Big Line Bundles Over Arithmetic Varieties

arXiv:math/0612424v1 [math.NT] 15 Dec 2006

Big Line Bundles Over Arithmetic Varieties Xinyi Yuan February 2, 2008

Contents 0 Introduction 0.1 Equidistribution over Algebraic Dynamics 0.2 A Generic Equidistribution Theorem . . . 0.3 Arithmetic Bigness . . . . . . . . . . . . . 0.4 Structure of this Paper . . . . . . . . . . .

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1 2 4 5 7

1 Arithmetic Bigness

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2 Arithmetic Volumes

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3 Analytic Parts

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4 Proof of the Main Theorem

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5 Equidistribution Theory 5.1 A Generic Equidistribution Theorem . . . . . . . . 5.2 Equidistribution at Infinite Places . . . . . . . . . . 5.3 Equidistribution at Finite Places . . . . . . . . . . 5.4 Equidistribution of Small Subvarieties . . . . . . . . 5.5 Equidistribution over Algebraic Dynamical Systems 5.6 Equidistribution over Multiplicative Groups . . . . 5.7 Equidistribution Theory . . . . . . . . . . . . . . .

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Introduction

In this paper, we prove a bigness theorem (Theorem 1.2) in the setting of Arakelov theory, which is an arithmetic analogue of a classical theorem of Siu. It is also an extension of the arithmetic Hilbert-Samuel formula implied by the arithmetic Riemann-Roch theorem of Gillet-Soul´e [GS2] and an estimate on analytic torsions of Bismut-Vasserot [BV].

1

This bigness result has a lot of consequences in the equidistribution theory initiated by Szpiro-Ullmo-Zhang [SUZ]. We will generalize to algebraic dynamical systems the archimedean equidistribution by [SUZ], the non-archimedean equidistribution by Chambert-Loir [Ch1], and the equidistribution of small subvarieties by Baker-Ih [BI] and Autissier [Au2]. The equidistribution theorem in [SUZ] was proved by a variational principle (cf. [Ch3]), where the key is to use the arithmetic Hilbert-Samuel formula to produce small sections. The formula works under the assumption that the curvature of the line bundle giving the polarization is strictly positive, since any small perturbation of the line bundle still have positive curvature. Such an assumption is also necessary in [Ch1]. However, in algebraic dynamics (e.g. multiplicative groups), the curvature is usually only semipositive and even a small perturbation may result in a somewhere negative curvature. Then the arithmetic Hilbert-Samuel is invalid in this case. Our bigness theorem solves this problem, since it works for negative curvatures. Our proof of the bigness theorem follows a strategy similar to the one used to prove the arithmetic Hilbert-Samuel formula by Abbes and Bouche [AB]. The crucial analytic part is the estimate of the distortion function of NL − jM in Theorem 3.3. It is implied by its ample case (Theorem 3.2) proved by Bouche [Bo] and Tian [Ti].

0.1

Equidistribution over Algebraic Dynamics

Projective Spaces Let K be a number field, and K be the algebraic closure of K. Fix an embedding K → C. Let Pn be the projective space over K, and φ : Pn → Pn be an endomorphism with coordinate φ = (f0 , f1 , · · · , fn ), where f0 , f1 , · · · , fn are homogeneous polynomials of degree q > 1 without non-trivial common zeros. For any algebraic point x = (z0 , z1 , · · · , zn ) ∈ Pn (K), the naive height of x is hnaive (x) :=

1 X log max{|z0 |v , |z1 |v , · · · , |zm |v }, [L : K] v

where L is a finite extension of K containing all the coordinates z0 , z1 , · · · , zm , and the summation is over all normalized valuations | · |v of L. The canonical height with respect to φ is defined by Tate’s limit 1 hnaive (φN (x)). N →∞ q N

hφ (x) = lim

One can show that the limit always exists. The canonical height has the following nice property: hφ (x) ≥ 0 and hφ (x) = 0 if and only if x if preperiodic. Here we say a point is preperiodic if its orbit {x, φ(x), φ2 (x), · · ·} is finite. To state our equidistribution theorem, we make some simple definitions related to sequences of algebraic points of Pn (K). 2

1. A sequence {xm }m≥1 of algebraic points is small if hφ (xm ) → 0 as m → ∞. 2. A sequence {xm }m≥1 of algebraic points is generic if no infinite subsequence of {xm } is contained in a proper closed subvariety of Pn . 3. Let {xm }m≥1 be a sequence of algebraic points and dµ a probability measure over the complex manifold Pn (C), i.e., a measure of total volume one. We say that the Galois orbits of {xm } are equidistributed with respect to dµ if the probability measure µxm := X 1 δx converges weakly to dµ over Pn (C), where O(xm ) is the orbit of #O(xm ) x∈O(xm )

xm under the Galois group Gal(K/K), and δx is the Dirac measure at x ∈ Pn (C).

We can also define the canonical probability measure dµφ over Pn (C) by Tate’s limit. It is a probability measure that satisfies φ∗ dµφ = q dim(X) dµφ and φ∗ dµφ = dµφ , which determine dµφ uniquely. The following theorem is a special case of Theorem 5.7 in this paper. Theorem. Suppose {xm }m≥1 is an infinite sequence of algebraic points in X which is generic and small. Then the Galois orbits of {xm } are equidistributed with respect to the canonical probability measure dµφ over Pn (C). Generalities We actually prove the equidistribution for any algebraic dynamical systems in Theorem 5.7. For a complete introduction of algebraic dynamics and related equidistribution we refer to [Zh4]. Let K be a number field. An algebraic dynamical system over K is a projective variety X over K endowed with an endomorphism φ : X → X which satisfies a polarization condition making it like the polynomial map over Pn . By Tate’s limit, we have the same notion of canonical height and canonical probability measure. And thus we have the dynamical equidistribution over X. See Section 5.5 for more details. Now we are going to consider three special cases: 1. Abelian varieties. When X is an abelian variety and φ = [2] is multiplication by 2, we get a dynamical system. A point is preperiodic if and only if it is torsion. The canonical height is exactly the Neron-Tate height, and the canonical probability measure is exactly the probability Haar measure over the complex torus X(C). Our equidistribution in this case is exactly the one in [SUZ], which was crucial in the proof of the Bogomolov conjecture by Ullmo [Ul] and Zhang [Zh3]. See [Zh5] for an abstract of this subject. 2. Multiplicative groups. When X = Gnm and φ = [2], we get a dynamical system over multiplicative groups. To compactify it, embed Gnm in Pn by the natural way and extend φ to a dynamics over Pn . Actually φ : (z0 , z1 , · · · , zn ) 7→ (z02 , z12 , · · · , zn2 ). The curvature is semipositive here, which can’t be handled by [SUZ]. That is why the proof 3

of the Bogomolov conjecture by Zhang [Zh1] and the proof of equidistribution by Bilu [Bi] were independent of each other and could not follow the idea of [Ul] and [Zh3]. However, our result here includes this case and leads to a uniform treatment following the case of abelian varieties in Section 5.6. 3. Almost split semi-abelian varieties. In [Ch2], Chambert-Loir proved equidistribution and Bogomolov conjecture over almost split semi-abelian varieties. The equidistribution was proved by choosing certain nice perturbation of the line bundle which preserves the semipositivity of the curvature. As in the multiplicative case, it can be handled by our uniform treatment once we apply Theorem 5.1 below.

0.2

A Generic Equidistribution Theorem

The above equidistribution theorem is implied by the following generic equidistribution theorem in Arakelov geometry. The basic references for Arakelov geometry are [Ar], [Fa], [GS1] and [Zh1]. Let K be a number field, X be a projective variety of dimension n − 1 over K, and L be a line bundle over X. Fix an embedding K → Cv for each place v, where Cv is the completion of the algebraic closure of Kv . We use the language of adelic metrized line bundles by Zhang ([Zh1], [Zh2]). Recall that an adelic metric over L is a Cv -norm k · kv over the fibre LCv (x) of each algebraic point x ∈ X(K) satisfying certain continuity and coherence conditions for each place v of K. f) is an OK -model of All the metrics we consider are induced by models. Suppose (X , L f is a Hermitian (X, L e ), i.e., X is an integral scheme projective and flat over OK and L f) gives (X, L e ). For any nonline bundle over X such that the generic fibre of (X , L fO ) e1 x∗ L archimedean place v, a point x ∈ X(K) extends to x˜ : Spec(OCv ) → XOCv . Then (˜ Cv gives a lattice in LCv (x), which induces a Cv -norm and thus an adelic metric. Such a metric fhas semipositive curvatures at all is called an algebraic metric. It is called semipositive if L archimedean places and non-negative degree on any complete vertical curve of X . An adelic metric over L is semipositive if it is the uniform limit of some sequence of semipositive algebraic metrics over L . Theorem 5.1 (Equidistribution of Small Points). Suppose X is a projective variety of dimension n − 1 over a number field K, and L is a metrized line bundle over X such that L is ample and the metric is semipositive. Let {xm } be an infinite sequence of algebraic points in X(K) which is generic and small. Then for any place v of K, the Galois orbits of the an with respect to the probability sequence {xm } are equidistributed in the analytic space XC v n−1 measure dµv = c1 (L )v / degL (X). We explain several terms in the theorem: 1. The definitions of a generic sequence and equidistribution are the same as before.

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2. Using the semipositive line bundle L , one can define the height hL (Y ) of any closed subvariety Y of X. A sequence {xm } of algebraic points in X(K) is called small if hL (xm ) → hL (X). an is the corresponding complex analytic space, and 3. For archimedean v, the space XC v n−1 the measure c1 (L )v is essentially the volume form by the Hermitian metric of L at v. See [Zh4, Proposition 3.1.5] for example. In this case, our theorem generalizes Theorem 3.1 by [SUZ].

4. For non-archimedean v, the theorem generalizes the recent work of Chambert-Loir an is the Berkovich space (cf. [Be]) associated to the variety XCv . [Ch1]. Here XC v Chambert-Loir constructs the v-adic canonical measure c1 (L )vn−1 and generalizes the equidistribution of [SUZ] to the v-adic case. We follow Chambert-Loir’s notion of canonical measures. 5. Another ingredient in our non-archimedean treatment is a theorem of Gubler [Gu] an can be approximated by model that any continuous real-valued function over XC v functions induced by certain formal models. In our case, all model functions are induced by arithmetic varieties, which puts the problem in the framework of arakelov theory. Finally, we obtain a proof analogous to the archimedean case, in which model functions play the role of smooth functions. Remark. The results in [SUZ] and [Ch1] assume the strict positivity of the metric at the place where equidistribution is considered.

0.3

Arithmetic Bigness

Our bigness theorem is the key to deal with negative curvatures. Here we state it and explain how it works. Siu’s Theorem Let X be a projective variety of dimension n defined over a field, and L be a line bundle over X. If L is ample, then when N is large enough, the Hilbert function h0 (L ⊗N ) = dim Γ(X, L ⊗N ) is a polynomial in N of degree n. Notice that ampleness is stable under pull-back via finite morphisms, but not via birational morphisms. Another useful notion for line bundles is bigness, which is stable under pull-back via dominant generically finite morphisms. The line bundle L is big if and only if there exists a constant c > 0 such that h0 (L ⊗N ) > cN n for all N large enough. See [La] for more details of bigness. Denote by c1 (L1 ) · · · c1 (Ln ) the intersection number of the line bundles L1 , · · · , Ln over X. The following is a basic theorem of Siu (cf. [La, Theorem 2.2.15]).

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Theorem (Siu). Let L and M be two ample line bundles over a projective variety X of dimension n. Then h0 (N(L − M )) ≥

c1 (L )n − n · c1 (L )n−1 c1 (M ) n N + O(N n−1 ). n!

