-dimensional arithmetic cycle ${\widehat{c}}_{1}(\overline{\mathscr{L}_{1}})\dots\widehat{c}_{1}(\overline{\mathscr{L}_{n}})\in{\widehat{\operatorname{CH}}}_{0}(\mathscr{X})$.

In fact, while the theory of Gillet & Soulé (1990) imposes regularity conditions on $\mathscr{X}$, the definition of arithmetic product of classes of the form ${\widehat{c}}_{1}(\overline{\mathscr{L}})$ requires less stringent conditions; in particular, the regularity of the generic fiber $\mathscr{X}_{\mathbf{Q}}$ is enough. See Faltings (1992) for such an approach, as well as Soulé’s notes in this volume. More generally, for every birational morphism $f\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0muZ^{\prime}\to Z$ such that $Z^{\prime}_{\mathbf{Q}}$ is regular, one has

[TABLE]

3 The height of a variety

3.1

Let $X$ be a proper $\mathbf{Q}$-scheme and let $L$ be a line bundle on $X$. The important case is when the line bundle $L$ is ample, an assumption which will often be implicit below; in that case, the pair $(X,L)$ is called a polarized variety.

3.2

Let $\mathscr{X}$ be a proper flat scheme over $\mathbf{Z}$ and let $\overline{\mathscr{L}}$ be a hermitian line bundle on $\mathscr{X}$ such that $\mathscr{X}_{\mathbf{Q}}=X$ and $\mathscr{L}_{\mathbf{Q}}=L$. Let $Z$ be a closed integral subscheme of $X$ and let $d=\dim(Z)$. Let $\mathscr{Z}$ be the Zariski-closure of $Z$ in $\mathscr{X}$; it is an integral closed subscheme of $\mathscr{X}$ and $\dim(\mathscr{Z})=d+1$.

\definame** 3.3****.**

The degree and the height of $Z$ relative to $\overline{\mathscr{L}}$ are defined by the formulas (provided $\deg_{\mathscr{L}}(Z)\neq 0$).

[TABLE]

Note that the degree $\deg_{\mathscr{L}}(Z)$ is computed on $X$, hence only depends on $L$. Moreover, the condition that $\deg_{\mathscr{L}}(Z)\neq 0$ is satisfied (for every $Z$) when $L$ is ample on $X$.

\propname** 3.4****.**

Let $f\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathscr{X}^{\prime}\to\mathscr{X}$ be a generically finite morphism of proper flat schemes over $\mathbf{Z}$, let $Z$ be a closed integral subscheme of $\mathscr{X}^{\prime}_{\mathbf{Q}}$ and let $d=\dim(Z)$. Assume that $L$ is ample on $X$ and that $\dim(f(Z))=d$. Then $\deg_{f^{*}\mathscr{L}}(Z)>0$ and

[TABLE]

Démonstration.

This follows readily from proposition 2.9 and its analogue for geometric degrees. Indeed, when one compares formula (3.3.2) for $Z$ and for $f(Z)$, both the numerator and the denominator get multiplied by $[\kappa(Z):\kappa(f(Z))]$. ∎

\exemname* 3.5**.*

For every $x\in X(\overline{\mathbf{Q}})$, let $[x]$ denote its Zariski closure in $X$. The function $X(\overline{\mathbf{Q}})\to\mathbf{R}$ given by $x\mapsto h_{\overline{\mathscr{L}}}([x])$ is a height function relative to the line bundle $\mathscr{L}_{\mathbf{Q}}$ on $X$ in the sense of Weil. Note that while the height functions defined by Weil which are associated with a line bundle $L$ on $X$ are only defined modulo the space of bounded functions on $X(\overline{\mathbf{Q}})$, the techniques of Arakelov geometry show that specifying a model $\mathscr{X}$ of $X$ and a hermitian line bundle $\overline{\mathscr{L}}$ on $\mathscr{X}$ such that $\mathscr{L}_{\mathbf{Q}}$ is isomorphic to $L$ allows to obtain a well-defined representative of this class, for which precise theorems, such as the equidistribution theorem below, can be proved.

\exemname* 3.6**.*

Let us assume that $X$ is an abelian variety over a number field $F$, with everywhere good reduction, and let $\mathscr{X}$ be an $\mathfrak{o}_{F}$-abelian scheme such that $\mathscr{X}_{F}=X$. Let $o$ be the origin of $X$ and let $\varepsilon_{o}\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\operatorname{Spec}(\mathfrak{o}_{F})\to\mathscr{X}$ be the corresponding section. Let $L$ be a line bundle on $X$ with a trivialisation $\ell$ of $L|_{o}$. There exists a unique line bundle $\mathscr{L}$ on $\mathscr{X}$ such that $\mathscr{L}_{F}=L$ and such that the given trivialisation of $L|_{o}$ extends to a trivialisation of $\varepsilon_{o}^{*}\mathscr{L}$. Moreover, for every embedding $\sigma\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0muF\hookrightarrow\mathbf{C}$ the theory of Riemann forms on complex tori endows $L_{\sigma}$ with a canonical metric $\left\|\cdot\right\|_{\sigma}$ whose curvature form $c_{1}(L_{\sigma},\left\|\cdot\right\|_{\sigma})$ is invariant by translation and such that $\left\|\ell\right\|_{\sigma}=1$; this is in fact the unique metric possessing these two properties. We let $\overline{\mathscr{L}}$ be the hermitian line bundle on $\mathscr{X}$ so defined.

The associated height function will be denoted by $\widehat{h}_{L}$: it extends the Néron–Tate height from $X(\overline{\mathbf{Q}})$ to all integral closed subschemes.

Assume that $L$ is even, that is $[-1]^{*}L\simeq L$. Then $[n]^{*}L\simeq L^{n^{2}}$ for every integer $n\geqslant 1$, and this isomorphism extends to an isomorphism of hermitian line bundles $[n]^{*}\overline{\mathscr{L}}\simeq\overline{\mathscr{L}}^{n^{2}}$. Consequently, for every integral closed subscheme $Z$ of $X$, one has the following relation

[TABLE]

Assume otherwise that $L$ is odd, that is $[-1]^{*}L\simeq L^{-1}$. Then $[n]^{*}L\simeq L^{n}$ for every integer $n\geqslant 1$; similarly, this isomorphism extends to an isomorphism of hermitian line bundles $[n]^{*}\overline{\mathscr{L}}\simeq\overline{\mathscr{L}}^{n}$. Consequently, for every integral closed subscheme $Z$ of $X$, one has the following relation

[TABLE]

\propname** 3.7****.**

Let $\mathscr{X}^{\prime}$ be a proper flat scheme over $\mathbf{Z}$ such that $\mathscr{X}^{\prime}_{\mathbf{Q}}=X$; let $\overline{\mathscr{L}^{\prime}}$ be a hermitian line bundle on $\mathscr{X}^{\prime}$ such that $\mathscr{L}^{\prime}_{\mathbf{Q}}=L$. Assume that $L$ is ample. Then there exists a real number $c$ such that

[TABLE]

for every integral closed subscheme $Z$ of $X$.

Démonstration.

One proves in fact the existence of a real number $c$ such that

[TABLE]

for every integral $d$-dimensional subvariety $Z$ of $X$, where $\mathscr{Z}$ and $\mathscr{Z}^{\prime}$ are the Zariski closures of $Z$ in $\mathscr{X}$ and $\mathscr{X}^{\prime}$ respectively. Considering a model $\mathscr{X}^{\prime\prime}$ that dominates $\mathscr{X}$ and $\mathscr{X}^{\prime}$ (for example, the Zariski closure in $\mathscr{X}\times_{\mathbf{Z}}\mathscr{X}^{\prime}$ of the diagonal), we may assume that $\mathscr{X}=\mathscr{X}^{\prime}$, hence $\mathscr{Z}=\mathscr{Z}^{\prime}$. A further reduction, that we omit here, allows us to assume that $\mathscr{L}$ is a nef line bundle, and that its hermitian metric is semipositive, and similarly for $\mathscr{L}^{\prime}$.

By multilinearity, the left hand side that we wish to bound from above is the absolute value of

[TABLE]

Then we view the section $1$ of $\mathscr{L}^{\prime}_{\mathbf{Q}}\otimes\mathscr{L}^{-1}_{\mathbf{Q}}$ as a meromorphic section $s$ of $\mathscr{L}^{\prime}\otimes\mathscr{L}^{-1}$. Note that its divisor is purely vertical, and its hermitian norm $\left\|s\right\|$ is a non-vanishing continuous function on $X(\mathbf{C})$. The definition of the arithmetic intersection numbers then leads us to estimate algebraic intersection numbers

[TABLE]

and an integral

[TABLE]

By positivity of the curvatures forms $c_{1}(\overline{\mathscr{L}})$ and $c_{1}(\overline{\mathscr{L}^{\prime}})$, the latter integral is bounded from above by

[TABLE]

The algebraic terms can be bounded as well. Observe that there exists an integer $n\geqslant 1$ such that $ns$ extends to a global section of $\mathscr{L}^{\prime}\otimes\mathscr{L}^{-1}$, and $ns^{-1}$ extends to a global section of its inverse $(\mathscr{L}^{\prime})^{-1}\otimes\mathscr{L})$. (This is the ultrametric counterpart to the fact that the section $s$ has non-vanishing norm on $X(\mathbf{C})$.) Consequently, $\operatorname{div}(ns|_{\mathscr{Z}})$ and $\operatorname{div}(ns^{-1}|_{\mathscr{Z}})$ are both effective, so that

[TABLE]

This inequality of cycles is preserved after taking intersections, so that

[TABLE]

where $\operatorname{div}(s|_{\mathscr{Z}})_{p}$ is the part of $\operatorname{div}(s|_{\mathscr{Z}})$ that lies above the maximal ideal $(p)$ of $\operatorname{Spec}(\mathbf{Z})$. There is a similar lower bound.

