The arithmetic Hodge index theorem for adelic line bundles

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DOI 10.1007/s00208-016-1414-1

Mathematische Annalen

The arithmetic Hodge index theorem for adelic line bundles

Xinyi Yuan¹ · Shou-Wu Zhang²

Received: 24 February 2015 / Revised: 27 March 2016 / Published online: 12 May 2016

Abstract In this paper, we prove index theorems for integrable metrized line bundles on projective varieties over complete fields and number fields respectively. As applications, we prove a non-archimedean analogue of the Calabi theorem and a rigidity theorem about the preperiodic points of algebraic dynamical systems.

Contents

1 Introduction . . . 1124

1.1 Local Hodge index theorem. . . 1124

1.2 Arithmetic Hodge index theorem . . . 1125

2 Local Hodge index theorem . . . 1126

2.1 Statements. . . 1126

2.2 Curves . . . 1128

2.3 Inequality . . . 1129

2.4 Equality . . . 1131

3 Arithmetic Hodge index theorem . . . 1134

3.1 Statements. . . 1134

3.2 Curves . . . 1136

3.3 Inequality . . . 1137

3.4 Equality . . . 1139

4 Algebraic dynamics . . . 1142

4.1 Statement of the main theorem . . . 1142

4.2 Semisimplicity . . . 1143

4.3 Admissible metrics . . . 1145

4.4 Preperiodic points . . . 1151

4.5 Consequences and questions . . . 1153

B

^{Xinyi Yuan}

yxy@math.berkeley.edu 1 Berkeley, USA 2

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Appendix 1 . . . 1155

Local intersections . . . 1155

Arithmetic intersections . . . 1158

Arithmetic positivity . . . 1161

Flat metrics. . . 1164

Lefschetz theorems. . . 1168

References. . . 1170

1 Introduction

The Hodge index theorem for line bundles on arithmetic surfaces proved by Faltings [13] and Hriljac [21] is one of the fundamental results in Arakelov theory. In [29], this index theorem has been generalized by Moriwaki to higher-dimensional arithmetic varieties. In this paper, we will give further generalizations of the index theorem for adelic metrized line bundles in the setting of [40]. These adelic metrics naturally appear in algebraic dynamical systems and moduli spaces of varieties. As applications, we will prove a non-archimedean analogue of the Calabi theorem of [6] and a rigidity theorem about the preperiodic points of algebraic dynamical systems. In more details, our results are explained as follows.

1.1 Local Hodge index theorem

Let K be an algebraic closed field endowed with a nontrivial complete (either archimedean or non-archimedean) absolute value| · |. Let X be a projective variety over K of dimensionn ≥ 1. Denote by X^anthe associated Berkovich analytic space (resp. complex analytic spaces) introduced in [1] ifKis non-archimedean (resp.

archimedean). Then there is a theory of integrable metrized line bundles and their intersection numbers developed in [8,17,19,40]. We will briefly review the theory in

“Appendix (Local intersections)”, and will refer to “Appendix (Local intersections)”

for our convention and terminology.

LetMbe an integrable metrized line bundle onXwithM =OX, andL1, . . . ,Ln−1

ben−1 semipositive metrized line bundles onX. Then we show in Sect.2the following inequality:

M²·L1· · ·Ln−1≤0.

Moreover, ifLi is ample andM isLi-boundedfor eachi, then the equality holds if and only ifM∈π^∗Pic(K ). Hereπ: X→ SpecK denotes the structure morphism.

WhenK =C, the above inequality is written in terms of distributions as

−

X(C) f∂∂¯

πi f ·c1(L1)· · ·c1(Ln−1)≤0, f := −log1_M.

IfXis smooth and the metrics ofMandLiare also smooth, this is a simple consequence of integration by parts.

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In general, for the inequality part, it is easily reduced to the smooth case whenKis archimedean or the model case whenKis non-archimedean. For the equality part, we use a non-archimedean analogue of Błocki’s method in [3], and the works of Gubler [19] and Chambert-Loir–Thuillier [10].

One application of our local Hodge index theorem is a non-archimedean analogue of the theorem of Calabi [6] on the uniqueness of semipositive metrics on an ample line bundle onX^anwith a given volume form.