In particular, L − M is big if c1 (L )n > n · c1 (L )n−1 c1 (M ). Here we write tensor product of line bundles additively, like the case of divisors. For example, N(L − M ) means (L ⊗ M ⊗(−1) )⊗N . Arithmetic Bigness One arithmetic analogue of the classical h0 is χsup (cf. [Fa]). See also Section 1 for an explanation. Our direct analogue of Siu’s theorem gives a nice expansion of χsup . Its accuracy allows it to play the role of the arithmetic Hilbert-Samuel formula in equidistribution. Let X be an arithmetic variety of dimension n, and let L be a hermitian line bundle over ⊗N X. We say that L is strongly big if there exists a constant c > 0 such that χsup (L ) > cN n for all N large enough. Note that there is a nice arithmetic theory of ample line bundles by Zhang [Zh1]. Namely, a hermitian line bundle L is ample if the following three conditions are satisfied: (a) LQ is ample in the classical sense; (b) L is relatively semipositive: the curvature of L is semipositive and deg(L |C ) ≥ 0 for any closed curve C on any special fibre of X over Spec(Z); (c) L is horizontally positive: the intersection number cˆ1 (L |Y )dim Y > 0 for any horizontal irreducible closed subvariety Y . Now we have the following main theorem which has the same appearance as Siu’s theorem: Theorem 1.2 (Main Theorem). Let L , M and E be three line bundles over an arithmetic variety X of dimension n. Assume L and M are ample. Then χsup (E + N(L − M )) ≥

cˆ1 (L )n − n · cˆ1 (L )n−1 cˆ1 (M ) n N + o(N n ). n!

In particular, L − M is strongly big if cˆ1 (L )n > n · cˆ1 (L )n−1 cˆ1 (M ). We can compare this theorem with the arithmetic Hilbert-Samuel formula. Actually the former is like a bigness version of the latter. Theorem (Arithmetic Hilbert-Samuel). Let L and E be two line bundles over an arithmetic variety X of dimension n. If L is relatively semipositive and LQ is ample, then χsup (E + NL ) =

cˆ1 (L )n n N + o(N n ) , N → ∞. n!

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(1)

The Hilbert-Samuel formula was originally proved by in Gillet-Soul´e [GS2] combining an estimate by Bismut-Vasserot [BV]. The above one is a refined version by Zhang [Zh1]. The original one was also proved by Abbes-Bouche [AB] using a more straight-forward method. We will extend the method in this paper to prove Theorem 1.2. Now let us see how the bigness theorem works in proving the equidistribution. The variational principle in [SUZ] is to consider the bundle L (ǫf ) = (L , e−ǫf k · kL ), the same line bundle L with metric multiplied by e−ǫf at v. Here f is any smooth function over the an , and ǫ > 0 is a small number. analytic space XC v The strategy is to write O(f ) = M 1 − M 2 for ample line bundles M 1 and M 2 , where O(f ) is the trivial line bundle with metric k1k = e−f . Then L (ǫf ) = (L + ǫM 1 ) − ǫM 2 is a difference of two ample line bundles and we can apply Theorem 1.2 to this difference. Note that ǫM 2 is small, and the leading term given by the theorem actually approximates cˆ1 (L (ǫf ))n up to an error O(ǫ2 ).

0.4

Structure of this Paper

The structure of this paper is as follows. In Section 1 we define two notions of bigness, state the main theorem (Theorem 1.2), and explore several basic properties of arithmetic bigness. Sections 2-4 give a proof of the main theorem. The outline of the proof is clear in Section 4. Some preliminary results on arithmetic volumes (resp. analytic estimate) are proved in Section 2 (resp. Section 3). Section 5 gives a detailed treatment of the equidistribution theory of small points, which reveals the importance of Theorem 1.2. The readers that are more interested in algebraic dynamics may assume Theorem 1.2 and jump directly to Section 5. At the end of Section 5, we summarize all related equidistribution theorems and conjectures to end this paper.

Acknowledgments I am very grateful to my advisor Shou-wu Zhang who introduced this subject to me. I am also indebted to him for lots of helpful conversations and encouragement during the preparation of this paper. I would like to thank Brian Conrad for clarification of many concepts in rigid analytic geometry. Finally, I would like to thank Zuoliang Hou, Xander Faber, Ming-lun Hsieh and Qi Li for their useful discussions. During the revision of this paper, Moriwaki [Mo3] proved the continuity of the volume functions of hermitian line bundles using the technique of the proof of Theorem 1.2 and Theorem 1.3 in this paper.

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1

Arithmetic Bigness

Notations and Conventions By an arithmetic variety X of dimension n, we mean an integral scheme X, projective and flat over Spec(Z) of absolute dimension n. We say that X is generically smooth if the generic fibre XQ is smooth. In this paper we don’t assume X to be generically smooth, and use generic resolution to relate the general case to the generically smooth case by Hironaka’s theorem. See [Zh1] for more on generic resolutions. In Theorem 2.2, one will see that resolution of singularities preserves bigness of line bundles very well, so we actually don’t need to worry about singularities on the generic fibre. A metrized line bundle L = (L , | · |) over X is an invertible sheaf L over X together with a hermitian metric | · | on each fibre of LC over XC . We say this metric is smooth if the pull-back metric over f ∗ L under any analytic map f : {z ∈ Cn−1 : |z| < 1} → XC is smooth in the usual sense. We call L a hermitian line bundle if its metric is smooth and invariant under complex conjugation. For a hermitian line bundle L , we say the metric or the curvature of L is semipositive if the curvature of f ∗ L with the pull-back metric under any analytic map f : {z ∈ Cn−1 : |z| < 1} → XC is semipositive definite. Let L 1 , L 2 , · · · , L n be n hermitian line bundles over X. Choose any generic resolue → X. Then the intersection number cˆ1 (L 1 )ˆ tion π : X c1 (L 2 ) · · · cˆ1 (L n ) is defined to be ∗ cˆ1 (π L 1 )ˆ c1 (π ∗ L 2 ) · · · cˆ1 (π ∗ L n ), where the latter is the usual arithmetic intersection number defined in [GS1]. This definition is independent of the generic resolution chosen (cf. [Zh1]). For any section s ∈ Γ(X, L )R = Γ(X, L ) ⊗Z R ⊂ Γ(XC , LC ), one has the supremum norm ksksup = supz∈XC |s(z)|. Define a basic invariance h0 (L ) = log # {s ∈ Γ(X, L ) : ksksup < 1} . Picking any Haar measure on Γ(X, L )R , define the arithmetic volume χsup (L ) = log

vol(Bsup ) , vol(Γ(X, L )R /Γ(X, L ))

where Bsup = {s ∈ Γ(X, L )R : ksksup < 1} is the corresponding unit ball. It is easy to see that this definition is independent of the Haar measure chosen. Zhang studied arithmetic ampleness in [Zh1]. Recall that a hermitian line bundle L is ample if the following three conditions are satisfied: (a) LQ is ample; (b) L is relatively semipositive: the curvature of L is semipositive and deg(L |C ) ≥ 0 for any closed curve C on any special fibre of X over Spec(Z); (c) L is horizontally positive: the intersection number cˆ1 (L |Y )dim Y > 0 for any horizontal irreducible closed subvariety Y . Note that the second condition in (b) means L is nef over any special fibre in the classical sense. By Kleiman’s theorem, it is equivalent to c1 (L |Y )dim Y ≥ 0 for any vertical irreducible closed subvariety Y . See [La, Theorem 1.4.9]. 8

The arithmetic Hilbert-Samuel formula is true for ample line bundles, and thus we can produce a lot of small sections. Furthermore, if X is generically smooth, then a NakaiMoishezon type theorem by Zhang [Zh1, Corollary 4.8] asserts that for any ample line bundle L and any hermitian line bundle E , the Z-module Γ(X, E + NL ) has a basis consisting of effective sections for N large enough. Here an effective section is a nonzero section with supremum norm less than or equal to 1. We call a line bundle effective if it admits an effective section. If the supremum norm of the section is less than 1, the section and the line bundle are said to be strictly effective. In the end, we state a fact telling that conditions (a) and (b) are not far from ampleness. More precisely, if L = (L , k · k) is such that LQ is ample and L is relatively semipositive, then the hermitian line bundle L (c) = (L , k · kc = k · ke−c ) is ample for c large enough. In fact, since LQ is ample, we can assume there exist sections s1 , · · · , sr ∈ Γ(X, L ) which are base-point free over the generic fibre. Fix a c such that s1 , · · · , sr are strictly effective in L (c). Now we claim that L (c) is ample. We need to show that cˆ1 (L (c)|Y )dim Y > 0 for any horizontal irreducible closed subvariety Y . Assume X is normal by normalization. We can find an sj such that div(sj ) does not contain Y , and thus Z dim Y dim Y −1 cˆ1 (L (c)|Y ) = cˆ1 (L (c)|div(sj )|Y ) − log ksj kc c1 (L )dim Y −1 YC

> cˆ1 (L (c)|div(sj )|Y )

dim Y −1

.

Now the proof can be finished by induction on dim Y . This fact is used in Lemma 5.3 when we apply Theorem 1.2.

Big Line Bundles Now we define two notions of arithmetic bigness, which are weaker than ampleness, but with more flexibility. Definition 1.1. Let X be an arithmetic variety of dimension n, and let L be a hermitian line bundle over X. We say that L is big if there exist a positive integer N0 and a positive number c such that h0 (NL ) > cN n for any integer N > N0 . We say that L is strongly big if there exist a positive integer N0 and a positive number c such that χsup (NL ) > cN n for any integer N > N0 . Remark. 1. Moriwaki [Mo2] defines that L is big if LQ is big in the classical sense and some positive power of L is strictly effective. It turns out that his definition is equivalent to ours. See Corollary 1.4 below. 9

2. Minkowski’s theorem gives h0 (NL ) ≥ χsup (NL ) + O(N n−1 ), and thus “strongly big” implies “big”. Its converse is not true in general. An example will be showed at the end of this section. 3. Either notion of bigness is invariant under dominant generically finite morphisms; i.e., the pull-back bundle of a big (resp. strongly big) line bundle via a dominant generically finite morphism is still big (resp. strongly big). 4. In two-dimensional case, Autissier [Au1, Proposition 3.3.3] proved a strong result for cˆ1 (L )2 2 general line bundles. Namely, χsup (NL ) ≥ N + o(N 2 ) for any hermitian line 2 bundle L over an arithmetic surface such that deg(LQ ) > 0. It tells us that L is strongly big if and only if deg(LQ ) > 0 and cˆ1 (L )2 > 0 by Corollary 1.4 below. The main theorem in this paper is the following: Theorem 1.2. Let L , M and E be three line bundles over an arithmetic variety X of dimension n. Assume L and M are ample. Then cˆ1 (L )n − n · cˆ1 (L )n−1 cˆ1 (M ) n χsup (E + N(L − M )) ≥ N + o(N n ). n! In particular, L − M is strongly big if cˆ1 (L )n > n · cˆ1 (L )n−1 cˆ1 (M ). This theorem will be proved in Section 4. But now we will state two properties of bigness. In the classical case, one has: big=ample+effective. More precisely, a line bundle is big if and only if it has a positive tensor power isomorphic to the tensor product of an ample line bundle and an effective line bundle. For the details see [La]. In the arithmetic case, we have a similar result. Theorem 1.3. A hermitian line bundle L is big if and only if NL = M + T for some positive integer N, some ample line bundle M and some effective line bundle T . We will prove this theorem in Section 4 after proving the main theorem. The proof is similar to some part of the proof of our main theorem. The key is to use the Riemann-Roch theorem in [GS3] to relate h0 to χsup . The following corollary gives more descriptions of big line bundles. And it also says that arithmetic bigness implies classical bigness over the generic fibre. Corollary 1.4. Let L be a hermitian line bundle over an arithmetic variety. The following are equivalent: (1) L is big. (2) NL = M + T for some positive integer N, some ample line bundle M and some effective line bundle T . (3) For any line bundle E , the line bundle NL + E is effective when N is large enough. 10