Adding all these contributions, this proves the proposition. We refer to Bost \bbletal (1994), §3.2.2, for more details. ∎

\propname** 3.8****.**

Let us assume that $L$ is ample. For every real number $B$, the set of integral closed subschemes $Z$ of $X$ such that $\deg_{L}(Z)\leqslant B$ and $h_{\overline{\mathscr{L}}}(Z)\leqslant B$ is finite.

The case of closed points is Northcott’s theorem, and the general case is Theorem 3.2.5 of Bost \bbletal (1994). The principle of its proof goes by reducing to the case where $X=\mathbf{P}^{N}$ and $\overline{\mathscr{L}}=\overline{\mathscr{O}(1)}$, and comparing the height $h_{\overline{\mathscr{L}}}(Z)$ of a closed integral subscheme $Z$ with the height of its Chow form. (That paper also provides a more elementary proof, relying on the fact that a finite set of sections of powers of $\mathscr{O}(1)$ are sufficient to compute by induction the height of any closed integral subscheme of $\mathbf{P}^{N}$ of given degree.)

4 Adelic metrics

4.1

Let $S=\{2,3,\dots,\infty\}$ be the set of places of $\mathbf{Q}$.

Each prime number $p$ is identified with the $p$-adic absolute value on $\mathbf{Q}$, normalized by $\mathopen{|}p\mathclose{|}_{p}=1/p$; these places are said to be finite. We denote by $\mathbf{Q}_{p}$ the completion of $\mathbf{Q}$ for this $p$-adic absolute value and fix an algebraic closure $\overline{\mathbf{Q}_{p}}$ of $\mathbf{Q}_{p}$. The $p$-adic absolute value extends uniquely to $\overline{\mathbf{Q}_{p}}$; the corresponding completion is denoted by $\mathbf{C}_{p}$: this is an algebraically closed complete valued field.

The archimedean place is represented by the symbol $\infty$, and is identified with the usual absolute value on $\mathbf{Q}$; it is also called the infinite place. For symmetry of notation, we may write $\mathbf{Q}_{\infty}=\mathbf{R}$ and $\mathbf{C}_{\infty}=\mathbf{C}$, the usual fields of real and complex numbers.

The notion of adelic metrics that we now describe is close in spirit to the one exposed by Peyre in this volume. However, Peyre is only interested in rational points and it suffices for him to consider local fields such as $\mathbf{Q}_{p}$, and “naïve $p$-adic analytic varieties”. We will however need to consider algebraic points. The topological spaces associated with varieties over $\mathbf{C}_{p}$ are not well behaved enough, and we will have to use a rigid analytic notion of $p$-adic varieties, specifically the one introduced by Berkovich.

4.2

Let $X$ be a proper scheme over $\mathbf{Q}$. Let $v\in S$ be a place of $\mathbf{Q}$.

Assume $v=\infty$. Then we set $X_{\infty}^{\mathrm{an}}=X(\mathbf{C}_{\infty})/F_{\infty}$, the set of complex points of $X$ modulo the action of complex conjugation $F_{\infty}$.

Assume now that $v=p$ is a finite place. Then we set $X_{p}^{\mathrm{an}}$ to be the analytic space associated by Berkovich (1990) to the $\mathbf{Q}_{p}$-scheme $X_{p}=X_{\mathbf{Q}_{p}}$. As a set, $X_{p}^{\mathrm{an}}$ is the quotient of the class of all pairs $(K,x)$, where $K$ is a complete valued field containing $\mathbf{Q}_{p}$ and $x\in X(K)$, modulo the identification $(K,x)\sim(K^{\prime},x^{\prime})$ if there exists a common valued field extension $K^{\prime\prime}$ of $K$ and $K^{\prime}$ such that $x$ and $x^{\prime}$ induce the same point of $X(K^{\prime\prime})$. Its topology is the coarsest topology such that for every open subscheme $U$ of $X$ and every $f\in\mathscr{O}_{X}(U)$, the subset $U_{p}^{\mathrm{an}}$ (consisting of classes $[K,x]$ such that $x\in U(K)$) is open in $X_{p}^{\mathrm{an}}$, and the function $[K,x]\mapsto\mathopen{|}f(x)\mathclose{|}$ from $U_{p}^{\mathrm{an}}$ to $\mathbf{R}_{+}$ is continuous.

The space $X_{p}^{\mathrm{an}}$ is a compact metrizable topological space, locally contractible (in particular locally arcwise connected). There is a canonical continuous map $X(\mathbf{C}_{p})\to X_{p}^{\mathrm{an}}$; it identifies the (totally discontinuous) topological space $X(\mathbf{C}_{p})/\operatorname{Gal}(\mathbf{C}_{p}/\mathbf{Q}_{p})$ with a dense subset of $X_{p}^{\mathrm{an}}$. It is endowed with a sheaf in local rings $\mathscr{O}_{X_{p}^{\mathrm{an}}}$; for every open subset $U$ of $X_{p}^{\mathrm{an}}$, every holomorphic function $f\in\mathscr{O}_{X_{p}^{\mathrm{an}}}(U)$ admits an absolute value $\mathopen{|}f\mathclose{|}\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0muU\to\mathbf{R}_{+}$.

We gather all places together and consider the topological space $X_{\mathrm{ad}}=\coprod_{v\in S}X_{v}^{\mathrm{an}}$, coproduct of the family $(X_{v}^{\mathrm{an}})_{v\in S}$. By construction, a function $\varphi$ on $X_{\mathrm{ad}}$ consists in a family $(\varphi_{v})_{v\in S}$, where $\varphi_{v}$ is a function on $X_{v}^{\mathrm{an}}$, for every $v\in S$.

4.3

Let $L$ be a line bundle on $X$; it induces a line bundle $L_{v}^{\mathrm{an}}$ on $X_{v}^{\mathrm{an}}$ for every place $v$.

A continuous $v$-adic metric on $L_{v}^{\mathrm{an}}$ is the datum, for every open subset $U$ of $X_{v}^{\mathrm{an}}$ and every section $s$ of $L_{v}^{\mathrm{an}}$ on $U$, of a continuous function $\left\|s\right\|\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0muU\to\mathbf{R}_{+}$, subject to the requirements:

For every open subset $V$ of $U$, one has $\left\|s|_{V}\right\|=\left\|s\right\||_{V}$; 2. 2.

For every holomorphic function $f\in\mathscr{O}_{X_{v}^{\mathrm{an}}}(U)$, one has $\left\|fs\right\|=\mathopen{|}f\mathclose{|}\left\|s\right\|$. 3. 3.

If $s$ does not vanish, then $\left\|s\right\|$ does not vanish as well.

If $L$ and $M$ are line bundles on $X$ equipped with $v$-adic metrics, then $L^{-1}$ and $L\otimes M$ admit natural $v$-adic metrics, and the canonical isomorphism $L^{-1}\otimes L\simeq\mathscr{O}_{X}$ is an isometry.

The trivial line bundle $\mathscr{O}_{X}$ admits a canonical $v$-adic metric for which $\left\|f\right\|=\mathopen{|}f\mathclose{|}$ for every local section of $\mathscr{O}_{X}$. More generally, for every $v$-adic metric $\left\|\cdot\right\|$ on $\mathscr{O}_{X}$, $\varphi=\log\left\|1\right\|^{-1}$ is a continuous function on $X_{v}^{\mathrm{an}}$, and any $v$-adic metric on $\mathscr{O}_{X}$ is of this form. The $v$-adically metrized line bundle associated with $\varphi$ is denoted by $\mathscr{O}_{X}(\varphi)$.

If $\overline{L}$ is a line bundle endowed with an $v$-adic metric and $\varphi\in\mathscr{C}(X_{v}^{\mathrm{an}},\mathbf{R})$, we denote by $\overline{L}(\varphi)$ the $v$-adically metrized line bundle $\overline{L}\otimes\mathscr{O}_{X}(\varphi)$. Explicitly, its $v$-adic metric is that of $\overline{L}$ multiplied by $e^{-\varphi}$.

\exemname* 4.4**.*

Let $\mathscr{X}$ be a proper flat scheme over $\mathbf{Z}$ such that $\mathscr{X}_{\mathbf{Q}}=X$, let $d$ be a positive integer and let ${\mathscr{L}}$ be a line bundle on $\mathscr{X}$ such that $\mathscr{L}_{\mathbf{Q}}=L^{d}$. Let us show that this datum endows $L$ with an $p$-adic metric, for every finite place $p\in S$.

Let thus fix a prime number $p$. There exists a canonical specialization map, $X_{p}^{\mathrm{an}}\to\mathscr{X}\otimes_{\mathbf{Z}}\mathbf{F}_{p}$; it is anticontinuous (the inverse image of an open subset is closed). For every open subset $\mathscr{U}\subset\mathscr{X}\otimes_{\mathbf{Z}}\mathbf{F}_{p}$, let $\mathopen{]}\mathscr{U}\mathclose{[}$ be the preimage of $\mathscr{U}$.