1.2 Arithmetic Hodge index theorem

LetKbe a number field andXbe a normal and geometrically integral projective variety over K of dimensionn ≥ 1. There is a theory of integrable metrized line bundles and their intersection numbers developed in [40]. We will briefly review the theory in “Appendix (Arithmetic intersections)”, and will refer to “Appendix (Arithmetic intersections)” for our convention and terminology.

Let M be an integrable adelicQ-line bundle on X, and L1, . . . ,Ln−1ben −1 nef adelic Q-line bundles on X. Assume that each Li is big and the equality M · L1· · ·Ln−1=0. Then in Sect.3, we prove the following inequality:

M²·L1· · ·Ln−1≤0.

Moreover, if Li is arithmetically positive andM is Li-bounded for eachi, then the equality holds if and only if M ∈ π^∗Pic(K )_Q. Hereπ : X → SpecK denotes the structure morphism.

WhenX is regular with a projective and flat modelX overOK, and(M,L1, . . . , Ln−1)comes from hermitian line bundles(M,L, . . . ,L)withLarithmetically ample, this result is due to Faltings [13] and Hriljac [21] forn = 1 and Moriwaki [29] for n >1. A slight extension of Moriwaki’s result gives the inequality part in the adelic situation. For the equality part in adelic situation, we will use the local index theorem, the vanishing result of curvatures of flat metrics by Gubler [19] (cf. Proposition5.17), the integration of Green’s functions on Berkovich spaces of Chambert-Loir–Thuillier [10], and Lefschetz-type theorems for normal varieties (cf. “Appendix (Lefschetz theorems)”).

1.2.1 Dynamical system

Let X be a projective variety overQ. Apolarizable algebraic dynamical system on X is a morphism f : X → X such that there is an ampleQ-line bundleLsatisfying f^∗L q Lfrom some rational numberq >1. A central invariant for such a dynamical system is the set Prep(f)ofpreperiodic points, i.e., points ofX(Q)with finite forward orbits under iterations of f. We show the following rigidity result about preperiodic points in Sect.4. Let f andg be two polarized algebraic dynamical systems on X, andZbe the Zariski closure of Prep(f)∩Prep(g). Then

Prep(f)∩Z(Q)=Prep(g)∩Z(Q).

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We prove the result by first constructing f-admissible metrics forevery line bundle on XifX is normal, and then applying our arithmetic index theorem.

2 Local Hodge index theorem

In this section, we prove a local Hodge index theorem over complete fields. As a consequence, we will prove a version of the Calabi theorem. When the base field isC, all of these are essentially due to the original work of Calabi [6] and the later extension by Błocki [3]. This local index theorem is a crucial ingredient in the proof of arithmetic Hodge index theorem in Sect.3.

2.1 Statements

First let us recall the theory of integrable metrized line bundles developed in [8,17, 19,40]. For more details, see “Appendix (Local intersections)”.

Let K be an algebraically closed field endowed with a complete and nontrivial absolute value| · |, which can be either archimedean or non-archimedean. Let X be a projective variety over K of dimensionn ≥ 1, and X^anthe associated Berkovich analytic space (resp. complex analytic space ifK =Cis archimedean). Then there are a categoryPic(X)of integrable metrized line bundlesL =(L, · )onXand the groupPic(X )of isometry classes of these bundles. There are also notions of model metrics and semipositive metrics. In particular, we call a metricsemipositiveif it is a uniform limit of model metrics induced by relatively nef models overOK. We call a metrized line bundlesemipositiveif the metric is semipositive, and call a metrized line bundleintegrableif it is the difference of two semipositive metrized line bundles.

WhenK was a discrete valuation field, these were defined in [40]. For general non- archimedean fields, they were defined by Gubler [17].

IfL0, . . . ,Lnare integrable metrized line bundles with nonzero rational sections 0, . . . , n such that∩|div(i)| = ∅, then there is a well-defined local intersection numberdiv( 0)· · ·div( n)using a limit process. Assume thatL0OX and0=1 under this identity. Then the intersection does not depend on the choice of1, . . . , n. Chambert-Loir [8] (whenKcontains a countable subfield) and Gubler [19] (in general) define a measurec1(L1)· · ·c1(Ln)onX^anso that

−

X^an

log0c1(L1)· · ·c1(Ln)=div( 0)· · ·div( n).