(4) LQ is big over XQ in the classical sense and NL is strictly effective for some positive integer N. Proof. (1) ⇐⇒ (2). It is Theorem 1.3. (3) =⇒ (2). It is trivial by setting E = −M . (2) =⇒ (3). Suppose NL = M + T as in (2). Then rNL + E = (E + rM ) + rT . Because M is ample, E + rM is effective for r large enough, and thus rNL + E is effective for r large enough. Replacing E by E + kL for k = 0, 1, · · · , N − 1, we see that N ′ L + E is effective when N ′ is large enough. Property (4) is Moriwaki’s definition of big line bundles, and (3) ⇔ (4) is Proposition 2.2 in [Mo2]. For convenience of readers, we still include it here. (2) =⇒ (4). Assume NL = M + T as in (2). It is easy to see that rNL is strictly effective for some integer r > 0. By NLQ = MQ + TQ , we see LQ is big by classical theory. (4) =⇒ (2). Assume that s ∈ Γ(X, NL ) with ksksup < 1. Since LQ is big, the line bundle −MQ + N ′ LQ is effective for some integer N ′ > 0. It follows that −M + N ′ L has a regular section t. Now ksr tksup ≤ kskrsup ktksup < 1 for r large enough. That means −M + (N ′ + rN)L is effective. Remark. Arithmetically big line bundles share many properties with the classical big line bundles. An important one is the continuity of the volume function 0

h (NL ) c ) = lim supN →∞ n vol(L , N /n!

which is proved by Moriwaki in the recent work [Mo3]. His proof follows the same strategy as the one we use to prove Theorem 1.2 and Theorem 1.3 here. To end this section, we give an example that a line bundle is big but not strongly big. Suppose X = P1Z = Proj Z[x0 , x1 ] and T = O(1). Pick a constant 0 < c < e−1 , and define a metric over T by |s(x0 , x1 )| , ks(x0 , x1 )k = p |x0 |2 + c|x1 |2

where s(x0 , x1 ) is considered as a homogeneous linear polynomial in x0 and x1 . It is easy to see that the metric is well defined. And the section s0 (x0 , x1 ) = x0 is effective. x1 be the usual affine coordinate on X − V (x0 ). Direct computation shows the Let z = x0 ic dz ∧ d¯ 1 z curvature form c1 (T ) = is positive and cˆ1 (T )2 = (1 + log c) < 0. 2 2 2π (1 + c|z| ) 2 Let M be any ample line bundle over X. For m > 0, the line bundle L = M + mT is big (ample+effective) and satisfies the arithmetic Hilbert-Samuel formula. But when m 1 1 is large enough, the leading coefficient cˆ1 (L )2 = (ˆ c1 (T ))2 in the arithmetic c1 (M ) + mˆ 2 2 Hilbert-Samuel formula is negative. We conclude that L is not strongly big.

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2

Arithmetic Volumes

In this section, we consider general normed modules and prove Theorem 2.1 which will be an important tool to read volume information from exact sequences. An important example in this class is the L2 -norm for a hermitian line bundle. As the first application, we show that strong bigness over an arithmetic variety is implied by strong bigness over its generic resolution in Theorem 2.2. By this, we need only work on generically smooth arithmetic varieties. By a normed Z-module M we mean a finitely generated Z-module M together with an R-norm on MR = M ⊗ R. For such an M, define h0 (M) = log #{m ∈ M : kmk < 1}. Denote by Mtor the torsion part of M, and by Mfree the free part of M. Define χ(M) = log

vol(B(M)) + log #Mtor , vol(MR /Mfree )

where B(M) = {m ∈ MR : kmk < 1} is the unit ball for the norm. Define χ(M) to be log #Mtor if M is torsion. Note that χ(M) does not depend on the Haar measure chosen over MR . The norm associated to M is quadratic if it is an inner product on MR . In this case we call M a quadratically normed Z-module. If m1 , m2 , · · · , mr is a Z-basis of Mfree , then χ(M) = log p

V (r) + log #Mtor , det(hmj , mk i)1≤j,k≤r

r r where V (r) = π 2 /Γ( + 1) is the volume of the unit ball in the Euclidean space Rr . 2 Let X be an arithmetic variety of dimension n and L be a line bundle over X. Then the supremum norm makes Γ(X, L )R a normed module. Apparently, it is not quadratic. However, one can define an L2 -norm which is quadratic and closely related to the supremum Z 1/2 2 |s(z)| dµ norm. Fix a measure dµ on X. One defines the L2 -norm by kskL2 = for

any s ∈ Γ(X, L )R . Then this L2 -norm makes Γ(X, L ) a quadratically normed module, and induces an associated arithmetic volume χL2 (L ). If X is generically smooth and the measure dµ is pointwise positive, Gromov’s norm inequality implies χsup (NL ) = χL2 (NL ) + O(N n−1 log N) (cf. [GS2]). This remains true if XQ is singular and the measure over X is the push-forward of a pointwise positive measure on a generic resolution of X. One can replace χsup by χL2 in Definition 1.1 or in Theorem 1.2. In Section 3, we will give a proof of Gromov’s inequality in the case of three line bundles. We consider quadratically normed modules because their volumes behave well under exact sequences. Theorem 2.1. The following are true for quadratically normed Z-modules: 12

(1) Suppose 0 → M ′ → M → M ′′ → 0 is an exact sequence of normed modules, i.e., the ′ ′′ sequence is exact as Z-modules, and the norms on MR and on MR are respectively the subspace norm and quotient norm induced from MR . Then χ(M) − χ(M ′ ) − χ(M ′′ ) = log V (rank(M)) − log V (rank(M ′ )) − log V (rank(M ′′ )). In particular, one has χ(M) ≤ χ(M ′ ) + χ(M ′′ ). (2) If M can be generated by elements with norms not greater than a positive constant c, then χ(M) ≥ log V (rank(M)) − rank(M) log c. (3) Let f : M ′ → M be an injection of Z-modules that is norm-contractive, i.e., kf (m′ )k ≤ ′ km′ k for all m′ ∈ MR . If M can be generated by elements with norms not greater than a positive constant c, then χ(M) − χ(M ′ ) ≥ log V (rank(M)) − log V (rank(M ′ )) − (rank(M) − rank(M ′ )) log c. ′′ ′ Proof. (1) MR is isomorphic to the orthogonal complement of MR in MR with the induced subspace norm. The result follows from the fact that log(vol(MR /Mfree)/#Mtor ) is additive if the volume elements are induced by the norms. (2) We can assume c = 1 and M is torsion free. By the condition we can find r = rank(M) elements m1 , m2 , · · · , mr ∈ M with kmj k ≤ 1 which form a Z-basis of a submodule M ′ . Since χ(M) ≥ χ(M ′ ), it suffices to prove χ(M ′ ) ≥ log V (rank(M ′ )). By V (r) , it remains to show that det(hmj , mk i) ≤ 1. The maχ(M ′ ) = log p det(hmj , mk i)1≤j,k≤r trix A = (hmj , mk i)1≤j,k≤r is symmetric and positive definite, so it has r positive eigenvalues r r X X x1 , x2 , · · · , xr . We have xj = tr(A) = hmj , mj i ≤ r. By the arithmetic-geometric j=1 r Y

mean inequality, det(A) =

j=1

xj ≤ 1.

j=1

(3) It suffices to show the case that M ′ is endowed with the induced subspace norm. Suppose M ′′ = M/M ′ endowed with the quotient norm. Apply (1), and apply (2) for M ′′ . Theorem 2.2. Let L and E be two hermitian line bundles over an arithmetic variety X of e → X be any birational morphism from another arithmetic variety dimension n. Let π : X e X to X. Then ⊗N ⊗N χsup (E ⊗ L ) ≥ χsup (π ∗ E ⊗ π ∗ L ) + o(N n ). Proof. For simplicity, we assume that E to be trivial. The general case is proved in the same way without any extra work. e → X is finite. In fact, consider the Stein Firstly, we can reduce to the case that π : X ′ p π e → X ′ → X, where X ′ = Spec(π∗ O e ) is finite over X. One has p∗ O e = OX ′ . factorization X X X For any hermitian line bundle T over X ′ , e p∗ T ) = Γ(X ′ , p∗ (p∗ T )) = Γ(X ′ , T ⊗ p∗ O e ) = Γ(X ′ , T ) Γ(X, X 13

e p∗ T ) = Γ(X ′ , T ) is actually an isometry by projection formula. The isomorphism Γ(X, e π ∗ L ⊗N ) = χsup (X ′ , π ′∗ L ⊗N ). So it suffices under the supremum norms. Therefore, χsup (X, to show the same result for the morphism π ′ : X ′ → X, which is finite. ⊗N ⊗N Secondly, it suffices to prove χL2 (L ) ≥ χL2 (π ∗ L ) + o(N n ) by choosing nice meae Suppose X e′ → X e is a generic resolution of X. e Fix a pointwise positive sures on X and X. ′ e , which induces push-forward measures over X e and X. These measures measure over X 2 define L -norms for line bundles over them. As discussed at the beginning of this section, a bound on χL2 is equivalent to the same bound on χsup . e → X is finite and endowed with measures as above, and we will Now assume π : X ⊗N ⊗N e π ∗ L ⊗N ) = prove χL2 (L ) ≥ χL2 (π ∗ L ) + o(N n ). The projection formula gives Γ(X, e π ∗ L ⊗N ) is an isometry Γ(X, L ⊗N ⊗ π∗ OXe ). And the natural injection Γ(X, L ⊗N ) → Γ(X, to its image under L2 -norms. The task is to bound the quotient. Pick a hermitian line bundle M over X satisfying the following two conditions: (1) L ⊗ M is arithmetically ample, and L ⊗ M is ample in the classical sense; (2) There exists an effective section s ∈ Γ(X, M ) which does not vanish at any associated point of the coherent sheaf π∗ OXe /OX over X. The finiteness of π implies that π ∗ (L ⊗ M ) is arithmetically ample, and π ∗ (L ⊗ M ) is ample in the classical sense. So χL2 (π ∗ (L ⊗ M )⊗N ) and χL2 ((L ⊗ M )⊗N ) satisfy the arithmetic Hilbert-Samuel formula. Consider the following commutative diagram of exact sequences 0 −→ (L ⊗ M )⊗N −→ (L ⊗ M )⊗N ⊗ π∗ OXe −→ (L ⊗ M )⊗N ⊗ (π∗ OXe /OX ) −→ 0 x x x N N N s s s 0 −→

L ⊗N

L ⊗N ⊗ π∗ OXe

−→

−→

L ⊗N ⊗ (π∗ OXe /OX )

−→ 0

which will induce a diagram for long exact sequences of cohomology groups over X. By the choice of s, the three vertical morphisms are injective. Thus the diagram implies an injection Γ(L ⊗N ⊗ π∗ OXe )/Γ(L ⊗N ) → Γ((L ⊗ M )⊗N ⊗ π∗ OXe )/Γ((L ⊗ M )⊗N )

(2)

which is norm-contractive. L∞ ⊗N e ∗ Since π ∗ (L ⊗ M ) is ample, the section ring ) is a finitely N =0 Γ(X, π (L ⊗ M ) generated Z-algebra. By picking a set of generators, one sees that there exists a constant e π ∗ (L ⊗ M )⊗N ) is generated by sections c > 0 such that Γ(X, (L ⊗ M )⊗N ⊗ π∗ OXe ) = Γ(X, with norms less than cN . And thus Γ((L ⊗ M )⊗N ⊗ π∗ OXe )/Γ((L ⊗ M )⊗N ) is generated by elements with norms less than cN . Applying Theorem 2.1(3) to the injection (2), we have

χ Γ(L ⊗N ⊗ π∗ OXe )/Γ(L ⊗N ) ≤ χ Γ((L ⊗ M )⊗N ⊗ π∗ OXe )/Γ((L ⊗ M )⊗N ) + O(N n−1 ). By Theorem 2.1(1), we obtain χL2 (π ∗ L

⊗N

) − χL2 (L

⊗N

) 14


= χ Γ(L ⊗N ⊗ π∗ OXe )/Γ(L ⊗N ) + O(N n−1 log N)
≤ χ Γ((L ⊗ M )⊗N ⊗ π∗ OXe )/Γ((L ⊗ M )⊗N ) + O(N n−1 log N) = χL2 (π ∗ (L ⊗ M )⊗N ) − χL2 ((L ⊗ M )⊗N ) + O(N n−1 log N) = o(N n ).