There exists a unique continuous metric on $L_{p}^{\mathrm{an}}$ such that for every open subscheme $\mathscr{U}$ of $\mathscr{X}\otimes_{\mathbf{Z}}\mathbf{Z}_{p}$ and every basis $\ell$ of $\mathscr{L}$ on $\mathscr{U}$, one has $\left\|\ell\right\|\equiv 1$ on $\mathopen{]}\mathscr{U}\otimes\mathbf{F}_{p}\mathclose{[}$. Explicitly, if $s$ is a section of $L_{p}^{\mathrm{an}}$ on an open subset $U$ of $\mathopen{]}\mathscr{U}\otimes\mathbf{F}_{p}\mathclose{[}$, there exists a holomorphic function $f\in\mathscr{O}_{X_{p}^{\mathrm{an}}}(U)$ such that $s^{d}=f\ell$ and $\left\|s\right\|=\mathopen{|}f\mathclose{|}^{1/d}$ on $U$.

Such $p$-adic metrics are called algebraic.

4.5

An adelic metric on $L$ is the datum, for every place $v\in S$, of a $v$-adic metric on the line bundle $L_{v}^{\mathrm{an}}$ on $X_{v}^{\mathrm{an}}$, subject to the additional requirement that there exists a model $(\mathscr{X},\mathscr{L})$ of $(X,L)$ inducing the given $p$-adic metric for all but finitely many primes $p$.

If $L$ and $M$ are line bundles on $X$ equipped with adelic metrics, then $L^{-1}$ and $L\otimes M$ admit natural adelic metrics, and the canonical isomorphism $L^{-1}\otimes L\simeq\mathscr{O}_{X}$ is an isometry.

The trivial line bundle $\mathscr{O}_{X}$ admits a canonical adelic metric for which $\left\|f\right\|=\mathopen{|}f\mathclose{|}$ for every local section of $\mathscr{O}_{X}$. More generally, for every adelic metric $\left\|\cdot\right\|$ on $\mathscr{O}_{X}$, and every place $v\in S$, then $\varphi_{v}=\log\left\|1\right\|_{v}^{-1}$ is a continuous function on $X_{v}^{\mathrm{an}}$, and is identically zero for all but finitely many places $v$; in other words, the function $\varphi=(\varphi_{v})\in\mathscr{C}(X_{\mathrm{ad}},\mathbf{R})$ has compact support. The adelically metrized line bundle associated with $\varphi$ is denoted by $\mathscr{O}_{X}(\varphi)$. Conversely, any adelic metric on $\mathscr{O}_{X}$ is of this form.

If $\overline{L}$ is a line bundle endowed with an adelic metric and $\varphi\in\mathscr{C}_{\mathrm{c}}(X_{\mathrm{ad}},\mathbf{R})$, we denote by $\overline{L}(\varphi)$ the adelically metrized line bundle $\overline{L}\otimes\mathscr{O}_{X}(\varphi)$. Explicitly, for every place $v$, its $v$-adic metric is that of $\overline{L}$ multiplied by $e^{-\varphi_{v}}$.

\remaname* 4.6**.*

Let $(\mathscr{X},\mathscr{L})$ and $(\mathscr{X}^{\prime},\mathscr{L}^{\prime})$ be two models of the polarized variety $(X,L)$. Since $X$ is finitely presented, there exists a dense open subscheme $U$ of $\operatorname{Spec}(\mathbf{Z})$ such that the isomorphism $\mathscr{X}_{\mathbf{Q}}=X=\mathscr{X}^{\prime}_{\mathbf{Q}}$ extends to an isomorphism $\mathscr{X}_{U}\simeq\mathscr{X}^{\prime}_{U}$. Then, up to shrinking $U$, we may assume that the isomorphism $\mathscr{L}_{\mathbf{Q}}=L=\mathscr{L}^{\prime}_{\mathbf{Q}}$ extends to an isomorphism $\mathscr{L}_{U}\simeq\mathscr{L}^{\prime}_{U}$. In particular, for every prime number $p$ such that $(p)\in U$, the $p$-adic norms on $L$ induced by $\mathscr{L}$ and $\mathscr{L}^{\prime}$ coincide.

4.7

Let $\overline{\operatorname{Pic}}(X_{\mathrm{ad}})$ be the abelian group of isomorphism classes of line bundles on $X$ endowed with adelic metrics. It fits within an exact sequence

[TABLE]

The morphism on the left is given by $u\mapsto(\log\mathopen{|}u\mathclose{|}_{v}^{-1})_{v\in S}$. It is injective up to torsion, as a consequence of Kronecker’s theorem: if $\mathopen{|}u\mathclose{|}_{v}=1$ for every place $v$, then there exists $m\geqslant 1$ such that $u^{m}=1$. Its image is the kernel of the morphism $\mathscr{C}(X_{\mathrm{ad}},\mathbf{R})\to\overline{\operatorname{Pic}}(X)$; indeed, an isometry $\mathscr{O}_{X}(\varphi)\to\mathscr{O}_{X}(\psi)$ is given by an element $u\in\Gamma(X,\mathscr{O}_{X}^{\times})$ such that $\psi_{v}+\log\mathopen{|}u\mathclose{|}_{v}^{-1}=\varphi_{v}$, for every place $v\in S$.

We denote by ${\widehat{c}}_{1}(\overline{L})$ the isomorphism class in $\overline{\operatorname{Pic}}(X_{\mathrm{ad}})$ of an adelically metrized line bundle $\overline{L}$ on $X$.

\remaname* 4.8**.*

Let $D$ be an effective Cartier divisor on $X$ and let $\mathscr{O}_{X}(D)$ be the corresponding line bundle; let $s_{D}$ be its canonical section. Assume that $\mathscr{O}_{X}(D)$ is endowed with an adelic metric.

Let $v\in S$ be a place of $\mathbf{Q}$. The function $g_{D,v}=\log\left\|s_{D}\right\|_{v}^{-1}$ is a continuous function on $X_{v}^{\mathrm{an}}\mathchoice{\mathbin{\vrule height=3.09999pt,width=6.93192pt,depth=-1.63612pt}}{\mathbin{\vrule height=3.09999pt,width=6.93192pt,depth=-1.63612pt}}{\mathbin{\vrule height=2.15277pt,width=3.65973pt,depth=-1.20554pt}}{\mathbin{\vrule height=0.86108pt,width=2.45418pt,depth=-1.03334pt}}\mathopen{|}D\mathclose{|}$, and is called a $v$-adic Green function for $D$. For every open subscheme $U$ of $X$ and any equation $f$ of $D$ on $U$, $g_{D,v}+\log\mathopen{|}f\mathclose{|}_{v}$ extends to a continuous function on $U_{v}^{\mathrm{an}}$. Conversely, this property characterizes $v$-adic Green functions for $D$.

The family $g_{D}=(g_{D,v})$ is called an adelic Green function for $D$.

\lemmname** 4.9**** (Chambert-Loir & Thuillier (2009), prop. 2.2).**

Let $\mathscr{X}$ be a proper flat integral scheme over $\mathbf{Z}$, let $\overline{\mathscr{L}}$ be a hermitian line bundle on $\mathscr{X}$. Let $X=\mathscr{X}_{\mathbf{Q}}$ and let $L=\mathscr{L}_{\mathbf{Q}}$, endowed with the algebraic adelic metric associated with $(\mathscr{X},\overline{\mathscr{L}})$. Assume that $\mathscr{X}$ is integrally closed in its generic fiber (for example, that it is normal).

Then the canonical map $\Gamma(\mathscr{X},\mathscr{L})\to\Gamma(X,L)$ is injective and its image is the set of sections $s$ such that $\left\|s\right\|_{v}\leqslant 1$ for every finite place $v\in S$.

Equivalently, effective Cartier divisors on $\mathscr{X}$ correspond to $v$-adic Green functions which are positive at all finite places $v$.

Démonstration.

Injectivity follows from the fact that $\mathscr{X}$ is flat, so that $X$ is schematically dense in $\mathscr{X}$. Surjectivity is a generalization of the fact that an integrally closed domain is the intersection of its prime ideals of height 1. ∎

4.10

Let $\left\|\cdot\right\|$ and $\left\|\cdot\right\|^{\prime}$ be two adelic metrics on $L$. The ratio of these metrics is a metric on the trivial line bundle, hence is of the form $\mathscr{O}_{X}(\varphi)$, for some function $\varphi=(\varphi_{v})\in\mathscr{C}_{\mathrm{c}}(X_{\mathrm{ad}},\mathbf{R})$. For all but finitely many places, $\varphi_{v}$ is identically [math]. For every place $v$, we let

[TABLE]

Since $X_{v}^{\mathrm{an}}$ is compact, this is a positive real number; it is equal to [math] for all but finitely many places $v$.

We then define the distance between the two given adelic metrics by

[TABLE]

The set of adelic metrics on a given line bundle $L$ is a real affine space, its underlying vector space is the subspace $\mathscr{C}_{\mathrm{c}}(X_{\mathrm{ad}},\mathbf{R})$ of $\mathscr{C}(X_{\mathrm{ad}},\mathbf{R})=\prod_{v}\mathscr{C}(X_{v}^{\mathrm{an}},\mathbf{R})$ consisting of families $(\varphi_{v})$ such that $\varphi_{v}\equiv 0$ for all but finitely many places $v\in S$.

The space $\mathscr{C}_{\mathrm{c}}(X_{\mathrm{ad}},\mathbf{R})$ is the union of the subspaces $\mathscr{C}_{U}(X_{\mathrm{ad}},\mathbf{R})$ of functions with (compact) support above a given finite set $U$ of places of $S$. We thus endow it with its natural inductive limit topology.