Assuming furtherc1(L1)· · ·c1(Ln)=0, then the local intersection does not depend on the choice of0, neither. This gives a well-defined intersection number

L0· · ·Ln:= −

X^an

log0c1(L1)· · ·c1(Ln).

For two integrable metrized line bundlesLandM onX, say thatM isL-bounded if there is a positive integermsuch that bothm L+Mandm L−Mare semipositive.

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Here we write tensor product of line bundles additively, som L−MmeansL^⊗^m⊗M^∨ (with the induced metric). We take this convention throughout this paper. The main result of this section is the following local index theorem.

Theorem 2.1 (Local Hodge index theorem)Let π : X → SpecK be an integral projective variety of dimension n ≥ 1. Let M be an integrable metrized line bundle on X with M OX, and L1, . . . ,Ln−1be n−1semipositive metrized line bundles on X . Then

M²·L1· · ·Ln−1≤0.

Moreover, if Li is ample and M is Li-bounded for each i , then the equality holds if and only if M ∈π^∗Pic(K ).

One application of our local index theorem is the following non-archimedean analogue of the Calabi theorem. We refer to Calabi [6] for the original theorem and to Kolodziej [23] and Błocki [3] for some regularity extensions.

Corollary 2.2 (Calabi Theorem)Let X be an integral projective variety over K . Let L be an ample line bundle on X , and · 1and · 2be two semipositive metrics on L. Then

c1(L, · 1)^dim^X =c1(L, · 2)^dim^X if and only if · 1

· 2

is a constant function on X^an.

Remark 2.3 There is an open problem about the existence of a metric on a line bundle with a given volume form, i.e., the non-archimedean analogue of the theorem of Yau [37]. Some partial results have been obtained by Liu [26] and Boucksom–Favre–

Jonsson [5].

We will prove Theorem2.1in the next three subsections. Here we see how it implies Corollary2.2.

Proof of Corollary2.2 Writen=dimX. Set f = −log(·1/·2)as a continuous function onX^an. Then the assumption implies

X^an

f c1(L, · 1)ⁿ=

X^an

f c1(L, · 2)ⁿ We need to show that f is constant.

DenoteL1=(L, · 1),L2=(L, · 2), andM =L1−L2. Then the difference of two sides of the above equality is just

n−1

an f c1(M)c1(L1)ⁱc1(L2)ⁿ⁻¹⁻ⁱ =0.

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Equivalently,

n−1

i=0

M²·Lⁱ₁·Lⁿ₂⁻¹⁻ⁱ =0.

By Theorem2.1, every term in the sum is non-positive. It forces M²·Lⁱ₁·Lⁿ₂⁻¹⁻ⁱ =0, ∀i =0, . . . ,n−1.

It follows that

M²·(L1+L2)ⁿ⁻¹=0.

Note thatM is(L1+L2)-bounded. Theorem2.1impliesM ∈ π^∗Pic(K ), which is

equivalent to the statement that f is a constant.

2.2 Curves

In this section, we prove Theorem2.1when dimX =1. Then the line bundlesLi do not show up. Considering the pull-backs of the line bundles on the normalization of X, the problem is reduced to the case thatXis smooth.

First, let us treat the case that K is non-archimedean. The result is obtained by combining many results of Gubler.

We start with the model case; i.e.,(X,M)is induced by an integral model(X,M) over the valuation ring OK of K. By the semistable reduction theorem of Bosch–

Lütkebohmert [4], any integral model ofXoverKis dominated by a semistable model overOK. Thus we only need to consider the caseX is semistable. Then the result is proved by Gubler [18, Theorem 7.17] following the methods of Faltings and Hriljac.

Now we extend it to the non-model case. As the limit case of the model case, we have the inequalityM²≤0 for general vertical metrized line bundles, and a Cauchy–

Schwarz inequality. To prove the equality part of the theorem, assumeM²=0. Then for any vertical integrable line bundleN onX, we have

(M·N)²≤(M²)(N²)=0.