3

Analytic Parts

In this section, we prove a volume comparison theorem in the first two subsections and show a Gromov type of norm inequality in the third subsection. This section is divided into three subsections according to different settings. Suppose X is a generically smooth arithmetic variety of dimension n. Let L and M be two hermitian line bundles with positive curvatures. Endow XC with the probability 1 1 c1 (L )n−1 = R c1 (L )n−1 induced by L , where measure dµ = dµL = n−1 degLQ (XQ ) c (L ) XC 1 1 the curvature form c1 (L ) = ∂ ∂¯ log |s| (s is any meromorphic section). πi Fix a nonzero section s ∈ Γ(M ). Then tensoring by s defines an injection Γ(NL − jM ) → Γ(NL − (j − 1)M ). This gives an induced quadratic norm ktks =

Z

2

2

|s(z)| |t(z)| dµ

1/2

for any t ∈ Γ(NL − jM ). We will compare this norm with the original L2 -norm ktkL2 =

Z

2

|t(z)| dµ

1/2

on Γ(NL − jM ). Let Bs and BL2 be the unit balls in Γ(NL − jM )R corresponding to these two norms. Then vol(BL2 )/vol(Bs ) is independent of the Haar measure chosen on Γ(NL − jM )R . The main result in this section is as follows: Proposition 3.1. If L and M have positive curvatures, and s ∈ Γ(M ) is effective, then Z  vol(BL2 ) 1 1 log( ) ≥ dim Γ(NL ) log |s(z)|dµ (1 + O( + )) vol(Bs ) j N as N, j → ∞.

15

Distortion Functions In this subsection, X denotes a compact complex manifold of complex dimension n with a probability measure dµ. Here X is not necessarily connected, but we require that each connected component is of dimension n. Many results over connected manifolds can be extended naturally to this case. For any hermitian line bundle L over X, the L2 -norm makes Γ(L ) a complex Hilbert space. Suppose s1 , s2 , · · · , sr form an orthonormal basis. Define the distortion function b(L ) : X → R by b(L )(z) = |s1 (z)|2 + |s2 (z)|2 + · · · + |sr (z)|2 which is independent of the basis. For convenience, define b(L ) to be zero everywhere if Γ(L ) = 0. The following theorem was proved independently by Bouche [Bo] and Tian [Ti]: Theorem 3.2. If L has positive curvature and the measure dµ over X is induced by L , then for any hermitian line bundle E , b(E + NL )(z) = dim Γ(E + NL )(1 + O(

1 )) N

uniformly on X as N → ∞. Now we generalize it to an estimate on NL − jM for hermitian line bundles L and M over X, which will be used to prove Proposition 3.1. Theorem 3.3. If L and M have positive curvatures, and the measure dµ over X is induced by L , then for any hermitian line bundle E , b(E + NL − jM )(z) ≤ dim Γ(E + NL )(1 + O(

1 1 + )) N j

uniformly on X as N, j → ∞. Proof. Assume E to be trivial as usual. For fixed z ∈ X, one can choose an orthonormal basis of Γ(jM ) under the measure dµM such that only one section in this basis is nonzero at z. Call this section sj . Then by Theorem 3.2, one has |sj (z)|2 = b(jM )(z) = dim Γ(jM )(1 + O(1/j)), ksj k2sup = supx∈X |sj (x)|2 ≤ supx∈X b(jM )(x) = dim Γ(jM )(1 + O(1/j)). Those imply |sj (z)|/ksj ksup = 1 + O(1/j). Note that this result actually does not depend on the measure on X. Next we use the measure dµ = dµL . For each such sj , consider the two norms k · kL2 and k · ksj on Γ(NL − jM ). By linear algebra, there exists a basis t1 , t2 , · · · , tr , which is orthonormal under k · kL2 and orthogonal under k · ksj . Since k · ksj is the induced norm under sj

Γ(NL − jM ) → Γ(NL ),

16

we can view sj t1 , sj t2 , · · · , sj tr as r orthogonal elements of Γ(NL ). Normalize them and apply Theorem 3.2 again: r X |sj (z)tk (z)|2 k=1

Since ks

j

tk k2L2

=

we have

Z

ksj tk k2L2

j

2

≤ b(NL )(z) = dim Γ(NL )(1 + O(

|s (x)tk (x)| dµ ≤ r X

ksj k2sup

Z

1 )). N

1 |tk (x)|2 dµ = ksj k2sup = |sj (z)|2 (1 + O( )), j

|tk (z)|2 ≤ dim Γ(NL )(1 + O(

k=1

1 1 + )). N j

Remark. Since any hermitian line bundle is the difference of two ample line bundles with positive curvatures, this theorem actually gives an upper bound of b(NL ) for any hermitian line bundle L .

The Proof Now we can prove Proposition 3.1. We follow the same strategy as [AB, Lemma 3.8]. Proof of Proposition 3.1. As in the proof of Theorem 3.3, pick a basis t1 , t2 , · · · , tr of Γ(NL − jM ) such that this basis is orthonormal under k · kL2 and orthogonal under k · ks . Then Z r r Y 1X vol(BL2 ) log |s(x)|2 · |tk (x)|2 dµ. ) = log ktk ks = log( vol(Bs ) 2 k=1 k=1 Z Since |tk (x)|2 dµ = 1, one can view |tk (x)|2 dµ as a probability measure on XC . Applying Jensen’s inequality to the function log, one gets Z Z 2 2 log |s(x)| · |tk (x)| dµ ≥ log |s(x)|2 · |tk (x)|2 dµ,

and thus

r Z vol(BL2 ) 1X log( log |s(x)|2 · |tk (x)|2 dµ ) ≥ vol(Bs ) 2 k=1 Z r X 1 = log |s(x)|2 · |tk (x)|2 dµ 2 Zk=1  1 1 1 2 ≥ dim Γ(NL ) log |s(z)| dµ (1 + O( + )) 2 j N  Z 1 1 log |s(z)|dµ (1 + O( + )), = dim Γ(NL ) j N

17

where the last inequality uses Theorem 3.3 and the assumption ksksup ≤ 1.

Gromov’s Inequality To end this section, we show a version of Gromov’s norm comparison theorem for there line bundles. The proof is the same as the original one in [GS2]. We still include it here. Theorem 3.4. Suppose X is a compact complex manifold of complex dimension n endowed with a measure. Let L , M and E be three hermitian line bundles over X. Then there exists a positive constant c such that kskL2 ≥ c(k + j)−n ksksup , ∀k, j > 0, s ∈ Γ(E + kL + jM ). Proof. Assume E is the trivial. One can find a finite open cover {Uα }α of X satisfying the following (1) {Uα }α trivializes L and M . (2) Each Uα is isomorphic to {z ∈ Cn : |z| < 3} under the coordinate zα : Uα → Cn . (3) The discs {x ∈ Uα : |zα (x)| < 1} defined by these coordinates still cover X. Suppose hα , h′α give the metrics of L and M , i.e., |s|2 = hα sα sα for a local section s = (sα ) of L , and similarly for M . View Dα = {zα ∈ Cn : |zα | ≤ 2} as a disc in R2n and hα as a smooth function on it. We can find a constant c bounding the length of the n gradient of  hα in {zα ∈ C : |zα | ≤ 2} for all α. Pick a constant c1 > 1 such that c c c1 > max , ∀x ∈ Dα , ∀α. , ′ hα (x) hα (x) For any x0 , x1 ∈ Dα , one has |hα (x1 ) − hα (x0 )| ≤ c|zα (x1 ) − zα (x0 )| and thus hα (x1 ) ≥ hα (x0 ) − c|zα (x1 ) − zα (x0 )| ≥ hα (x0 )(1 − c1 |zα (x1 ) − zα (x0 )|) Now consider the norms of s = (sα ) ∈ Γ(kL + jM ). Suppose ksksup = |s(x0 )| for a point x0 ∈ X. Suppose x0 is contained in {x ∈ Uα : |zα (x)| < 1}. Now the neighborhood U = {x ∈ Uα : |zα (x) − zα (x0 )| < c−1 1 } is contained in Dα since c1 > 1. We have hα (x) ≥ hα (x0 )(1 − c1 |zα (x) − zα (x0 )|) for all x ∈ U and the same for h′α . Finally, we come to our estimate. For simplicity, assume zα (x0 ) = 0. Then ksk2L2 Z |s(x)|2 dµ ≥ UZ hα (z)k h′α (z)j sα (z)sα (z)dV (z) ≥ c′ −1 n {z∈C : |z| 0 is a vertical prime divisor lying over a prime pk in Spec(Z). Here Y1 , · · · , Yr are not necessarily distinct. Now we divide the problem into two cases. Case 1: dim X > 2. Since YQ is smooth, Y0 is reduced and irreducible by [Ha, Corollary 7.9]. And each Yk for k ≥ 0 is locally principal by our assumption. 21

Let Mk = O(Yk ) be the corresponding line bundle and sk be the section defining Yk . Endow Mk for k > 0 with the metric |sk | = 1, and M0 with the metric |s0 | = |s|. Then M = M 0 + M 1 + · · · + M r is an isometry. Denote N k = M 1 + · · · + M k for k = 1, · · · , r and N 0 = 0 to be the trivial metrized line bundle. The norm k · ks can be considered as a norm on Γ(NL − jM + Nk ) under the identity Γ(NL − jM + Nk )R = Γ(NL − jM )R . Note that Γ(Yk+1, NL − jM + Nk+1 ) is torsion, so the exact sequence 0 → Γ(NL − jM + Nk ) → Γ(NL − jM + Nk+1 ) → Γ(Yk+1, NL − jM + Nk+1 ) implies that for k = 0, · · · , r − 1,

= ≥ ≥ = =

χs (NL − jM + Nk ) − χs (NL − jM + Nk+1 ) − log # (Γ(NL − jM + Nk+1 )/Γ(NL − jM + Nk )) − log #Γ(Yk+1 , NL − jM + Nk+1 ) − log #Γ(Yk+1 , NL ) − dim Γ(Yk+1, NL ) log pk+1 cˆ1 (L )n−1 · Yk+1 n−1 N + O(N n−2). − (n − 1)!

To complete the proof, it suffices to show χs (NL − jM + N r ) − χL2 (NL − (j − 1)M ) ≥ −

cˆ1 (L )n−1 · Y0 n−1 N + O(N n−2 log N). (n − 1)!

Denote Γ = Γ(X, NL − (j − 1)M )/s0 Γ(X, NL − jM + Nr ). Then we have two exact sequences: ⊗s

0 → Γ(X, NL − jM + Nr ) →0 Γ(X, NL − (j − 1)M ) → Γ → 0, 0 → Γ → Γ(Y0 , NL − (j − 1)M ). Two norms are induced on Γ: the quotient norm k · kq and the subspace norm k · ksub . By Theorem 2.1(1), one has χs (NL − jM + N r ) − χL2 (NL − (j − 1)M ) + χq (Γ) ≥ 0. So it suffices to show χq (Γ) ≤

cˆ1 (L )n−1 · Y0 n−1 N + O(N n−2 log N). (n − 1)!