\exemname* 4.11** (Algebraic dynamics).*

We expose here the point of view of Zhang (1995b) regarding the canonical heights of Call & Silverman (1993) in the context of algebraic dynamics. Let $X$ be a proper $\mathbf{Q}$-scheme, let $f\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0muX\to X$ be a morphism, let $L$ be a line bundle on $X$ such that $f^{*}L\simeq L^{q}$, for some integer $q\geqslant 2$. We fix such an isomorphism $\varepsilon$. The claim is that there exists a unique adelic metric on $L$ for which the isomorphism $\varepsilon$ is an isometry.

Let us first fix a place $v$ and prove that there is a unique $v$-adic metric on $L$ for which $\varepsilon$ is an isometry. To that aim, let us consider, for any $v$-adic metric $\left\|\cdot\right\|$ on $L$, the induced $v$-adic metric on $f^{*}L$ and transfer it to $L^{q}$ via $\varepsilon$. This furnishes a $v$-adic metric $\left\|\cdot\right\|^{f}$ on $L$ such that $\varepsilon$ is an isometry from $(L,f^{*}\left\|\cdot\right\|)$ to $(L,\left\|\cdot\right\|^{f})^{q}$, and it is the unique $v$-adic metric on $L$ satisfying this property. Within the real affine space of $v$-adic metrics on $L$, normed by the distance $\delta_{v}$, and complete, the self-map $\left\|\cdot\right\|\mapsto\left\|\cdot\right\|^{f}$ is contracting with Lipschitz constant $1/q$. Consequently, the claim follows from Banach’s fixed point theorem.

We also note that there exists a dense open subscheme $U$ of $\operatorname{Spec}(\mathbf{Z})$, a model $(\mathscr{X},\mathscr{L})$ of $(X,L)$ over $U$ such that $f\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0muX\to X$ extends to a morphism $\varphi\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathscr{X}\to\mathscr{X}$ and the isomorphism $\varepsilon\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0muf^{*}L\simeq L^{q}$ extends to an isomorphism $\varphi^{*}\mathscr{L}\simeq\mathscr{L}^{q}$, still denoted by $\varepsilon$. This implies that for every finite place $p$ above $U$, the canonical $v$-adic metric is induced by the model $(\mathscr{X},\mathscr{L})$.

Consequently, the family $(\left\|\cdot\right\|_{v})$ of $v$-adic metrics on $L$ for which $\varepsilon$ is an isometry is an adelic metric.

5 Arithmetic ampleness

\definame** 5.1****.**

Let $\mathscr{X}$ be a proper scheme over $\mathbf{Z}$ and let $\overline{\mathscr{L}}$ be a hermitian line bundle on $\mathscr{X}$. One says that $\overline{\mathscr{L}}$ is relatively semipositive if:

For every vertical integral curve $C$ on $\mathscr{X}$, one has $\deg_{\mathscr{L}}(C)\geqslant 0$; 2. 2.

For every holomorphic map $f\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbf{D}\to\mathscr{X}(\mathbf{C})$, the curvature of $f^{*}\overline{\mathscr{L}}$ is semipositive.

If $\overline{\mathscr{L}}$ is relatively semipositive, then $\mathscr{L}_{\mathbf{Q}}$ is nef. (The degree of a curve $Z$ in $X_{\mathbf{Q}}$ with respect to $L_{\mathbf{Q}}$ of the integral over $Z(\mathbf{C})$ of the curvature of $\overline{\mathscr{L}}$, hence is positive.)

\exemname* 5.2**.*

Let us consider the tautological line bundle $\mathscr{O}(1)$ on the projective space $\mathbf{P}^{N}_{\mathbf{Z}}$. Its local sections correspond to homogeneous rational functions of degree $1$ in indeterminates $T_{0},\dots,T_{N}$. It is endowed with a natural hermitian metric such that, if $f$ is such a rational function, giving rise to the section $s_{f}$, and if $x=[x_{0}:\dots:x_{N}]\in\mathbf{P}^{N}(\mathbf{C})$, one has the formula

[TABLE]

(By homogeneity of $f$, the right hand side does not depend on the choice of the system of homogeneous coordinates for $x$.) The corresponding hermitian line bundle $\overline{\mathscr{O}(1)}$ is relatively semipositive. It is in fact the main source of relatively semipositive hermitian line bundles, in the following way.

Let $\mathscr{X}$ be a proper scheme over $\mathbf{Z}$ and let $\overline{\mathscr{L}}$ be a hermitian line bundle on $\mathscr{X}$. One says that $\overline{\mathscr{L}}$ is relatively ample if there exists an embedding $\varphi\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathscr{X}\hookrightarrow\mathbf{P}^{N}_{\mathbf{Z}}$, a metric with strictly positive curvature on $\mathscr{O}_{\mathbf{P}^{N}}(1)$ and an integer $d\geqslant 1$ such that $\overline{\mathscr{L}}^{d}\simeq\varphi^{*}\overline{\mathscr{O}_{\mathbf{P}^{N}}(1)}$.

\propname** 5.3****.**

Let $X$ be a proper scheme over $\mathbf{Q}$ and let $L_{0},\dots,L_{d}$ be line bundles on $X$.

Let $\mathscr{X},\mathscr{X}^{\prime}$ be proper flat schemes over $\mathbf{Z}$ such that $X=\mathscr{X}_{\mathbf{Q}}=\mathscr{X}^{\prime}_{\mathbf{Q}}$, let $\overline{\mathscr{L}_{0}},\dots,\overline{\mathscr{L}_{d}}$ (respectively $\overline{\mathscr{L}^{\prime}_{0}},\dots,\overline{\mathscr{L}^{\prime}_{d}}$) be semipositive hermitian line bundles on $\mathscr{X}$ (respectively $\mathscr{X}^{\prime}$) such that $\mathscr{L}_{j,\mathbf{Q}}=\mathscr{L}^{\prime}_{j,\mathbf{Q}}=L_{j}$. We write $\overline{L_{0}},\dots,\overline{L_{d}}$ (respectively $\overline{L^{\prime}_{0}},\dots,\overline{L^{\prime}_{d}}$) for the corresponding adelically metrized line bundles on $X$.

Then for every closed subscheme $Z$ of $\mathscr{X}$, one has

[TABLE]

where the factor $c_{1}(L_{j})$ is omitted in the $j$th term.

Démonstration.

Using the projection formula, we first reduce to the case where $\mathscr{X}=\mathscr{X}^{\prime}$ is normal. We then write

[TABLE]

and bound the $j$th term as follows. Let $s_{j}$ be the regular meromorphic section of $\mathscr{O}_{X}=\mathscr{L}^{\prime}_{j}\otimes(\mathscr{L}_{j})^{-1}$ corresponding to $1$. By definition, one has

[TABLE]

Moreover, all components of $\operatorname{div}(s_{j}|_{Z})$ are vertical. For every $j\in\{0,\dots,d\}$ and every $v\in S$, let $\delta_{j,v}=\delta_{v}(\overline{L_{j}},\overline{L^{\prime}_{j}})$ (this is zero for all but finitely many places $v$). Using the fact that $\mathopen{|}\log\left\|s_{j}\right\|_{v}\mathclose{|}\leqslant\delta_{v}(\overline{L_{j}},\overline{L^{\prime}_{j}})$ for every place $v\in S$, the normality assumption on $\mathscr{X}$ implies that

[TABLE]

Since the line bundles $\mathscr{L}_{k}$ and $\mathscr{L}^{\prime}_{k}$ are semipositive, this implies the bound

[TABLE]

Similarly, the curvature forms $c_{1}(\overline{L_{k}})$ and $c_{1}(\overline{L^{\prime}_{k}})$ are semipositive, so that the upper bound $\log\left\|s_{j}\right\|^{-1}\leqslant\delta_{j,\infty}$ implies

[TABLE]

Adding these contributions, we get one of the desired upper bound, and the other follows by symmetry. ∎

\definame** 5.4****.**

An adelic metric on a line bundle $L$ on $X$ is said to be semipositive if it is a limit of a sequence of semipositive algebraic adelic metrics on $L$.

Let $\overline{\operatorname{Pic}}{}^{+}(X)$ be the set of all isomorphism classes of line bundles endowed with a semipositive metric. We endow it with the topology for which the set of adelic metrics on a given line bundle is open and closed in $\overline{\operatorname{Pic}}{}^{+}(X)$ and endowed with the topology induced by the distance of adelic metrics.

It is a submonoid of $\overline{\operatorname{Pic}}(X)$; moreover, its image in $\operatorname{Pic}(X)$ consists of (isomorphism classes of) nef line bundles on $X$. I thank the referee for pointing out an example (Demailly \bbletal, 1994, 1.7) of a nef line bundle on a complex projective variety admitting no smooth semipositive metric, as well as for the numerous weaknesses of exposition he/she pointed out.

\coroname** 5.5****.**

Let $Z$ be an integral closed subscheme of $X$, let $d=\dim(Z)$. The arithmetic degree map extends uniquely to a continuous function $\overline{\operatorname{Pic}}{}{}^{+}(X)^{d+1}\to\mathbf{R}$. This extension is multilinear and symmetric.

Démonstration.

This follows from proposition 5.3 and from the classical extension theorem of uniformly continuous maps. ∎

\definame** 5.6****.**

Let $X$ be a projective $\mathbf{Q}$-scheme and let $L$ be a line bundle on $X$. An adelic metric on $L$ is said to be admissible if there exists two line bundles endowed with semipositive adelic metrics, $\overline{M_{1}}$ and $\overline{M_{2}}$, such that $\overline{L}\simeq\overline{M_{1}}\otimes\overline{M_{2}}{}^{-1}$.