Thus for any vertical integrable line bundleN, M·N =0.

We first prove that the measurec1(M)=0. Note that the generic fibersM andN are isomorphic toOX. Use the regular sections 1M ofMand 1NofNto compute the intersection numberM·N. By the induction formula of Chambert-Loir, Gubler, and Chambert-Loir–Thuillier recalled in “Appendix (Local intersections)”, we have

0=M·N =div(1 M)·div(1 N)

=div(1 N)·div(1 M)= −

X^an

log1Nc1(M).

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Here we have also used the symmetry of the intersection number in [17, Proposition 9.3]. Note that log1Ncan be any model function onXân. The spaceCmod(Xân)of model functions onXânis dense in the spaceC(Xân)of continuous functions onXân with respect to the supremum norm. This is part of [17, Theorem 7.12], which actually works in the current generality. Therefore, the measurec1(M)=0.

It remains to prove that M is constant from the vanishing of the measure. The following argument is suggested by an anonymous referee of a previous version of this paper.

Take two pointsP1,P2∈ X(K). Take N to be the line bundleN =O(P1−P2) with a flat metric as in Definition5.10and Theorem5.11. Denote bythe canonical rational section ofNwith divisorP1−P2. Use the rational sections 1 ofMandofN to compute the intersection numberM·N. Apply the induction formula of Chambert- Loir, Gubler, and Chambert-Loir–Thuillier in “Appendix (Local intersections)” again.

We have

div(1)·div()= −

X^an

log1c1(N)=0, and

div() ·div(1) =log1(P2)−log1(P1)−

X^an

logc1(M)

=log1(P2)−log1(P1).

By the symmetry of the intersection number in [17, Proposition 9.3] again, we have log1(P2)=log1(P1).

This is true for anyP1,P2∈ X(K). Note thatX(K)is dense in X^an. Hence, log1 is constant on X^an. This finishes the proof of Theorem2.1in the non-archimedean case.

The theorem in the archimedean case follows from a similar argument. The theorem is clear in the case of smooth metrics by integration by parts. In fact, denoting f =

−log1_M in the smooth case, we have

M²= −

X(C) f∂∂¯ πi f =

X(C)

1

πi∂f ∧ ¯∂f ≤0

with equality holds only if f is constant. By taking limits, this implies thatM²≤ 0 in the case of integrable metrics. The proof of the remaining part is also similar.

2.3 Inequality

Now we prove the inequality part of Theorem2.1in the general case by induction onn =dimX. We have already treated the casen =1, so we assumen ≥2 in the following discussion. First, we can make the following two assumptions:

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(1) Xis smooth by de Jong’s alteration [11] and the projection formula in [8,17,40];

(2) eachLi is arithmetically positive by adding some multiple of an arithmetically positive integrable line bundleAon X as follows. Take a small rational number >0 and setL_i = Li +A. Then the inequalityM²·L1· · ·Ln−1 ≤0 is the limit of the inequalityM²·L₁· · ·L_n₋₁≤0.

When K = C, by approximation, it suffices to prove the inequality under the assumption that all metrics are smooth, and the curvature forms of Li are positive definite. Let f = −log1_M. Thenc1(M)= −^∂_π^∂^¯_i f and

M²·L1· · ·Ln−1= −

X(C) f∂∂¯

πi f c1(L1)· · ·c1(Ln−1)

=

X(C)

1

πi∂f ∧ ¯∂f ∧c1(L1)· · ·c1(Ln−1)≤0.

WhenKis non-archimedean, by approximation, it is reduced to prove the inequality M²·L1· · ·Ln−1≤0

under the following assumptions:

(1) Xis smooth andXis a normal integral model ofX overOK;

(2) M,L1, . . . ,Ln−1are line bundles onXwith generic fibersM,L1,· · · ,Ln−1; (3) Liis relatively ample onX.

We claim a Bertini-type result that, there is a positive integer m and a nonzero sections∈ H⁰(X,mLn−1)such thatY :=div(s)is horizontal onX with a smooth generic fiberYK.

Assuming the claim, then

M²·L1· · ·Ln−1= 1

mM|²_Y·L1|_Y· · ·Ln−2|_Y. Thus it is non-positive by induction.