We have an injection Γ(Y0 , NL − (j − 1)M ) → Γ(Y0 , NL ) given by tensoring by s⊗(j−1) |Y0 , which is norm-contractive since ksksup < 1. By the ampleness theorem of Zhang, for N large

22

enough, Γ(Y0 , NL ) is generated by elements with L2 -norms less than 1. Therefore, we can apply Theorem 2.1(3) to the injection Γ → Γ(Y0 , NL ) and obtain χsub (Γ) ≤ χL2 (Γ(Y0, NL )) + log V (rankΓ) − log V (rankΓ(Y0 , NL )) = χL2 (Γ(Y0, NL )) + O(rankΓ(Y0 , NL ) log rankΓ(Y0 , NL )) =

cˆ1 (L )n−1 · Y0 n−1 N + O(N n−2 log N). (n − 1)!

The following Lemma 4.4 completes the proof. Case 2: dim X = 2. If dim X = 2, then Y0 is reduced by the smoothness but not necessarily irreducible. The only difference is that we can’t simply apply the arithmetic Hilbert-Samuel formula to χ (Y , NL ) because of the irreducibility of Y0 . But we can use the injection Γ(Y0 , NL ) → LL2 0 l l l Γ(Y0 , NL ), where {Y0 }l goes through all irreducible components of Y0 . This injection has finite cokernel since it is an isomorphism after tensoring by C. Since (Y0 )C is a finite set of points, it is easy to see that there exists a constant c > 0 −1 independent of N such L thatl c k · k1 ≤ k · k2 ≤ ck · k1 , where k · k1 and k · k2 are the norms for Γ(Y0 , NL ) and l Γ(Y0 , NL ). Therefore, we have X M χL2 (Y0l , NL ) + O(log N) Γ(Y0l , NL )) + O(log N) = χL2 (Y0 , NL ) ≤ χL2 ( =

X

l

l

(ˆ c1 (L ) ·

Y0l )N

+ O(log N) = (ˆ c1 (L ) · Y0 )N + O(log N).

l

This matches with L thel result in Case 1. Actually it is an equality since the cokernel of Γ(Y0 , NL ) → l Γ(Y0 , NL ) is bounded by a constant independent of N. Lemma 4.4. χq (Γ) − χsub (Γ) ≤ O(N n−2 log N).

Proof. Denote the quotient map by φ : Γ(X, NL − (j − 1)M ) → Γ. Applying Theorem 3.4, we get for any γ ∈ Γ, kγkq =

inf

t∈φ−1 (γ)

ktkL2 ≥ cN −n

inf

t∈φ−1 (γ)

ktksup ≥ cN −n

inf

t∈φ−1 (γ)

kt|Y0 kL2 = cN −n kγksub .

We have Bq (Γ) ⊆ c−1 N n Bsub (Γ), and vol(Bq (Γ)) ≤ (c−1 N n )rankΓ vol(Bsub (Γ)). Therefore, χq (Γ) − χsub (Γ) = log

vol(Bq (Γ)) ≤ (rankΓ) log(c−1 N n ) = O(N n−2 log N). vol(Bsub (Γ))

Lemma 4.1 is a simple summation of the inequalities in Lemma 4.2 and Lemma 4.3. 23

Proof of Lemma 4.1. By Lemma 4.2 and Lemma 4.3,

≥ = = = = =

χL2 (NL − jM ) − χL2 (NL − (j − 1)M ) Z  cˆ1 (L )n−1 · Y n−1 N n−1 − N + rankΓ(NL ) log |s(z)|dµ + O( + N n−2 log N) (n − 1)! j Z n−1 degLQ (XQ ) N n−1 cˆ1 (L )n−1 · Y n−1 N log |s(z)|dµ + O( − N + + N n−2 log N) (n − 1)! (n − 1)! j n−1 n−1 Z n−1 N · Y n−1 N cˆ1 (L ) log |s(z)|c1 (L )n−1 + O( N + + N n−2 log N) − (n − 1)! (n − 1)! j   Z N n−1 N n−1 n−1 n−1 cˆ1 (L ) + N n−2 log N) + O( · Y − log |s(z)|c1 (L ) − (n − 1)! j N n−1 N n−1 n−1 2 − · (Y, − log |s| ) + O( cˆ1 (L ) + N n−2 log N) (n − 1)! j n−1 n−1 N N cˆ1 (L )n−1 cˆ1 (M ) + O( + N n−2 log N). − (n − 1)! j

Proof of Theorem 1.3 Let us now sketch a proof of Theorem 1.3. Let M be normed free Z-module defined in Section 2. Then χ(M) and h0 (M) are related by Gillet-Soul´e [GS3]. Fix a Z-basis m1 , · · · , mr of M. This basis identifies MR with Rr . Then B(M) = {m ∈ MR : kmk < 1} is a convex symmetric body in Rr . Define ) ( r r X X bi mi ∈ B(M) , ai bi < 1, ∀ m = h1 (M) = log # (a1 , · · · , ar ) ∈ Zr : i=1

i=1

which is the logarithm of the number of lattice points in the dual of B(M) in Rr . Then [GS3, Theorem 1] gives the following Riemann-Roch type theorem: h0 (M) − h1 (M) = χ(M) + O(r log r),

where r is the rank of M, and the error term O(r log r) depends only on r. It has many consequences: (1) If M has two norms k · k1 and k · k2 such that k · k1 ≤ k · k2 , then h02 (M) ≤ h01 (M) ≤ h02 (M) + χ1 (M) − χ2 (M) + O(r log r). The first inequality is trivial and the second one is implied by h11 (M) ≤ h12 (M).

24

(2) If M has two norms k · k1 and k · k2 such that k · k1 = αk · k2 for some α > 0, then h01 (M) = h02 (M) + O(r log α + r log r). This is implied by (1). See [GS3, Proposition 4]. (3) Let 0 → M ′ → M → M ′′ → 0 be an exact sequence of normed modules, then h0 (M) ≤ h0 (M ′ ) + h0 (M ′′ ) + O(r ′ log r ′ ), where r ′ = rank(M ′ ). Denote L(M) = {m ∈ M : kmk < 1}, and similarly for L(M ′ ) and L(M ′′ ). Consider the induced map p : L(M) → L(M ′′ ). For any y ∈ L(M ′′ ) and x ∈ p−1 (y), the set p−1 (y) − x = {z − x : z ∈ p−1 (y)} is contained in L2 (M ′ ) = {m ∈ M ′ : kmk < 2}. This gives #p−1 (y) ≤ #L2 (M ′ ), and thus #L(M) ≤ (#L2 (M ′ )) · (#L(M ′′ )). Take logarithm and use (2). (4) Let L be a hermitian line bundle over an arithmetic variety X of dimension n. By Gromov’s inequality together with (1) and (2), the h0 -numbers for Γ(X, NL ) induced by the supremum norm and the L2 -norm are equivalent, i.e., h0sup (NL ) = h0L2 (NL ) + O(N n−1 log N). (5) If L is ample and X is generically smooth, it is easy to see that h1 (Γ(X, NL )) = 0 when Γ(X, NL ) has a basis consisting of effective sections. Thus all h0sup (NL ), h0L2 (NL ), χsup (NL ) and χL2 (NL ) are equivalent, with the same expansion cˆ1 (L )n n N + O(N n−1 log N). n! The same is true for E + NL . Proof of Theorem 1.3. The “if” part is easy. Suppose rL = M + T with r positive, M ample and T effective. We need to show that L is big. Let E be any line bundle. Picking a strictly effective section s ∈ Γ(X, T ), the injection Γ(X, E + NM ) → Γ(X, E + NrL ) defined by tensoring by s⊗N is norm-contractive. It follows that h0 (E + NrL ) ≥ h0 (E + NM ) ≥

cˆ1 (M )n n N + o(N n ). n!

Set E = kL for k = 0, 1, · · · , r − 1, we get h0 (NL ) ≥

1 cˆ1 (M )n n N + o(N n ). n r n! 25

So L is big. Now we show the other direction. Suppose L is big, and we need to show that it is ′ the sum of an ample line bundle and an effective line bundle. Write L = L − M for ′ ample line bundles L and M with positive curvatures. It suffices to show that NL − M = ′ NL −(N +1)M is effective for N large enough. By a similar process as in Theorem 2.2, one can reduce the problem to the case with the three assumptions. By the result (4) above, it is equivalent to consider L2 -norms and norms induced by them. With the help of the above results again, it is easy to see that Lemma 4.1 still works with an error term O(N n−1 log N) if we replace χL2 by h0L2 everywhere. Especially, it works for j = N + 1, i.e., ′

′

′

′

h0L2 (NL − (N + 1)M ) − h0L2 (NL − NM ) ≥ O(N n−1 log N). Thus

h0 (NL − (N + 1)M ) ≥ h0 (NL − NM ) + O(N n−1 log N) ′

is positive when N is large enough, and then NL − (N + 1)M is effective.

5

Equidistribution Theory

As an application of Theorem 1.2, some equidistribution theorems are generalized in this section. The equidistribution theory we are going to consider originated in the paper [SUZ] of Szpiro-Ullmo-Zhang. They proved an equidistribution theorem [BV, Theorem 3.1] over complex analytic spaces for line bundles of positive curvatures over generically smooth arithmetic varieties, and it was extended to certain cases by Ullmo [Ul, Theorem 2.4] and Zhang [Zh3, Theorem 2.1] to prove the Bogomolov conjecture. Recently, Chambert-Loir [Ch1] defined the canonical measures over Berkovich spaces, and proved an equidistribution theorem over the Berkovich spaces [Ch1, Theorem 3.1]. It is a non-archimedean analogue of Szpiro-Ullmo-Zhang’s theory. All the above results assume the strict positivity of the metrized line bundle at the place where equidistribution is considered, except for the case of curves in [Ch1] which makes use of Autissier’s theorem. See Remark (3) of Definition 1.1 for Autissier’s expansion. As we have seen in the introduction, we can remove the strict positivity condition with the asymptotic result in Theorem 1.2. We will put the two generalized results in Theorem 5.1 as conclusions at different places. We also have Theorem 5.2, an algebraic version of Theorem 5.1. Our proof follows the original idea of Szpiro-Ullmo-Zhang. This section consists of five subsections. We state the main equidistribution theorems (Theorem 5.1 and Theorem 5.2) in the first subsection, and prove them in the second (resp. third) subsection in the archimedean (resp. non-archimedean) case. In the fourth subsection, we extend Theorem 5.1 to equidistribution of small subvarieties as what Baker-Ih [BI] and Autissier [Au2] did for the equidistribution of Szpiro-Ullmo-Zhang. In the fifth subsection, we consider the consequences of these theorems in algebraic dynamical systems. In the sixth subsection, we apply our equidistribution to multiplicative groups and get a proof of the Bogomolov conjecture following the strategy of [Ul] and [Zh3]. Finally, in the seventh subsection, we give a list of all related equidistribution statements. 26

5.1

A Generic Equidistribution Theorem

Let K be a number field, and X be a projective variety over K. For each place v, denote by Kv the v-adic completion of K, and by Cv the completion of the algebraic closure K v of Kv . We endow Kv with the normalized absolute value | · |v , and Cv the unique extension of that absolute value. There are two canonical analytic v-spaces: an an is the usual associated to the variety XCv . Namely, XC 1. The Cv -analytic space XC v v complex analytic space Xv (C) if v is archimedean, and the Berkovich space associated to XCv if v is non-archimedean. See [Be] for an introduction of Berkovich spaces. See also Zhang’s simple description in Section 5.3. an an is the usual associated to the variety XKv . Namely, XK 2. The Kv -analytic space XK v v complex analytic space Xv (C) if v is complex archimedean, the quotient of the usual complex analytic space Xv (C) by the complex conjugate if v is real archimedean, and the Berkovich space associated to XKv if v is non-archimedean.