More generally, we say that a $v$-adic metric on $L$ is admissible if it is the $v$-adic component of an admissible adelic metric on $L$. The set of all admissible adelically metrized line bundles on $X$ is denoted by $\overline{\operatorname{Pic}}{}^{{\mathrm{adm}}}(X)$; it is the subgroup generated by $\overline{\operatorname{Pic}}{}^{+}(X)$.

By construction, the arithmetic intersection product extends by linearity to $\overline{\operatorname{Pic}}{}^{\mathrm{adm}}(X)$. We use the notation $\mathop{\widehat{\operatorname{deg}}}({\widehat{c}}_{1}(\overline{L}_{0})\dots\widehat{c}_{1}(\overline{L}_{d})\mid Z)$ for the arithmetic degree of a $d$-dimensional integral closed subscheme $Z$ of $X$ with respect to admissible adelically metrized line bundles $\overline{L}_{0},\dots,\overline{L}_{d}$.

This gives rise to a natural notion of height parallel to that given in definition 3.3.

\exemname* 5.7**.*

Let us retain the context and notation of example 4.11. Let us moreover assume that $L$ is ample and let us prove that the canonical adelic metric on $L$ is semipositive.

We make the observation that if $\left\|\cdot\right\|$ is an algebraic adelic metric on $L$ induced by a relatively semipositive hermitian line bundle $\overline{\mathscr{L}}$ on a proper flat model $\mathscr{X}$ of $X$, then the metric $\left\|\cdot\right\|^{f}$ is again relatively semipositive. Indeed, the normalization of $\mathscr{X}$ in the morphism $f\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0muX\to X$ furnishes a proper flat scheme $\mathscr{X}^{\prime}$ over $\mathbf{Z}$ such that $\mathscr{X}^{\prime}_{\mathbf{Q}}=X$ and a morphism $\varphi\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathscr{X}^{\prime}\to\mathscr{X}$ that extends $f$. Then $\varphi^{*}\overline{\mathscr{L}}$ is a relatively semipositive hermitian line bundle on $\mathscr{X}^{\prime}$, model of $L^{d}$, which induces the algebraic adelic metric $\left\|\cdot\right\|^{f}$ on $L$.

Starting from a given algebraic adelic metric induced by a relatively semipositive model (for example, a relatively ample one), the proof of Picard’s theorem invoked in example 4.11 proves that the sequence of adelic metrics obtained by the iteration of the operator $\left\|\cdot\right\|\mapsto\left\|\cdot\right\|^{f}$ converges to the unique fixed point. Since this iteration preserves algebraic adelic metrics induced by a relatively semipositive model, the canonical adelic metric on $L$ is semipositive, as claimed.

For a generalization of this construction, see theorem 4.9 of Yuan & Zhang (2017).

6 Measures

\definame** 6.1****.**

Let $X$ be a projective $\mathbf{Q}$-scheme. A function $\varphi\in\mathscr{C}(X_{\mathrm{ad}},\mathbf{R})$ is said to be admissible if the adelically metrized line bundle $\mathscr{O}_{X}(\varphi)$ is admissible.

The set $\mathscr{C}_{\mathrm{adm}}(X_{\mathrm{ad}},\mathbf{R})$ of admissible functions $(\varphi_{v})$ is a real vector subspace of $\mathscr{C}_{\mathrm{c}}(X_{\mathrm{ad}},\mathbf{R})$. One has an exact sequence

[TABLE]

analogous to (4.7.1).

More generally, we say that a function $\varphi_{v}\in\mathscr{C}(X_{v}^{\mathrm{an}},\mathbf{R})$ is admissible if it is the $v$-adic component of an admissible function $\varphi=(\varphi_{v})$. This defines a real vector subspace $\mathscr{C}_{\mathrm{adm}}(X_{v}^{\mathrm{an}},\mathbf{R})$ of $\mathscr{C}(X_{v}^{\mathrm{an}},\mathbf{R})$.

\propname** 6.2**** (Gubler, 1998, theorem 7.12).**

For every place $v\in S$, the subspace $\mathscr{C}_{\mathrm{adm}}(X_{v}^{\mathrm{an}},\mathbf{R})$ is dense in $\mathscr{C}(X_{v}^{\mathrm{an}},\mathbf{R})$.

The space $\mathscr{C}_{\mathrm{adm}}(X_{\mathrm{ad}},\mathbf{R})$ of admissible functions is dense in $\mathscr{C}_{\mathrm{c}}(X_{\mathrm{ad}},\mathbf{R})$.

Démonstration.

Observe that $X_{v}^{\mathrm{an}}$ is a compact topological space. By corollary 7.7 and lemma 7.8 of Gubler (1998), the subspace of $\mathscr{C}_{{\mathrm{adm}}}(X_{v}^{\mathrm{an}},\mathbf{R})$ corresponding to algebraic $v$-adic metrics on $L$ separates points and is stable under $\sup$ and $\inf$. The first part of the proposition thus follows from Stone’s density theorem.

The second part follows from the first one and a straightforward argument. ∎

\theoname** 6.3****.**

Let $Z$ be an integral closed subscheme of $X$, let $d=\dim(Z)$, let $\overline{L_{1}},\dots,\overline{L_{d}}$ be admissible adelically metrized line bundles on $X$.

There exists a unique measure $c_{1}(\overline{L_{1}})\dots c_{1}(\overline{L_{d}})\delta_{Z}$ on $X_{\mathrm{ad}}$ such that

[TABLE]

for every compactly supported admissible function $\varphi_{0}$ on $X_{\mathrm{ad}}$. 2. 2.

This measure is supported on $Z_{\mathrm{ad}}$; for every place $v$ of $S$, the mass of its restriction to $X_{v}^{\mathrm{an}}$ is equal to

[TABLE]

If $\overline{L_{1}},\dots,\overline{L_{d}}$ are semipositive, then this measure is positive. 3. 3.

The induced map from $\overline{\operatorname{Pic}}^{\mathrm{adm}}(X)^{d}$ to the space $\mathscr{M}(X_{\mathrm{ad}})$ of measures on $X_{\mathrm{ad}}$ is $d$-linear and symmetric. 4. 4.

Every admissible function is integrable for this measure.

Démonstration.

Let us first assume that $\overline{L_{1}},\dots,\overline{L_{d}}$ are semipositive. It then follows from the definition of the arithmetic intersection degrees that the map

[TABLE]

is a positive linear form on $\mathscr{C}_{\mathrm{adm}}(X_{\mathrm{ad}},\mathbf{R})$. By the density theorem, it extends uniquely to a positive linear form on $\mathscr{C}_{c}(X_{\mathrm{ad}},\mathbf{R})$, which then corresponds to an inner regular, locally finite, positive Borel measure on $X_{\mathrm{ad}}$.

The rest of the theorem follows from this. ∎

\remaname* 6.4**.*

At archimedean places, the construction of the measure $c_{1}(\overline{L_{1}})\dots c_{1}(\overline{L_{d}})\delta_{Z}$ shows that it coincides with the measure defined by Bedford & Taylor (1982) and Demailly (1985). 2. 2.

At finite places, it has been first given in Chambert-Loir (2006). By approximation, the definition of the measure in the case of a general semipositive $p$-adic metric is then deduced from the case of algebraic metrics, given by a model $(\mathscr{X},\mathscr{L})$. In this case, the measure $c_{1}(\overline{L_{1}})\dots c_{1}(\overline{L_{d}})\delta_{Z}$ on $X_{p}^{\mathrm{an}}$ has finite support. Let us describe it when $Z=X$ and the model $\mathscr{X}$ (the general case follows). For each component $\mathscr{Y}$ of $\mathscr{X}\otimes\mathbf{F}_{p}$, there exists a unique point $y\in X_{p}^{\mathrm{an}}$ whose specialization is the generic point of $\mathscr{Y}$. The contribution of the point $y$ to the measure is then equal to

[TABLE]

where $m_{\mathscr{Y}}$ is the multiplicity of $\mathscr{Y}$ in the special fiber, that is, the length of the ideal $(p)$ at the generic point of $\mathscr{Y}$.

\exemname* 6.5**.*

Let $X$ be an abelian variety of dimension $d$ over a number field $F$. Let $\overline{L}$ be an ample line bundle equipped with a canonical adelic metric; let us then describe the measure $c_{1}(\overline{L})^{d}$ on $X_{v}^{\mathrm{an}}$, for every place $v\in S$. For simplicity, we assume that $F=\mathbf{Q}$.

First assume $v=\infty$. Then $X_{\infty}^{\mathrm{an}}$ is the quotient, under complex conjugation, of the complex torus $X(\mathbf{C})$, and the canonical measure on $X_{\infty}^{\mathrm{an}}$ is the direct image of the unique Haar measure on $X(\mathbf{C})$ with total mass $\deg(c_{1}(L)^{d}\mid X)$. 2. 2.

The situation is more interesting in the case of a finite place $p$.

If $X$ has good reduction at $p$, that is, if it extends to an abelian scheme $\mathscr{X}$ over $\mathbf{Z}_{p}$, then the canonical measure is supported at the unique point of $X_{p}^{\mathrm{an}}$ whose specialization is the generic point of $\mathscr{X}\otimes\mathbf{F}_{p}$.