It remains to prove the Bertini-type result. We use an argument from [28]. By our definition of models, there is a noetherian local subring(R, ℘)ofOKwithm∩R=℘, wheremdenotes the maximal ideal ofOK, so that(X,M,L1, . . . ,Ln−1)are base changes of models(X⁰,M⁰,L⁰₁, . . . ,L⁰_n₋₁)overR. WriteLforL⁰_n₋₁for simplicity.

We first find a sections0 ∈ H⁰(X⁰,mL)for sufficiently largemsuch that div(s0) is horizontal. In fact, denote byV1, . . . ,Vr the irreducible components of the special fiber ofX⁰. Take a distinct closed pointxiinV_i⁰for everyi, and denote byk(xi)the residue field ofxi, viewed as a skyscraper sheaf onX⁰. By the ampleness ofL, for sufficiently largem, the natural map

H⁰(X⁰,mL)−→

i

H⁰(X⁰, (mL)⊗k(xi))

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is surjective. By lifting an element on the right-hand side which is non-zero at every component, we get a sections0∈H⁰(X⁰,mL)non-vanishing at anyxi. Thus div(s0) is horizontal. Once we haves0, consider the subset

s0 =s0+℘·H⁰(X⁰,mL)⊂H⁰(X⁰,m L).

Here℘ denotes the maximal ideal of R as above, X⁰ denotes the generic fiber of X, and L = L|X⁰. Then any element ofs₀ has a horizontal zero locus. It is easy to check that any polynomial on H⁰(X⁰,m L)which vanishes ons₀ is identically 0. Hence, the image ofs₀ inP(H⁰(X⁰,m L))is Zariski dense. Assume thatm Lis very ample. By Bertini’s theorem, there is a Zariski open subset ofP(H⁰(X⁰,m L)) whose elements correspond to smooth and irreducible hyperplane sections. This open set intersects the image ofs0. Hence, we can find an elements⁰∈s0 with smooth and irreducible zero locus on X⁰. The pullback ofs⁰toX satisfies all the required properties.

The following Cauchy–Schwarz inequality is a direct consequence of the inequality part of Theorem2.1.

Corollary 2.4 Let M and M be two vertical integrable line bundles on X , and L1, . . . ,Ln−1be n−1semipositive line bundles on X . Then

(M·M·L1· · ·Ln−1)²≤(M²·L1· · ·Ln−1)(M²·L1· · ·Ln−1).

2.4 Equality

Now we prove the equality part of Theorem2.1in the general case by induction on n=dimX. The casen=1 is proved, so we assumen≥2.

LetM,L1, . . . ,Ln−1be integrable line bundles onX such that the following conditions hold:

(1) Mis trivial onX;

(2) MisLi-bounded for everyi; (3) M²·L1· · ·Ln−1=0.

There are two key lemmas in our proof. The first one is inspired by Błocki [3].

Lemma 2.5 For any semipositive integrable line bundles L⁰_i on X with underlying bundle L⁰_i equal to Li, and any integrable integrable line bundle M with trivial underlying line bundle M, the following identities hold:

M·M·L⁰₁· · ·L⁰_n₋₁=0, M²·M·L⁰₁· · ·L⁰_n₋₂=0.

Proof The second equality follows from the first one by takingL¯⁰_n₋₁= ¯Ln−1±M¯ with small. So it suffices to prove the first inequality. Note that condition (2) implies

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that every Li is semipositive. ReplacingLi by a large multiple if necessary, we can assume that bothLi±Mare semipositive for eachi. DenoteL^±_i =Li ±M.

First, we have the following equality

M²·L⁽₁¹⁾· · ·L⁽_n₋ⁿ₁⁻¹⁾ =0

for any sign function : {1, . . . ,n−1} → {+,−}. In fact, sinceMis Li-bounded, there is a constantt >0 such that

L^±_{i t} :=Li −t L^±_i =(1−t)Li∓t M is semipositive for anyi. Write

M²·L1· · ·Ln−1=M²·t L⁽₁¹⁾· · ·t L⁽_n₋ⁿ₁⁻¹⁾+

n−1

=1

M²·

i≤

L⁽_{i t}ⁱ⁾·

j>

L⁽_i ⁱ⁾.