Both spaces are Hausdorff, compact, and finite disjoint unions of path-connected coman an /Gal(K v /Kv ) as topological spaces. = XC ponents. They are related by XK v v an an . We simply call the and XK We will state an equidistribution theorem over each of XC v v former the geometric case and the latter the algebraic case. One will see at the end of this subsection that the geometric case implies the algebraic case and that the algebraic cases over all finite extensions of K imply the geometric case. Geometric Case Let K be a number field and X be a projective variety of dimension n − 1 over K. Fix an embedding K → Cv for each place v. We will consider equidistribution of small algebraic an for each place v. points over XC v We use the language of adelically metrized line bundles by Zhang ([Zh1], [Zh2]). Recall that an adelic metric over a line bundle L of X is a Cv -norm k · kv over the fibre LCv (x) of each algebraic point x ∈ X(K) satisfying certain continuity and coherence conditions for each place v of K. The metric is semipositive if it is the uniform limit of a sequence of metrics induced by fj ) of (X, L ej ) such that each L fj is a relatively semipositive arithmetic integral models (Xj , L line bundle over Xj . A metrized line bundle is integrable if it is isometric to the difference of two semipositive metrized line bundles. The intersection number of integrable line bundles is uniquely defined by that limit process. Fix an integrable line bundle L over X. The height of X is defined to be hL (X) =

cˆ1 (L )n . n degL (X)

The height of an algebraic point x ∈ X(K) is defined to be hL (x) =

cˆ1 (L |x¯ ) , deg(x)

27

where x¯ is the closure of x in X, and deg(x) is the degree of the residue field of x¯ over K. Denote by O(x) the Galois orbit of x, the orbit of x under the action of Gal(K/K). Then O(x) is a set of algebraic points of order deg(x). One has 1 X X (− log ks(z)kv ). hL (x) = deg(x) v z∈O(x)

an for any place v. Define the probability We can also view O(x) as a finite subset of XC v measure associated to x by X 1 µv,x = δz , deg(x) z∈O(x)

an . XC v

where δz is the Dirac measure of z in Associated to L , there is a v-adic canonical measure c1 (L )vn−1 of total volume degL (X) an for any place v. When v is archimedean, the measure c1 (L )vn−1 is over the space XC v simply the usual differential form in the smooth case and extended to the general case by resolution of singularities and some limit process. For limits of volume forms we refer to [Zh4, Proposition 3.1.5]. When v is non-archimedean, the canonical measure c1 (L )vn−1 is defined by Chambert-Loir in [Ch1]. We will describe it in more details when we prove equidistribution at non-archimedean places. Now we recall some related definitions of equidistribution, which was stated in Section 0.1 in dynamical case. The only difference if that the height of X is not zero anymore.

1. A sequence {xm }m≥1 of algebraic points in X(K) is small if hL (xm ) → hL (X) as m → ∞. 2. A sequence {xm }m≥1 of algebraic points in X(K) is generic if no infinite subsequence of {xm } is contained in a proper closed subvariety of X. 3. Let {xm }m≥1 be a sequence of algebraic points in X(K) and dµ a probability measure an for a place v of K. We say the Galois orbits of {xm } are over the analytic space XC v equidistributed with respect to dµ if the probability measure {µv,xm } associated to the an ; i.e., sequence converges weakly to dµ over XC v Z X 1 f (x) → f (x)dµ an #O(xm ) XC x∈O(xm )

v

an → C. for any continuous function f : XC v

The equidistribution theorem in this subsection is the following: Theorem 5.1 (Equidistribution of Small Points). Suppose X is a projective variety of dimension n − 1 over a number field K, and L is a metrized line bundle over X such that L is ample and the metric is semipositive. Let {xm } be an infinite sequence of algebraic points in X(K) which is generic and small. Then for any place v of K, the Galois orbits of the an with respect to the probability sequence {xm } are equidistributed in the analytic space XC v n−1 measure dµv = c1 (L )v / degL (X). 28

Algebraic Case As in the geometric case, let K be a number field, X be a projective variety of dimension n − 1 over K, and L be an integrable line bundle over X. We are going to consider an for any place v. equidistribution of small closed points over XK v View X (resp. XKv ) as a scheme of finite type over K (resp. Kv ). When we talk about points in X or XKv here, we always mean closed points in the corresponding schemes. Note that in the geometric case points are algebraic points. When Kv ∼ = C, there is no difference between closed points and algebraic points in XKv . The height of X is still cˆ1 (L )n hL (X) = . n degL (X) The height of a closed point x ∈ X is defined to be hL (x) =

cˆ1 (L |x ) deg(x)

where deg(x) is still the degree of the residue field of x over K. For any closed point x ∈ X, the base change xKv splits into finitely many closed points in the scheme XKv . They form a set Ov (x), called the Galois orbit of x. We can also view an . Define the probability measure associated to x by Ov (x) as a finite subset of XK v µv,x =

X 1 deg(z)δz deg(x) z∈Ov (x)

an , and deg(z) is the degree of the residue field of z where δz is the Dirac measure of z in XC v over Kv . There is still a v-adic canonical measure c1 (L )vn−1 of total volume degL (X) over the space an for any place v. Actually the v-adic canonical measure here is just the push-forward XK v an an . → XK measure of the one in the geometric case under the natural map XC v v With analogous definitions of small sequences, generic sequences, and equidistribution, we have the following algebraic version of Theorem 5.1:

Theorem 5.2 (Equidistribution of Small Points). Suppose X is a projective variety of dimension n − 1 over a number field K, and L is a metrized line bundle over X such that L is ample and the metric is semipositive. Let {xm } be an infinite sequence of closed points in X which is generic and small. Then for any place v of K, the Galois orbits of the sequence an with respect to the canonical measure {xm } are equidistributed in the analytic space XK v n−1 dµv = c1 (L )v / degL (X). Equivalence an an , the push-forward measures of µv,xm and dµv = c1 (L )vn−1 → XK Via the projection XC v v an an . Thus it is easy to see that Theorem 5.1 over XCv give exactly their counterparts over XK v implies Theorem 5.2.

29

Conversely, Theorem 5.2 implies Theorem 5.1. The results of Theorem 5.2 for all finite extensions K ′ of K imply the equidistribution of Theorem 5.1. In fact, considering the base change XK ′ of X, Theorem 5.2 implies Z Z f µv,xm → f dµv , an XC v

an XC v

an an an → XK → C that is the pull-back via XC for any continuous function f : XC ′ of a v v v ′ an an continuous function over XKv′ for some extension of the valuation v to K . Here all XK ′ v an . form a projective system of analytic spaces with limit XC v A classical result says that any finite extension of Kv is isomorphic to some Kv′ above. For a proof see, for example, Exercise 2 in Page 30 of [Se]. Now it suffices to show that the an . We need the vector space of all such f is dense in the ring of continuous functions of XC v following Stone-Weierstrass Theorem.

Theorem (Stone-Weierstrass). Let X be a compact Hausdorff space, C(X) be the ring of real-valued continuous functions of X, and V ⊂ C(X) be an R-vector space. Then V is dense in C(X) under the supremum norm if the following two conditions hold: (1) For any f, g ∈ V , the functions max{f (x), g(x)} and min{f (x), g(x)} belong to V . (2) For any distinct points x 6= y in X, there exists f ∈ V such that f (x) 6= f (y). an . Applying the theorem, we only need to check that for any distinct Let us go back to XC v an points x, y ∈ XCv , there exist a finite extension E of Kv , and a continuous function f over XEan such that f takes different values at the images of x and y in XEan . This is equivalent to finding an E such that x and y have different images in XEan . an have the same image in XEan for any finite extension E over Assume that x, y ∈ XC v Kv . We are going to show that x = y. The problem is local. Assume M (A) is an affinoid an b v ) → M (A ⊗ containing the image of x and y. The natural map M (A⊗C subdomain of XK v b v to A ⊗ E. And thus the E) is just the restriction of multiplicative semi-norms from A⊗C semi-norms x and y have the same restriction on A ⊗ E for any E by the assumption. But S b b E A ⊗ E = A ⊗ K v is dense in A⊗Cv . It follows that x and y are the same on A⊗Cv . That completes the proof.

5.2

Equidistribution at Infinite Places

Now we are going to prove Theorem 5.1 and Theorem 5.2 for any archimedean place v. We will show Theorem 5.1, and this is enough by the equivalence relation developed at the end of last subsection. The proof follows the original idea in [SUZ] and [Zh3], except that we use Theorem 1.2 to produce small sections instead of the arithmetic Hilbert-Samuel formula. an is the usual complex space Xv (C). Assume v is archimedean. Then Cv = C, and XC v an an is called smooth Embed XCv in a compact complex manifold Y. A continuous function on XC v if it can be extended to a smooth function on Y . As in [Zh3], by the Stone-Weierstrass an can be approximated uniformly by smooth functions. theorem, continuous functions on XC v 30

It suffices to show lim

m→∞

Z

f µxm =

an XC v

1 degL (X)

Z

an XC v

f c1 (L )vn−1

an . for any smooth real-valued function f on XC v an For any real function g on XCv and any metrized line bundle M = (M , k · k) over X, define the twist M (g) = (M , k · k′ ) to be the line bundle M over X with the metric ksk′v = kskv e−g and ksk′w = kskw for any w 6= v. We first prove a lemma.

Lemma 5.3. Assume the above condition, i.e., L is a semipositive metrized line bundle an . For ǫ > 0, the adelic volume over X and f a smooth real-valued function on XC v χ(NL (ǫf )) ≥

cˆ1 (L (ǫf ))n + O(ǫ2 ) n N + o(N n ), n!

where the error term O(ǫ2 ) is independent of N. Proof. See [Zh2] for the definition and basic property of adelic volumes for metrized line bundles. By a limit process, it suffices to prove the case that L is induced by a single f) of (X, L ). semipositive model (X , L One has O(f ) = M 1 − M 2 for ample line bundles M 1 and M 2 over X . This is the very reason that we consider smooth functions. fis relatively semipositive, there exists a constant c > 0 such that L f(c) is ample. Since L f(c+ǫf ) = (L f(c)+ǫM 1 )−ǫM 2 is the difference of two ample line bundles. Applying Then L Theorem 1.2, one gets n n−1 f f f(c + ǫf )) ≥ cˆ1 (L (c) + ǫM 1 ) − n · cˆ1 (L (c) + ǫM 1 ) cˆ1 (ǫM 2 ) N n + o(N n ) χsup (N L n! n 2 f cˆ1 (L (c + ǫf )) + O(ǫ ) n = N + o(N n ) n! f(ǫf ))n + cn deg (X) + O(ǫ2 ) cˆ1 (L L = N n + o(N n ). n!

By definition, it is easy to see that

f(c + ǫf )) − χsup (N L f(ǫf )) = cNrankΓ(N L f) = c χsup (N L

degL (X) n N + o(N n ). (n − 1)!

f(ǫf )). Thus we get the bound for χsup (N L

With this lemma, the proof of Theorem 5.1 is the same as the original ones. In fact, by adelic Minkowski’s theorem (cf. Appendix C of [BG]), the lemma implies the existence of a nonzero small section s ∈ Γ(X, NL ) for any fixed archimedean place w0 such that log ksk′w0

  cˆ1 (L (ǫf ))n + O(ǫ2 ) 2 ≤− N + o(N) = −hL (ǫf ) (X) + O(ǫ ) N + o(N), n degL (X) 31

and log ksk′w ≤ 0 for all w 6= w0 . Here k · k′w denotes the metric of L (ǫf ). Computing the heights of the sequence by this section, we get lim inf hL (ǫf ) (xm ) ≥ hL (ǫf ) (X) + O(ǫ2 ). By definition, hL (ǫf ) (xm ) = hL (xm ) + ǫ hL (ǫf ) (X) = hL (X) + ǫ

Z

f µv,xm ,

an XC v

1

degL (X)

Z

an XC v

f c1 (L )vn−1 + O(ǫ2 ).

Since lim hL (xm ) = hL (X),

m→∞

we have lim inf

Z

f µv,xm ≥

an XC v

1 degL (X)

Z

an XC v

f c1 (L )vn−1 .