Let us assume, on the contrary, that $X$ has (split) totally degenerate reduction. In this case, the uniformization theory of abelian varieties shows that $X_{p}^{\mathrm{an}}$ is the quotient of a torus $({\mathbf{G}_{\mathrm{m}}}^{d})^{\mathrm{an}}$ by a lattice $\Lambda$. The definition of $({\mathbf{G}_{\mathrm{m}}}^{d})^{\mathrm{an}}$ shows that this analytic space contains a canonical $d$-dimensional real vector space $V$, and $V/\Lambda$ is a real $d$-dimensional torus $S(X_{p}^{\mathrm{an}})$ contained in $X_{p}^{\mathrm{an}}$, sometimes called its skeleton. Gubler (2007) has shown that the measure $c_{1}(\overline{L})^{d}$ on $X_{p}^{\mathrm{an}}$ coincides with the Haar measure on $S(X_{p}^{\mathrm{an}})$ with total mass $\deg(c_{1}(L)^{d}\mid X)$.

The general case is a combination of these two cases, see Gubler (2010).

\remaname* 6.6**.*

At finite places, the theory described in this section defines measures $c_{1}(\overline{L}_{1})\dots c_{1}(\overline{L}_{d})\delta_{Z}$ without defining the individual components $c_{1}(\overline{L}_{1})$,…, $c_{1}(\overline{L}_{d})$, and $\delta_{Z}$. In Chambert-Loir & Ducros (2012), we propose a theory of real differential forms and currents on Berkovich analytic spaces that allows a more satisfactory analogy with the theory on complex spaces. In particular, we provide an analogue of the Poincaré–Lelong equation, and semipositive metrized line bundles possess a curvature current $c_{1}(\overline{L})$ (curvature form in the “smooth” case) whose product can be defined and coincides with the measures of the form $c_{1}(\overline{L}_{1})\dots c_{1}(\overline{L}_{d})\delta_{Z}$ that we have discussed.

7 Volumes

7.1

Let $X$ be a proper $\mathbf{Q}$-scheme and let $\overline{L}$ be a line bundle endowed with an adelic metric.

The Riemann-Roch space $\mathrm{H}^{0}(X,L)$ is a finite dimensional $\mathbf{Q}$-vector space. For every place $v\in S$, we endow it with a $v$-adic semi-norm:

[TABLE]

for $s\in\mathrm{H}^{0}(X,L)$. If $X$ is reduced, then this is a norm; let then $B_{v}$ be its unit ball.

In the case where the metric on $L$ is algebraic, given by a model $\mathscr{L}$ on a normal model $\mathscr{X}$ of $X$, the real vector space $\mathrm{H}^{0}(X,L)_{\mathbf{R}}$ admits a natural lattice $\mathrm{H}^{0}(\mathscr{X},\mathscr{L})$ and the norm $\left\|\cdot\right\|_{\infty}$ at the archimedean place. The real space $\mathrm{H}^{0}(X,L)_{\mathbf{R}}$ can then be endowed with the unique Haar measure for which $B_{\infty}$ has volume $1$, and we can then consider the covolume of the lattice $\mathrm{H}^{0}(\mathscr{X},\mathscr{L})$. Minkowski’s first theorem will then eventually provide non-trivial elements of $\mathrm{H}^{0}(\mathscr{X},\mathscr{L})$ of controlled norm at the archimedean place.

The following construction, due to Bombieri & Vaaler (1983), provides the analogue in the adelic setting.

Let $\mathbf{A}$ be the ring of adeles of $\mathbf{Q}$ and let $\mu$ be a Haar measure on $\mathrm{H}^{0}(X,L)\otimes\mathbf{A}$. Then $\prod_{v\in S}B_{v}$ has finite strictly positive volume in $\mathrm{H}^{0}(X,L)_{\mathbf{A}}$, and one defines

[TABLE]

This does not depend on the choice of the Haar measure $\mu$.

One also defines

[TABLE]

This is a finite set. We then let

[TABLE]

The following inequality is an adelic analogue of Blichfeldt’s theorem, itself a close companion to the adelic analogue of Minkowski’s first theorem proved by Bombieri & Vaaler (1983). We refer to corollaire 2.12 of Gaudron (2009) for details.

\lemmname** 7.2****.**

One has

[TABLE]

7.3

The volume and the $\chi$-volume of $\overline{L}$ are defined by the formulas:

[TABLE]

One thus has the inequality

[TABLE]

In fact, it has been independently shown by Yuan (2009) and Chen (2010) that the volume is in fact a limit.

The relation between volumes and heights follows from the following result.

\lemmname** 7.4****.**

Assume that $L$ is ample. Then, for every real number $t$ such that

[TABLE]

the set of closed points $x\in X$ such that $h_{\overline{L}}(x)\leqslant t$ is not dense for the Zariski topology.

Démonstration.

Consider the adelically metrized line bundle $\overline{L}(-t)$, whose metric at the archimedean place has been multiplied by $e^{t}$. It follows from the definition of the $\chi$-volume that

[TABLE]

Indeed, for every finite place $p$, changing $\overline{L}$ to $\overline{L}(-t)$ does not modify the balls $B_{p}$ in $\mathrm{H}^{0}(X,\overline{L}^{n})\otimes_{\mathbf{Q}}\mathbf{Q}_{p}$, while it dilates it by the ratio $e^{-nt}$ at the archimedean place, so that its volume is multiplied by $e^{-nt\dim(\mathrm{H}^{0}(X,L^{n}))}$.

Consequently,

[TABLE]

In particular, there exists an integer $n\geqslant 1$ and a nonzero section $s\in\mathrm{H}^{0}(X,\overline{L}^{n})$ such that $\left\|s\right\|_{p}\leqslant 1$ for all finite places $p$, and $\left\|s\right\|_{\infty}\leqslant e^{-nt}$. Let now $x\in X$ be a closed point that is not contained in $\mathopen{|}\operatorname{div}(s)\mathclose{|}$; one then has

[TABLE]

whence the lemma. ∎

\theoname** 7.5****.**

Assume that $\overline{L}$ is semipositive. Then one has

[TABLE]

This is the arithmetic Hilbert–Samuel formula, due to Gillet & Soulé (1988); Bismut & Vasserot (1989) when $X$ is smooth and the adelic metric of $\overline{L}$ is algebraic. Abbes & Bouche (1995) later gave an alternative proof. In the given generality, the formula is a theorem of Zhang (1995a, b).

\theoname** 7.6****.**

The function $\overline{L}\mapsto\mathop{\widehat{\operatorname{vol}}}\nolimits(X,\overline{L})$ extends uniquely to a continuous function on the real vector space $\overline{\operatorname{Pic}}(X)\otimes_{\mathbf{Q}}\mathbf{R}$. 2. 2.

If $\mathop{\widehat{\operatorname{vol}}}\nolimits(X,\overline{L})>0$, then $\mathop{\widehat{\operatorname{vol}}}\nolimits$ is differentiable at $\overline{L}$. 3. 3.

If $\overline{L}$ is semipositive, then $\mathop{\widehat{\operatorname{vol}}}\nolimits$ and $\mathop{\widehat{\operatorname{vol}}}\nolimits\nolimits_{\chi}$ are differentiable at $\overline{L}$, with differential

[TABLE]

This theorem is proved by Chen (2011) as a consequence of results of Yuan (2008, 2009). It essentially reduces from the preceding one in the case $\overline{L}$ is defined by an ample line bundle on a model of $X$, and its metric has strictly positive curvature. Reaching the “boundary” of the cone of semipositive admissible metrized line bundles was the main result of Yuan (2008) who proved that for every admissible metrized line bundle $\overline{M}$ and every large enough integer $t$, one has

[TABLE]

It is this inequality, an arithmetic analogue of an inequality of Siu, will be crucial for the applications to equidistribution in the next section.

8 Equidistribution

The main result of this section is the equidistribution theorem 8.4. It has been first proved in the case $v=\mathbf{C}$ by Szpiro \bbletal (1997), under the assumption that the given archimedean metric is smooth and has a strictly positive curvature form, and the general case is due to Yuan (2008). However, our presentation derives it from a seemingly more general result, lemma 8.2, whose proof, anyway, closely follows their methods. Note that for the application to Bogomolov’s conjecture in §9, the initial theorem of Szpiro \bbletal (1997) is sufficient.

\definame** 8.1****.**

Let $X$ be a proper $\mathbf{Q}$-scheme, let $\overline{L}$ be an ample line bundle on $X$ endowed with an admissible adelically metric. Let $(x_{n})$ be a sequence of closed points of $X$.

One says that $(x_{n})$ is generic if for every strict closed subscheme $Z$ of $X$, the set of all $n\in\mathbf{N}$ such that $x_{n}\in Z$ is finite; in other words, this sequence converges to the generic point of $X$. 2. 2.

One says that $(x_{n})$ is small (relative to $\overline{L}$) if

[TABLE]

\lemmname** 8.2****.**

Let $X$ be a proper $\mathbf{Q}$-scheme, let $d=\dim(X)$, let $\overline{L}$ be a semipositive adelically metrized line bundle on $X$ such that $L$ is ample. Let $(x_{n})$ be a generic sequence of closed points of $X$ which is small relative to $\overline{L}$. For every line bundle $\overline{M}$ on $X$ endowed with an admissible adelic metric, one has

[TABLE]

Démonstration.

Since $L$ is ample, $L^{t}\otimes M$ is ample for every large enough integer $t$, and the classical Hilbert-Samuel formula implies that

[TABLE]

when $t\to\infty$. Since $\overline{L}$ is semipositive and $L$ is ample, the main inequality of Yuan (2008) implies that

[TABLE]

Consequently, when $t\to\infty$, one has

[TABLE]

The sequence $(x_{n})$ is generic, hence lemma 7.4 furnishes the inequality:

[TABLE]

We observe that

[TABLE]

so that, when $t\to\infty$, we have

[TABLE]

Applying this inequality for $\overline{M}^{-1}$ shows that $\limsup_{n}h_{\overline{M}}(x_{n})$ is bounded above by its right hand side. The lemma follows. ∎

8.3

Let $X$ be a proper $\mathbf{Q}$-scheme. Let $v\in S$ be a place of $\mathbf{Q}$.