Then the inequality part of Theorem2.1implies that every term on the right hand side is non-positive. It forces

M²·L⁽₁¹⁾· · ·L⁽_n₋ⁿ₁⁻¹⁾=0.

Second, we claim that for anyand any sign functionon{, . . . ,n−1}, M²·L⁰₁· · ·L⁰₋₁·L⁽⁾ · · ·L⁽_n₋ⁿ₁⁻¹⁾=0. (1) When=n, it gives

M²·L⁰₁· · ·L⁰_n₋₁=0.

By the Cauchy–Schwarz inequality in Corollary2.4, it follows that M·M·L⁰₁· · ·L⁰_n₋₁=0.

We now prove the claim by induction on . We already have the case = 1.

Assume the equality (1) forand for all sign functions on{, . . . ,n−1}. Letbe a sign function on{+1, . . . ,n−1}which has two extensions⁺, ⁻on{, . . . ,n−1} by^±()= ±. Applying Corollary2.4again toM,M^± =L^± −L⁰, and semipositive bundlesL¯⁰₁, . . . ,L¯⁰₋₁,L¯^±⁽⁾, . . . ,L¯⁽_n₋ⁿ₁⁻¹⁾, we have four equalities:

M·M^δ¹ ·L⁰₁· · ·L⁰₋₁·L^δ²· · ·L⁽_n₋ⁿ₁⁻¹⁾=0, δ1= ±, δ2= ±.

Take the difference for two equations with fixedδ1to obtain M·M^δ¹·L⁰₁· · ·L⁰₋₁·M·L⁽⁺₊₁¹⁾· · ·L⁽_n₋ⁿ₁⁻¹⁾=0.

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It is just

M²·L⁰₁· · ·L⁰₋₁·(L^δ¹ −L⁰)·L⁽⁺₊₁¹⁾· · ·L⁽_n₋ⁿ⁻₁¹⁾=0.

The left-hand side splits to a difference of two terms. One term is zero by the induction assumption for. It follows that the other term is also zero, which gives

M²·L⁰₁· · ·L⁰₋₁·L⁰·L⁽⁺₊₁¹⁾· · ·L⁽_n₋ⁿ₁⁻¹⁾=0.

It is exactly the case+1. The proof is complete.

The second key lemma is the following induction result.

Lemma 2.6 Assume that Ln−1is ample. For any closed subvariety Y of codimension one in X , we have

M|²Y ·L1|Y· · ·Ln−2|Y =0.

Proof ReplacingLn−1by a multiple if necessary, we can assume that there is a global sections of Ln−1vanishing onY. Write div(s) =

iaiYi withai > 0 and prime divisorsYi. Use the non-archimedean induction formula developed by Chambert-Loir–

Thuillier and Gubler reviewed in “Appendix (Local intersections)”. We have

M²·L1· · ·Ln−1= t

i=1

aiM|²Y_i ·L1|Y_i · · ·Ln−2|Y_i

−

X^an

logsc1(M)²c1(L1)· · ·c1(Ln−2).

The integral vanishes by Lemma2.5. In fact, takeMto be any vertical metrized line bundle onX and writeφ= −log1_M as a real-valued function on X^an. Then the lemma gives

X^an

φc1(M)²c1(L1)· · ·c1(Ln−2)=0.

It implies that the measurec1(M)²c1(L1)· · ·c1(Ln−2)=0. Hence the integration of

−logsis also zero. By the proved inequality on everyYi, we have M|²Y_i ·L1|Y_i· · ·Ln−2|Y_i =0.

Since one ofYi isY, the lemma is proved.

Now are ready to prove the equality part of Theorem2.1. LetMandLibe as in the theorem. We need to prove thatM is constant, or equivalently log1_M is constant onX^an. By induction onn, Lemma2.6immediately gives the following result.

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Lemma 2.7 For any integral closed subvariety Y of codimension one in X , the restric- tion M|Y is constant on Y .