Replacing f by −f in the inequality, we get the other direction and thus Z Z 1 f c1 (L )vn−1 . lim f µv,xm = an an deg (X) XC XC L v

v

5.3

Equidistribution at Finite Places

In this subsection, we prove Theorem 5.1 and Theorem 5.2 for any non-archimedean place v. We will show Theorem 5.2, the algebraic case instead of the geometric case. Then Theorem 5.1 is implied by the argument at the end of Section 5.1. The proof here is parallel to the archimedean case, so the task is to initiate a process which can be run in the same way as in the archimedean case. The key is Gubler’s theorem that continuous functions over Berkovich spaces can be approximated by model functions which will be defined later. One can also strengthen Lemma 3.4 and Lemma 3.5 in [Ch1] to prove the result here. Canonical Measures an is the Berkovich space associated to the variety XKv for nonThe analytic space XK v archimedean v. The canonical measure c1 (L )vn−1 is defined by Chambert-Loir [Ch1] using ideas from the archimedean case. For example, if L0 , · · · , Ld are line bundles over X with vadic metrics, and Z is a closed subvariety of X of dimension d, then the local height formula (for sj ∈ Γ(X, Lj ) intersecting properly over Z)

c 0 ) · · · div(s c d )|Z )v (div(s

c 1 ) · · · div(s c d )|div(s | ) )v − = (div(s 0 Z

Z

an XK v

32

log ks0 kv c1 (L1 )v · · · c1 (Ld )v δZKanv

holds as in the archimedean case. And one also has the global height X c 0 ) · · · div(s c d )|Z )v , (div(s (ˆ c1 (L 0 ) · · · cˆ1 (L d ))|Z = v

where the sum is over all places v of K. Denote by OKv the valuation ring of Kv , and by kv the residue field. If the v-adic metric f) with X normal, then the canonical measure on L is defined by a single OKv -model (X , L an has a simple expression over XK v c1 (L )vn−1 =

r X

mi degLf(Yi )δηi ,

i=1

where Y1 , · · · , Yr are the irreducible components of the special fibre Xkv , and m1 , · · · , mr are an of the generic point of Yj under their multiplicities, and ηj is the unique preimage in XK v an . Locally, η is the semi-norm given by the valuation of the → X the reduction map XK j kv v local ring of the scheme at the generic point of Yj . an have properties similar to the algebraic case. The canonical measures over XC v Model Functions Let B be a Kv -Berkovich space which is Hausdorff, compact and strictly Kv -analytic. There is a notion of formal OKv -model for B, which is an admissible formal OKv -scheme with generic fibre B. For the basics of formal models we refer to [Ra] and [BL]. Let M be a line bundle over B. Among the Kv -metrics over M, there are some called formal metrics by Gubler [Gu]. They are induced by formal models of (B, M). Definition 5.4. A continuous function over B is called a model function if it is equal to − log k1k1/l for some nonzero integer l and some formal metric k · k over the trivial bundle of B. It is easy to see that all model functions form a vector space. The following theorem is due to Gubler [Gu, Theorem 7.12]. Theorem (Gubler). The vector space of model functions on B is uniformly dense in the ring of real-valued continuous functions on B. an is the Now let’s come back to our situation: X is a projective space over K and XK v corresponding Berkovich space at v. To compute heights, we work on global projective OK model of (X, OX ) in the usual sense, i.e. a pair (X , M ) consisting of an integral scheme X projective and flat over OK with generic fibre X, and a line bundle M over X which extends OX . A global projective OK -model gives a formal OKv -model by completion with respect to the ideal sheaf (̟) where ̟ is a uniformizer of OKv . Thus it induces a formal metric over OX , which is compatible with the adelic metric defined by Zhang. Now we are going to show that all formal metrics arise in this way.

33

an are induced by global projecLemma 5.5. All formal metrics over the trivial bundle of XK v tive OK -models. Thus all model functions are induced by global projective OK -models.

Proof. Let (X , M ) be any formal OKv -model. Fix a global projective OK -model X0 of c0 its completion at v. Then X c0 gives another formal OK -model of X and denote by X v an . By Raynaud’s result ([BL], Theorem 4.1) on the category of formal models, there XK v ′ ′ c exist two admissible formal blowing-ups X → X0 and φ : X → X , both of which induce an . Then (X ′ , M ′ ) induces the same formal metric as (X , M ), where isomorphisms over XK v ′ ∗ M = φ M. c0 . By formal GAGA Denote by I the coherent ideal sheaf for the blowing-up X ′ → X (EGAIII.1, Section 5), I comes from a coherent ideal sheaf of the projective OKv -variety (X0 )OKv . We still denote it by I . We can also consider I as a coherent ideal sheaf of X0 , since I contains some power of the maximal ideal (̟) of OKv . Let X ′′ be the blowing-up of X0 with respect to I . Then X ′′ is a global projective OK -model, and the completion at v of X ′′ gives X ′ . Now it remains to find a model of M ′ over X ′′ . One can descend M ′ to a line bundle over XO′′Kv by formal GAGA, and we still denote it by M ′ . Let D be a Cartier divisor on XO′′Kv defined by any rational section of M ′ . Since M ′ is trivial over the generic fibre of XO′′Kv , there exists a positive integer r > 0 such that ̟ r ⊗ D is effective. Replacing D by ̟ r ⊗ D, we assume that D is effective. Let J be the ideal sheaf of D in XO′′Kv , which can also be considered as a coherent ideal sheaf of X ′′ . If J is invertible over X ′′ , then (X ′′ , J ⊗(−1) ) is a desired global projective model which gives the same metric as (X ′ , M ′) does. Otherwise, consider the blowing-up π : X ′′′ → X ′′ with respect to J . Then π −1 J is invertible over X ′′′ , and (X ′′′ , (π −1J )⊗(−1) ) gives what we want. (In fact, one can show that J is invertible over X ′′ by this blowing-up.) Remark. It is possible to work directly on global projective OK -models and show that the model functions defined by them are uniformly dense, which will be enough for our application. Of course, it still follows Gubler’s idea in proving the density theorem. Use the Stone-Weierstrass theorem. Pick any initial projective model, blow-up it suitably to get separation of points, and use certain combinatorics and blowing-ups to prove the model functions are stable under taking maximum and minimum. A Description of the Berkovich Space an in an elementary Using model functions, Zhang [Zh4] constructed the Berkovich space XK v way. For any projective variety X over K, let V be the vector space of all model functions coming from projective OKv -model of X. Each element of V is considered as a map from |XKv | to R, where |XKv | is purely the underlying space of the scheme. Now take R(XKv ) to be the completion under the supremum norm of the ring generated by V . Then we have an XK = Hom(R(XKv ), R), v

34

where Hom is taking all continuous homomorphisms. In fact, by the density of model functions, R(XKv ) is exactly the ring of continuous an an . The . Therefore its spectrum recovers XK functions over the compact Hausdorff space XK v v an same construction is valid for XCv . Proof of Equidistribution Now we are ready to prove Theorem 5.2 when v is non-archimedean. By the density theorem proved above, it suffices to show Z Z 1 lim f µv,xm = f c1 (L )vn−1 m→∞ X an an deg (X) XK L K v

v

for any model function f = − log k1kv induced by a projective OK -model (X , M ). Denote by O(f ) the trivial line bundle OX with the adelic metric given by the model (X , M ), i.e., the metric such that k1kv = e−f and k1kw = 1 for any w 6= v. Define the twist L (ǫf ) = L + ǫO(f ) for any positive rational number ǫ. Note that we even have exactly the same notation as in the archimedean case. Over X , the line bundle M is a difference of two ample line bundles. It follows that O(f ) is a difference of two ample metrized line bundles. This tells why we spend so much energy proving the density of model functions induced by global models. Now everything including Lemma 5.3 follows exactly in the same way. In particular, we have lim inf hL (ǫf ) (xm ) ≥ hL (ǫf ) (X) + O(ǫ2 ). By the definition of our metrics and intersections, Z hL (ǫf ) (xm ) = hL (xm ) + ǫ f µv,xm , an XK v

hL (ǫf ) (X) = hL (X) + ǫ

1

degL (X)

Z

an XK v

f c1 (L )vn−1 + O(ǫ2 ).

We still have the condition lim hL (xm ) = hL (X).

m→∞

The variational principle follows exactly in the same way, so we conclude that Z Z 1 f c1 (L )vn−1 . lim f µv,xm = an degL (X) XKan XK v

v

5.4

Equidistribution of Small Subvarieties

Szpiro-Ullmo-Zhang’s equidistribution theorem was generalized to equidistribution of small subvarieties by Baker-Ih [BI] and Autissier [Au2]. Now we will generalize our theory to small subvarieties in the same manner. We follow the proof of [Au2], which still makes use of the 35

variational principle. We will only formulate the result in the geometric case, though it is immediate for both cases. Suppose we are in the situation of Theorem 5.1. By a subvariety of X, we mean a reduced and irreducible closed subscheme defined over K. For any subvariety Y of X, define its height to be cˆ1 (L )dim Y +1 |Y , hL (Y ) = (dim Y + 1) degL (Y ) where Y is the closure of Y in the scheme X. Then Y K splits into a finite set of subvarieties in XK . We denote this set by O(Y ), and an is a closed call it the Galois orbit of Y . For any Z ∈ O(Y ), the associated analytic space ZC v an subspace of XCv . Thus we can also view O(Y ) as a finite set of closed analytic subspace of an for any place v. XC v Now define the probability measure associated to Y by µv,Y =

X

1

degL (Y ) Z∈O(Y )

Y , δZCanv c1 (L |Z )dim v

Y an Y where c1 (L |Z )dim , and δZCanv c1 (L |Z )dim is the v-adic canonical measure over ZC sends a v v v Z Y an → C to continuous function f : XC . f c1 (L |Z )dim v v an ZC v

We need an additional assumption: hL (Y ) ≥ hL (X) for any subvariety Y of X. We will see later that for dynamical systems hL (X) = 0 and hL (Y ) ≥ 0 is always true. If L is an ample metrized line bundle, the assumption is equivalent to hL (x) ≥ hL (X) for any point x of X by the successive minima of Zhang [Zh2, Theorem 1.10]. With the same notions of small sequences, generic sequences and equidistribution as in section 5.1, we have Theorem 5.6 (Equidistribution of Small Subvarieties). Suppose X is a projective variety of dimension n − 1 over a number field K, and L is a metrized line bundle over X such that L is ample and the metric is semipositive. Assume hL (Y ) ≥ hL (X) for any subvariety Y of X. Let {Ym } be an infinite sequence of subvarieties of X which is generic and small. Then for any place v of K, the Galois orbits of the sequence {Ym } are equidistributed in the an with respect to the canonical measure dµv = c1 (L )vn−1 / degL (X). analytic space XC v Proof. We will sketch a proof of this theorem. Many things will only be written down formally. But it is not hard to make them rigorous following the way we treat Theorem 5.1 and Theorem 5.2. As in the case of small points, we consider the perturbation L (ǫf ) where f is a smooth function for archimedean v and a model function for non-archimedean v. We have seen that there exists a nonzero small section s ∈ Γ(X, NL ) for any fixed archimedean place w0 such that   ′ 2 log kskw0 ≤ −hL (ǫf ) (X) + O(ǫ ) N + o(N), 36

and log ksk′w ≤ 0 for all w 6= w0 . Here k · k′w denotes the metric of N(L (ǫf )), and k · kw denotes the metric of NL . We are going to estimate hL (Ym ) by this section, which is more delicate than the case of points. By the definition of intersections, for any d-dimensional irreducible subvariety Y of X not contained in div(s),

hL (Y ) =

cˆ1 (L )d+1 |Y (d + 1) degL (Y )

1 = N(d + 1) degL (Y ) 1 = d+1 1 = d+1 ≥

1 d+1

cˆ1 (L )d |div(s|Y ) −

1 dhL (div(s|Y )) − N

XZ w

XZ w

Y

an Cw

log kskw c1 (L )dw

log kskw µw,Y an XC w

!