Let $x\in X$ be a closed point. Let $F=\kappa(x)$; this is a finite extension of $\mathbf{Q}$, and there are exactly $[F:\mathbf{Q}]$ geometric points on $X(\mathbf{C}_{v})$ whose image is $x$, permuted by the Galois group $\operatorname{Gal}(\mathbf{C}_{v}/\mathbf{Q}_{v})$. We consider the corresponding “probability measure” in $X(\mathbf{C}_{v})$, giving mass $1/[F:\mathbf{Q}]$ to each of these geometric points, and let $\delta_{v}(x)$ be its image under the natural map $X(\mathbf{C}_{p})\to X_{v}^{\mathrm{an}}$.

By construction, $\delta_{v}(x)$ is a probability measure on $X_{v}^{\mathrm{an}}$ with finite support, a point of $X_{v}^{\mathrm{an}}$ being counted proportionaly to the number of its liftings to a geometric point supported by $x$.

The space of measures on $X_{v}^{\mathrm{an}}$ is the dual of the space $\mathscr{C}(X_{v}^{\mathrm{an}},\mathbf{R})$; it is endowed with the topology of pointwise convergence (“weak-* topology”).

\theoname** 8.4****.**

Let $X$ be a proper $\mathbf{Q}$-scheme, let $d=\dim(X)$, let $\overline{L}$ be a semipositive adelically metrized line bundle on $X$ such that $L$ is ample. Let $(x_{n})$ be a generic sequence of closed points of $X$ which is small relative to $\overline{L}$. Then for each place $v\in S$, the sequence of measures $(\delta_{v}(x_{n}))$ on $X_{v}^{\mathrm{an}}$ converges to the measure $c_{1}(\overline{L})^{d}/\deg(c_{1}(L)^{d}\mid X)$.

Démonstration.

Let $\mu_{\overline{L}}$ denote the probability measure $c_{1}(\overline{L})^{d}/\deg_{L}(X)$ on $X_{v}^{\mathrm{an}}$ and let $f\in\mathscr{C}(X_{v}^{\mathrm{an}},\mathbf{R})$ be an admissible function, extended by zero to an element of $\mathscr{C}_{\mathrm{adm}}(X_{\mathrm{ad}},\mathbf{R})$. We apply lemma 8.2 to the metrized line bundle $\overline{M}=\mathscr{O}_{X}(f)$ whose underlying line bundle on $X$ is trivial. For every closed point $x\in X$, one has

[TABLE]

Moreover,

[TABLE]

It thus follows from lemma 8.2 that

[TABLE]

The case of an arbitrary continuous function on $X_{v}^{\mathrm{an}}$ follows by density. ∎

9 The Bogomolov conjecture

9.1

Let $X$ be an abelian variety over a number field $F$, that is, a projective connected algebraic variety over $F$ which is endowed with an additional structure of an algebriac group. For every integer $n$, we write $[n]$ for the multiplication by $n$-morphism on $X$.

Let us first explain how the theory of canonical adelic metrics allows to extend the Néron–Tate height to arbitrary integral closed subschemes. For alternative and independent presentations, see Philippon (1991), Gubler (1994), Bost \bbletal (1994).

Let $L$ be a line bundle on $X$ trivialized at the origin.

If $L$ is even ($[-1]^{*}L\simeq L$), the theorem of the cube implies that there exists, for every integer $n$, a unique isomorphism $[n]^{*}L\simeq L^{n^{2}}$ which is compatible with the trivialization at the origin. If $n\geqslant 2$, then by example 4.11, it admits a unique adelic metric for which this isomorphism is an isometry. is an isometry; these isomorphisms are then all isometries.

Similarly, if $L$ is odd ($[-1]^{*}L\simeq L^{-1})$, then it admits a unique adelic metric for which the canonical isomorphisms $[n]^{*}L\simeq L^{n}$ are isometries, for all integer $n$.

In general, one can write $L^{2}\simeq(L\otimes[-1]^{*}L)\otimes(L\otimes[-1]^{*}L^{-1})$, as the sum of an even and an odd line bundle, and this endows $L$ with an adelic metric. This adelic metric is called the canonical adelic metric on $L$ (compatible with the given trivialization at the origin).

If $L$ is ample and even, then the canonical adelic metric on $L$ is semipositive. This implies that the canonical adelic metric of an arbitrary even line bundle is admissible.

Assume that $L$ is odd. Fix an even ample line bundle $M$. Up to extending the scalars, there exists a point $a\in X(F)$ such that $L\simeq\tau_{a}^{*}M\otimes M^{-1}$, where $\tau_{a}$ is the translation by $a$ on $X$. Then there exists a unique isomorphism $L\simeq\tau_{a}^{*}M\otimes M^{-1}\otimes M_{a}^{-1}$ which is compatible with the trivialization at the origin, and this gives rise to an isometry $\overline{L}\simeq\tau_{a}^{*}\overline{M}\otimes\overline{M}^{-1}\otimes\overline{M}_{a}^{-1}$. In particular, the adelic metric of $\overline{M}$ is admissible. In fact, it follows from a construction of Künnemann that it is even semipositive, see Chambert-Loir (1999).

9.2

In particular, let us consider an ample even line bundle $L$ on $X$ endowed with a canonical adelic metric. This furnishes a height

[TABLE]

for every integral closed subscheme $Z$ of $X$, where $d=\dim(Z)$.

In fact, if $(\mathscr{X},\overline{\mathscr{L}})$ is any model of $(X,L)$, one has

[TABLE]

which shows the relation of the point of view of adelic metrics with Tate’s definition of the Néron–Tate height, initially defined on closed points. This formula also implies that $h_{\overline{L}}$ is positive.

More generally, if $Z$ is an integral closed subscheme of $X_{\overline{F}}$, we let $h_{\overline{L}}(Z)=h_{\overline{L}}([Z])$, where $[Z]$ is its Zariski-closure in $X$ (more precisely, the smallest closed subscheme of $X$ such that $[Z]_{\overline{F}}$ contains $Z$).

\lemmname** 9.3****.**

The induced height function $h_{\overline{L}}\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0muX(\overline{F})\to\mathbf{R}$ is a positive quadratic form. It induces a positive definite quadratic form on $X(\overline{F})\otimes\mathbf{R}$. In particular, a point $\in X(\overline{F})$ satisfies $h_{\overline{L}}(x)=0$ if and only if $x$ is a torsion point.

Démonstration.

For $I\subset\{1,2,3\}$, let $p_{I}\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0muX^{3}\to X$ be the morphism given by $p_{I}(x_{1},x_{2},x_{3})=\sum_{i\in I}x_{i}$. The cube theorem asserts that the line bundle

[TABLE]

on $X^{3}$ is trivial, and admits a canonical trivialisation. The adelic metric of $\overline{L}$ endows it with an adelic metric which satisfies $[2]^{*}\mathscr{D}_{3}(\overline{L})\simeq\mathscr{D}_{3}(\overline{L})^{4}$, hence is the trivial metric. This implies the following relation on heights:

[TABLE]

on $X(\overline{F})^{3}$. Consequently,

[TABLE]

is a symmetric bilinear form on $X(\overline{F})$. Using furthermore that $h_{\overline{L}}(-x)=h_{\overline{L}}(x)$ for all $x$, we deduce that $h_{\overline{L}}$ is a quadratic form on $X(\overline{F})$.

Since $L$ is ample, $h_{\overline{L}}$ is bounded from below. The formula $h_{\overline{L}}(x)=h_{\overline{L}}(2x)/4$ then implies that $h_{\overline{L}}$ is positive. By what precedes, it induces a positive quadratic form on $X(\overline{F})_{\mathbf{R}}$.

Let us prove that it is in fact positive definite. By definition, it suffices that its restriction to the subspace generated by finitely many points $x_{1},\dots,x_{m}\in X(\overline{F})$ is positive definite. Let $E$ be a finite extension of $F$ such that $x_{1},\dots,x_{m}\in X(E)$. On the other hand, Northcott’s theorem implies that for every real number $t$, the set of $(a_{1},\dots,a_{m})\in\mathbf{Z}^{m}$ such that $h_{\overline{L}}(a_{1}x_{1}+\dots+a_{m}x_{m})\leqslant t$ is finite. One deduces from that the asserted positive definiteness. ∎

\definame** 9.4****.**

A torsion subvariety of $X_{\overline{F}}$ is a subvariety of the form $a+Y$, where $a\in X(\overline{F})$ is a torsion point and $Y$ is an abelian subvariety of $X_{\overline{F}}$.

\theoname** 9.5****.**

1. a)

Let $Z$ be an integral closed subscheme of $X_{\overline{F}}$. One has $h_{\overline{L}}(Z)=0$ if and only if $Z$ is a torsion subvariety of $X_{\overline{F}}$. 2. b)

Let $Z$ be an integral closed subscheme of $X_{\overline{F}}$ which is not a torsion subvariety. There exists a strictly positive real number $\delta$ such that the set

[TABLE]

is not Zariski-dense in $Z_{\overline{F}}$.