Go back to the line bundleMonX. Let|X|0be the set of closed points ofX, which is naturally a dense subset ofX^an. It suffices to prove that log1_Mis constant on|X|0. By the lemma, it suffices to prove that any two pointsx1,x2of|X|0 are connected by integral subvarieties of codimension one inX. By [20, Chapter III, Corollary 7.9], any hyperplane section is connected. It follows that any hyperplane section passing throughx1andx2connects them. This finishes the proof.

3 Arithmetic Hodge index theorem

In this section, we prove a Hodge index theorem for adelic metrized line bundles on projective varieties over number fields, generalizing the following theorem in the context of arithmetic intersection theory of Gillet–Soulé [15].

Theorem 3.1 ([13,21,29])Let K be a number field andπ : X → SpecOK be an arithmetic variety with a regular generic fiber of dimension n ≥ 1. Let L be an arithmetically ample hermitianQ-line bundle onX. LetMbe a hermitianQ-line bundle onXsuch that c1(MK)·c1(LK)ⁿ⁻¹=0on the generic fiberXK. Then

c1(M)²·c1(L)ⁿ⁻¹≤0,

and the equality holds if and only ifM=π^∗M0for some hermitianQ-line bundle M0on SpecOK.

The theorem was due to Faltings [13] and Hriljac [21] forn = 1 and Moriwaki [29] for generaln.

3.1 Statements

LetX a projective variety overQ¯. In [40] and as reviewed in “Appendix (Arithmetic intersections)”, we have defined a categoryPic(X)of integrable metrized line bundles as certain limits of hermitian line bundles on models ofXoverOK, the ring of integers of number fieldsKinQ, and their intersection pairing. Then¯ Pic(X) denotes the group of isomorphism classes of objects ofPic(X), andPic(X )_Q =Pic(X )⊗_ZQis the group of integrable adelicQ-line bundles. We also refer to “Appendix (Arithmetic intersections)” for the notions “arithmetically positive” and “Li-bounded.” The main result of this section is the following Hodge index theorem for such metrized line bundles.

Theorem 3.2 Letπ : X → SpecQ¯ be a normal and integral projective variety of dimension n≥1. Let M be an integrable adelicQ-line bundle on X , and L1, . . . ,Ln−1

be n−1nef adelicQ-line bundles on X . Assume M·L1· · ·Ln−1=0and that each Li is big. Then

M²·L1· · ·Ln−1≤0.

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Moreover, if Liis arithmetically positive and M is Li-bounded for each i , then the equality holds if and only if

M ∈π^∗Pic( Q)¯ _Q.

Remark 3.3 The assumption “M isLi-bounded for eachi” is necessary for the condition of the equality in Theorem3.2. For a counter-example, assumen =2 andL¯ is induced by an arithmetically ample line bundleLon an integral modelX of X over OK, the ring of integers in a number field. Letα: X → X be the blowing-up of a closed point inX, andMthe (vertical) line bundle onXassociated to the exceptional divisor endowed with the trivial hermitian metric1 =1. ThenMK ·LK =0 and M²·α^∗L=0 by the projection formula. ButMis not coming from any line bundle on the base SpecOK. In this case,Mis notL-bounded if we convert the objects to the adelic setting.

Remark 3.4 A similar proof of Theorem3.2should give a function field analogue. In fact, letKbe an algebraic closure of the function field of a projective and smooth curve B over an algebraically closed fieldk. Fixing a point x ∈ X(K)gives a morphism i :X →alb(X)to the Albanese variety ofXoverK. See discussion in Sect. A.4. Let Abe aK/k-trace of alb(X), i.e,Ais an abelian variety defined overksuch thatAK

can be embedded into alb(X)as a maximal abelian subvariety defined overk. Then we have a surjective homomorphism alb(X)→ A⊗k K which induces a morphism j : X → AK over K. Then the theorem should hold overK with the last inclusion M ∈π^∗Pic(K )_Qreplaced by

M ∈π^∗Pic(K )_Q+ j^∗Pic⁰(A)_Q, where j^∗denotes the following natural composition

Pic⁰(A)_Q −→Pic(A)_Q−→Pic(A×k B)_Q−→Pic(A K)_Q −→Pic(X )_Q. The second arrow is the pull-back via the projection A×k B → Ato the first factor, and the last arrow is the pull-back via j : X→ AK.