!

Z Z 1 X ′ dhL (div(s|Y )) − log kskw µw,Y − ǫ f µv,Y an N w XCan X Cv w ! Z dhL (X) + hL (ǫf ) (X) − O(ǫ2 ) − oN (1) − ǫ f µv,Y

1 = hL (X) + d+1

ǫ

Z

f dµv − ǫ

an XC v

Z

an XC v

!

!

f µv,Y − O(ǫ2 ) − oN (1) . an XC v

Now it goes in the same way as the case of points. Set Y = Ym and use hL (Ym ) → hL (X). The above implies Z Z lim inf f µv,Ym ≥ f dµv . an XC v

Replacing f by −f , we conclude that Z lim

an XC v

f µv,Ym =

an XC v

5.5

Z

f dµv . an XC v

Equidistribution over Algebraic Dynamical Systems

The equidistribution theorems treated in previous subsections have direct consequences in algebraic dynamics. For a complete introduction to the basics and equidistribution of algebraic dynamics, we refer to [Zh4]. And we will only state the equidistribution of small points in the geometric case. Let K be a number field. Let X be a projective variety over K, and φ : X → X be a morphism polarized by an ample line bundle L over X, meaning that φ∗ L ∼ = L ⊗q for some integer q > 1. Then (X, φ, L ) is called an algebraic dynamical system. 37

Fix an isomorphism α : φ∗ L ∼ = L ⊗q . By [Zh2], there exists a unique semipositive metric over L which makes α an isometry. Actually, it can be obtained by Tate’s limit like the canonical height and the canonical measure in Section 0.1. This metric is called the canonical metric. Denote by L the line bundle L endowed with this metric. For any place v of K, one has the canonical measure c1 (L )vn−1 and the canonical probability measure an dµv,φ := c1 (L )vn−1 / degL (X) over XC . v Using the canonical metric, we define the canonical height of a subvariety Y by ˆ L (Y ) = h (Y ) = h L

cˆ1 (L )dim Y +1 |Y , (dim Y + 1) degL (Y )

as in the previous subsection. It is the same as the one defined by Tate’s limit. Now we use the same notions of small sequences, generic sequences and equidistribution as in Section 5.1. A closed subvariety Y of X is called preperiodic if the orbit ˆ L (X) = h (X) = 0 since X is preperiodic, and {Y, φ(Y ), φ2 (Y ), · · ·} is finite. Note that h L thus a small sequence really has heights going to zero. Now the following theorem is just a dynamical version of Theorem 5.1. Theorem 5.7 (Dynamical Equidistribution of Small Points). Let (X, φ, L ) be an algebraic dynamical system over a number field K, and {xm } be an infinite sequence of algebraic points of X which is generic and small. Then for any place v of K, the Galois orbits of the sequence an with respect to the canonical probability {xm } are equidistributed in the analytic space XC v measure dµv,φ = c1 (L )vn−1 / degL (X). Remark. Following the formulation in Section 5.4, we have equidistribution of small subvarieties over a dynamical system. It will be included in Section 5.7. As in [SUZ], this result gives the equivalence between the dynamical Bogomolov conjecture and the strict equidistribution of small points. Conjecture (Dynamical Bogomolov Conjecture). Let Y be an irreducible closed subvariety of X which is not preperiodic. Then there exists a positive number ǫ > 0, such that the set ˆ L (x) < ǫ} is not Zariski dense in Y . {x ∈ Y (K) : h Remark. The known cases of this conjecture are: the case of multiplicative groups by Zhang [Zh1], the case of abelian varieties proved by Ullmo [Ul] and Zhang [Zh3], and the almost split semi-abelian case proved by Chambert-Loir [Ch2]. The general case without any group structure is widely open. A sequence {xm }m≥1 of algebraic points in X is call strict if no infinite subsequence of {xm } is contained in a proper preperiodic subvariety of X. The strict equidistribution is the following: Conjecture (Dynamical Strict Equidistribution of Small Points). Let {xm } be an infinite sequence of algebraic points of X which is strict and small. Then for any place v of K, the an with respect Galois orbits of the sequence {xm } are equidistributed in the analytic space XC v n−1 to the canonical probability measure dµv,φ = c1 (L )v / degL (X). 38

Corollary 5.8. Dynamical Bogomolov Conjecture ⇐⇒ Dynamical Strict Equidsitribution of Small Points. By a result of Bedford-Taylor and Demailly, the support of the canonical measure is empty or Zariski dense in YCanv for archimedean v. See [Zh4, Theorem 3.1.6] for example. With this result, the corollary is easily implied by Theorem 5.7.

5.6

Equidistribution over Multiplicative Groups

All the above equidistribution theorems are also true for the multiplicative group Gnm , though they are excluded from our definition of dynamics because they are not projective. In fact, the usual embedding to Pn induces a dynamical system over Pn . Over abelian varieties, the Bogomolov conjecture was proved ([Ul], [Zh3]) using the equidistribution theory in [SUZ]. However, in the multiplicative case, the proof of the Bogomolov conjecture ([Zh1]) and the proof of the equidistribution ([Bi]) were independent of each other. And the methods were quite different from the case of abelian varieties. In this subsection, we will give a uniform treatment for multiplicative groups. The reason that Szpiro-Ullmo-Zhang’s method does not work for multiplicative groups is because the canonical metric is not strictly positive. It is not a problem in this paper. Fix a number field K, and an integer n > 0. We are going to work on the space Gnm over K. Embed Gnm in Pn by (z1 , z2 , · · · , zn ) 7→ (1, z1 , z2 , · · · , zn ). For any point x ∈ X(K), the canonical height h(x) of x is the same as the naive height of x in Pn ; i.e., h(x) =

1 X log max{|1|v , |z1 |v , · · · , |zm |v }, [L : K] v

where L is any finite extension of K containing z1 , z2 , · · · , zn , and the summation is over all normalized valuations | · |v of L. Fix an embedding K → C, and | · | always mean the usual absolute value over C. The canonical probability measure dµ of (Gnm )C = (C∗ )n is supported on the unit polycircle S n defined by |z1 | = |z2 | = · · · |zn | = 1, where it coincides with the probability Haar measure of S n . Now we have the following equidistribution theorem. Theorem (Bilu). Let {xm } be an infinite sequence of algebraic points of Gnm which is generic and small. Then the Galois orbits of the sequence {xm } are equidistributed in (C∗ )n with respect to the canonical probability measure dµ. Multiplication by 2 over Gnm extends to a dynamical system φ : Pn → Pn given by (z0 , z1 , · · · , zn ) 7→ (z02 , z12 , · · · , zn2 ). And φ is polarized by L = (O(1), k · k) with metric ks(z0 , z1 , · · · , zn )k =

|s(z0 , z1 , · · · , zn )| , max{|z0 |, |z1 |, · · · , |zn |}

where s(z0 , z1 , · · · , zn ) is represented by a linear homogeneous polynomial and the above definition is independent of the homogeneous coordinate. 39

This metric is the uniform limit of the smooth positive metric ks(z0 , z1 , · · · , zn )kt =

|s(z0 , z1 , · · · , zn )| |t

1

(|z0 + |z1 |t + · · · + |zn |t ) t

,

as t → ∞. And the canonical measure c1 (L )dim Z |Z over any subvariety Z is the weak limit of the smooth measure c1 (O(1), k · kt )dim Z |Z . See [Zh4, Proposition 3.1.5]. One checks that the canonical measure c1 (L )n is exactly the one supported on S n . Then the above theorem is a special case of Theorem 5.7. And we can sketch the expected proof of the following Bogomolov conjecture. Theorem (Zhang). Let Y be an irreducible closed subvariety of Gnm which is not torsion. Then there exists a positive number ǫ > 0, such that the set {x ∈ Y (K) : h(x) < ǫ} is not Zariski dense in Y . We claim that it can be proved by the method of Zhang [Zh3] proving the Bogomolov conjecture over abelian varieties. Note that we still have equidistribution, though our varieties here are not compact. We have a nice compactification as above. Denote X = Gnm . As in [Zh3], after some reduction, the map α : Y m → X m−1 given by −1 (z1 , z2 , · · · , zm ) 7→ (z1 z2−1 , z2 z3−1 , · · · , zm−1 zm )

is a generic embedding when m is large enough. Assuming the theorem is not true, by [Zh3] there is a generic and small sequence {yi} in m Y . Then {µyi } converges weakly to a probability measure µ over Y m (C). The sequence {α(yi)} is generic and small in α(Y m ), and its equidistribution gives a measure µ′ over α(Y m )(C). It turns out that µ = α∗ µ′ . Now let us explain the meaning of this pull-back. We know that X is polarized by the line bundle L we defined above. Thus X m−1 is polarized by M = p∗1 L ⊗ · · · ⊗ p∗m−1 L , where pj : X m−1 → X is the j-th projection. Then µ′ = c1 (M )d |α(Y m )(C) / degM (α(Y m )), where d = dim α(Y m ) = dim Y m . And α∗ µ′ is the canonical probability measure of α∗ M over Y m (C). If {M t }t is a sequence of smooth semipositive metrics which converges uniformly to the metric of M , then α∗ µ′ is the limit of the pull-back differential form α∗ c1 (M t )d |α(Y m )(C) / degM (α(Y m )) as t → ∞. To find a contradiction, we still need a smooth point z ∈ Y (C) such that the canonical measure µ′′ = c1 (L )dim Y |Y (C) / degL (Y ) of Y is nonzero at z. Such a point exists since the support of µ′′ is Zariski dense in Y (C). This is implied by the result of Bedford-Taylor and Demailly we mentioned in the last subsection. See [Zh4, Theorem 3.1.6]. Now everything follows the same way. Let qj : Y m → Y be the j-th projection. Then ∗ ′′ µ = q1∗ µ′′ · · · qm µ as a limit of smooth forms in the sense of [Zh4, Proposition 3.1.5]. The measure µ is nonzero at (z, z, · · · , z) since µ′′ is nonzero at z. The morphism α is singular at (z, z, · · · , z) since it maps the whole diagonal to the identity point (1, 1, · · · , 1). It follows that the pull-back of any differential form by α is zero at (z, z, · · · , z). And thus α∗ µ′ is zero at (z, z, · · · , z) which gives a contradiction. 40

Corollary (Strict Equidistribution of Small Points). Let {xm } be an infinite sequence of algebraic points of Gnm which is generic and strict. Then the Galois orbits of the sequence {xm } are equidistributed in (C∗ )n with respect to the canonical probability measure dµ.

5.7

Equidistribution Theory

Up to now, we have seen many equidistribution theorems. We have repeated some similar statements again and again, but there are still some that we have not mentioned. The aim of this section is to formulate all related equidistribution statements. Of course, most of them are included or implied by some others. There are three parts in an equidistribution statement: (A) An algebraic variety X over a number field K usually with a polarization. There are four cases: an abelian variety, a multiplicative group, an almost split semi-abelian variety, a dynamical system, or just a projective variety polarized by an adelic line bundle satisfying the condition of Theorem 5.1. (B) A small sequence of points or subvarieties of X with one of the following assumptions: generic, or strict. Here we don’t have the notion of a strict sequence for the last case of part (A). When consider small subvarieties, we have to impose in the last case of (A) some condition like: h(Y ) ≥ h(X) for any subvariety Y of X. an and (C) An analytic space that we consider equidistribution. There are two options: XC v an XKv for each place v of K.

It is clear that the above data give 10 generic equidistributions and 8 strict equidistributions. All the generic equidistribution of small points are implied by Theorem 5.1. And every strict equidistribution is equivalent to the Bogomolov conjecture over the same variety. The only unknown case is when we take a dynamical system in part (A). There are similar statements for a general semi-abelian variety. Strictly speaking, a general semi-abelian variety does not give a dynamical system. And the height of the compactification is negative in general (cf. [Ch2]). All the above statements are unknown in this case.

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44