Assertion a) has been independently conjectured by Philippon (1991, 1995) and Zhang (1995b). Assertion b) has been conjectured by Bogomolov (1980) in the particular case where $Z$ is a curve of genus $g\geqslant 2$ embedded in its jacobian variety; for this reason, it is called the “generalized Bogomolov conjecture”. The equivalence of a) and b) is a theorem of Zhang (1995b). In fact, the implication a)$\Rightarrow$b) already follows from theorem 7.5 and lemma 7.4. Assume indeed that a) holds and let $Z$ be an integral closed subscheme of $X$ which is not a torsion subvariety; by a), one has $h_{\overline{L}}(Z)>0$, and one may take for $\delta$ any real number such that $0<\delta<h_{\overline{L}}(Z)$.

Theorem 9.5 has been proved by Zhang (1998), following a breakthrough of Ullmo (1998) who treated the case of a curve embedded in its jacobian; their proof makes use of the equidistribution theorem. Soon after, David & Philippon (1998) gave an alternative proof; when $Z$ is not a translate of an abelian subvariety, their proof provides a strictly positive lower bound for $h_{\overline{L}}(Z)$ (in a)) as well as an explicit real number $\delta$ (in b)) which only depends on the dimension and the degree of $Z$ with respect to $L$.

As a corollary of theorem 9.5, one obtains a new proof of the Manin–Mumford conjecture in characteristic zero, initially proved by Raynaud (1983).

\coroname** 9.6****.**

Let $X$ be an abelian variety over an algebraically closed field of characteristic zero, let $Z$ be an integral closed subscheme of $X$ which is not a torsion subvariety. Then the set of torsion points of $X$ which are contained in $Z$ is not Zariski-dense in $Z$.

Démonstration.

A specialization argument reduces to the case where $X$ is defined over a number field $F$. In this case, the torsion points of $X$ are defined over $\overline{F}$ and are characterized by the vanishing of their Néron–Tate height relative to an(y) ample line bundle $L$ on $X$. It is thus clear that the corollary follows from theorem 9.5, b). ∎

9.7

The proof of theorem 9.5, b), begins with the observation that the statement does not depend on the choice of the ample line $L$ on $X$. More precisely, if $\overline{M}$ is another symmetric ample line bundle on $X$ endowed with a canonical metric, then there exists an integer $a\geqslant 1$ such that $\overline{L}^{a}\otimes\overline{M}^{-1}$ is ample, as well as $\overline{M}^{a}\otimes\overline{L}^{-1}$. Consequently, $h_{\overline{L}}\geqslant a^{-1}h_{\overline{M}}$ and $h_{\overline{M}}\geqslant a^{-1}h_{\overline{L}}$. From these two inequalities, one deduces readily that the statement holds for $\overline{L}$ if and only if it holds for $\overline{M}$.

For a similar reason, if $f\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0muX^{\prime}\to X$ is an isogeny of abelian varieties, then the statements for $X$ and $X^{\prime}$ are equivalent. Let indeed $Z$ be an integral closed subvariety of $X_{\overline{F}}$ and let $Z^{\prime}$ be an irreducible component of $f^{-1}(Z)$. Then $Z$ is a torsion subvariety of $X_{\overline{F}}$ if and only if $Z^{\prime}$ is a torsion subvariety of $X^{\prime}_{\overline{F}}$. On the other hand, the relation $h_{f^{*}\overline{L}}(x)=h_{\overline{L}}(f(x))$ shows that $h_{f^{*}\overline{L}}$ has a strictly positive lower bound on $Z^{\prime}$ outside of a strict closed subset $E^{\prime}$ if and only if $h_{\overline{L}}$ has a strictly positive lower bound on $Z$ outside of the strict closed subset $f(E^{\prime})$.

9.8

Building on that observation, one reduces the proof of the theorem to the case where the stabilizer of $Z$ is trivial.

Let indeed $X^{\prime\prime}$ be the neutral component of this stabilizer and let $X^{\prime}=X/X^{\prime\prime}$; this is an abelian variety. By Poincaré’s complete reducibility theorem, there exists an isogeny $f\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0muX^{\prime}\times X^{\prime\prime}\to X$. This reduces us to the case where $X=X^{\prime}\times X^{\prime\prime}$ and $Z=Z^{\prime}\times X^{\prime\prime}$, for some integral closed subscheme $Z^{\prime}$ of $X^{\prime}_{\overline{F}}$. We may also assume that $\overline{L}=\overline{L}^{\prime}\boxtimes\overline{L}^{\prime\prime}$. It it then clear that the statement for $(X^{\prime},Z^{\prime})$ implies the desired statement for $(X,Z)$.

\lemmname** 9.9****.**

Assume that $\dim(Z)>0$ and that its stabilizer is trivial. Then, for every large enough integer $m\geqslant 1$, the morphism

[TABLE]

is birational onto its image but not finite.

Démonstration.

Let $m$ be an integer and let $x=(x_{1},\dots,x_{m})$ be an $\overline{F}$-point of $Z^{m}$. Then a point $y=(y_{1},\dots,y_{m})\in Z(\overline{F})^{m}$ belongs to the same fiber as $x$ if and only if $y_{2}-y_{1}=x_{2}-x_{1},\dots$, that is, if and only if, $y_{1}-x_{1}=y_{2}-x_{2}=\dots=y_{m}-x_{m}$. This identifies $f^{-1}(f(x))$ with the intersection $(Z-{x_{1}})\cap\dots\dots\cap(Z-{x_{m}})$ of translates of $Z$. If $m$ is large enough and $x_{1},\dots,x_{m}$ are well chosen in $Z$, then this intersection is equal to stabilizer of $Z$ in $X_{\overline{F}}$, hence is reduced to a point. In that case, the morphism $f$ has a fiber reduced to a point, hence it is generically injective.

On the other hand, the preimage of the origin $(o,\dots,o)$ contains the diagonal of $Z^{m}$, which has strictly positive dimension by hypothesis. ∎

9.10

For the proof of theorem 9.5, b), we now argue by contradiction and assume the existence of a generic sequence $(x_{n})$ in $Z(\overline{F})$ such that $h_{\overline{L}}(x_{n})\to 0$.

Having reduced, as explained above, to the case where the stabilizer of $Z$ is trivial, we consider an integer $m\geqslant 1$ such that the morphism $f\nobreak\mskip 4.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0muZ^{m}\to X^{m-1}_{\overline{F}}$ is birational onto its image, but not finite.

Since the set of strict closed subschemes of $Z$ is countable, one can construct a generic sequence $(y_{n})$ in $Z^{m}$ where $y_{n}$ is of the form $(x_{i_{1}},\dots,x_{i_{m}})$. One has $h_{\overline{L}}(y_{n})\to 0$, where, by abuse of language, we write $h_{\overline{L}}$ for the height on $X^{m}$ induced by the adelically metrized line bundle $\overline{L}\boxtimes\dots\boxtimes\overline{L}$ on $X^{m}$. This implies that $h_{\overline{L}}(Z)=0$, hence the sequence $(y_{n})$ is small.

For every integer $n$, let $z_{n}=f(y_{n})$. By continuity of a morphism of schemes, the sequence $(z_{n})$ is generic in $f(Z^{m})$. Moreover, we deduce from the quadratic character of the Néron–Tate height $h_{\overline{L}}$ that $h_{\overline{L}}(z_{n})\to 0$. In particular, $h_{\overline{L}}(f(Z^{m}))=0$, and the sequence $(z_{n})$ is small.

Fix an archimedean place $\sigma$ of $F$. Applied to the sequences $(y_{n})$ and $(z_{n})$, the equidistribution theorem 8.4 implies the following convergences:

[TABLE]

where, by $\propto$, I mean that both sides are proportional. (The proportionality ratio is the degree of $Z^{m}$, resp. of $f(Z^{m})$, with respect to the indicated measure.) Since $f(y_{n})=z_{n}$, we conclude that the measures

[TABLE]

on $f(Z^{m})$ are proportional.

Recall that the archimedean metric of $\overline{L}$ has the property that it is smooth and that its curvature form $c_{1}(\overline{L})$ is a smooth strictly positive $(1,1)$-form on $X_{\sigma}(\mathbf{C})$. Consequently, on a dense smooth open subscheme of $f(Z^{m})$ above which $f$ is an isomorphism, both measures are given by differential forms, which thus coincide there. We can pull back them to $Z^{m}$ by $f$ and obtain a proportionality of differential forms

[TABLE]

on $Z_{\sigma}(\mathbf{C})^{m}$. At this point, the contradiction appears: the differential form on the left is strictly positive at every point, while the one on the right vanishes at every point of $Z_{\sigma}^{m}(\mathbf{C})$ at which $f$ is not smooth.

This concludes the proof of theorem 9.5.

\remaname* 9.11**.*

The statement of 9.5 can be asked in more general contexts that allow for canonical heights. The case of toric varieties has been proved by Zhang (1995a), while in that case the equidistribution result is first due to Bilu (1997). The case of semiabelian varieties is due to David & Philippon (2002), by generalization of their proof for abelian varieties; I had proved in Chambert-Loir (1999) the equidistribution result for almost-split semi-abelian varieties, and the general case has just been announced by Kühne (2018).

The general setting of algebraic dynamics $(X,f)$ is unclear. For a polarized dynamical system as in 4.11, the obvious and natural generalization proposed in Zhang (1995b) asserts that subvarieties of height zero are exactly those whose forward orbit is finite. However, Ghioca and Tucker have shown that it does not hold; see Ghioca \bbletal (2011) for a possible rectification. The case of dominant endomorphisms of $(\mathbf{P}^{1})^{n}$ is a recent theorem of Ghioca \bbletal (2017).

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