Denote by Pic^τ(X)the group of isomorphism classes of numerically trivial line bundles onX. Assume thatMis aflatadelic line bundle onXin the sense of “Appendix (Flat metrics)”, i.e.,Mis numerically trivial and the adelic metric is flat at every place.

Then by Proposition5.19,M²·L1· · ·Ln−1does not depend on the choice of the flat metric ofM and the metrics ofL1, . . . ,Ln−1. Then it is reasonable to write

M,ML₁,...,L_n₋₁ =M²·L1· · ·Ln−1.

The definition extends to a quadratic form on M ∈ Pic^τ(X)_Rby bilinearity. In this case, we present anR-version of the theorem as follows.

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Theorem 3.5 Let X be a normal projective variety of dimension n ≥1overQ. Let¯ M ∈Pic^τ(X)_Rand L1, . . . ,Ln−1be n−1nefQ-line bundles on X . Then

M,ML₁,...,L_n₋₁ ≤0.

Moreover, if each Li is ample, then the equality holds if and only if M=0.

The inequality part of Theorem3.5is implied by that of Theorem3.2. In fact, it suffices to assumeM ∈Pic^τ(X)_Qfor the inequality.

To apply Theorem3.2, we need eachLi to be big. Thereafter, take an ample line bundle Aand a positive rational number. Define

Li, =Li+A.

LetL1,, . . . ,Ln−1,be any nef adelic line bundles extendingL1,, . . . ,Ln−1,. Then we have

M,ML1,...,Ln−1 =lim

→0M,ML1,,...,Ln−1, =lim

→0M²·L1,· · ·Ln−1, ≤0.

3.2 Curves

In this section, we prove Theorems3.2and3.5when dimX =1. Then degM =0 and the line bundlesLido not show up.

We first consider Theorem3.2. ThenM has a flat metric structure M0as defined in “Appendix (Flat metrics)”. By Proposition5.17(2)(3),M0is actually induced by a hermitian line bundle M on any regular integral model X/OK of X such that c1(M) = 0 onX(C)and thatMis perpendicular toPic(X )vert. Define a vertical adelicQ-line bundleN by

M =M0+N. NoteM0·N =0 by the flatness ofM0. It follows that

M²=M²₀+N²=M²₀+

v

N²_v,

whereN_vbe the restriction of NtoX⊗K K_vfor each placevof K. Now Theorems 3.2and3.5follow from the index theorem of Faltings and Hriljac forM0and the local index theorem in Theorem2.1forN_vas follows:

(1) By the result Faltings and Hriljac,M²₀=M²≤0 where the equality is attained if and only ifM0∈π^∗Pic(K )_Q.

(2) By Theorem2.1,N²≤0 and the equality is attained if and only ifN_vis constant for everyv.

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Now we consider Theorem 3.5. Then M ∈ Pic⁰(X)_R. It still has a flat metric structureM0, as anR-linear combinations of flat adelic line bundles. By linearity, the result of Faltings and Hriljac still gives

M²₀= −2hNT(M).

Here the Neron–Tate height functionhNT:Pic⁰(X)_R→Ris extended from Pic⁰(X) by bilinearity. By [34, §3.8],hNT is positive definite on Pic⁰(X)_R. This proves the theorem forn=1.

3.3 Inequality

We already known that the inequality part of Theorem3.5is implied by that of Theorem 3.2. Now we prove the inequality of Theorem3.2by induction onn =dimX. We have already treated the casen =1, so we assumen≥2 in the following discussion.

We make the following two further assumptions:

(1) X is smooth by the resolution of singularities of Hironaka and the projection formula in [40];

(2) eachLi is arithmetically positive in the sense of Definition5.3.

For the second assumption, fix an arithmetically positive adelic line bundle Aon X. Take a small rational number >0. SetL_i =Li+AandM=M +δA. Here δis a number such that

M·L₁· · ·L_n₋₁=(M+δA)·L₁· · ·L_n₋₁=0. It determines

δ= −M·L₁· · ·L_n₋₁ A·L₁· · ·L_n₋₁. As→0, we haveδ→0 since

M·L₁· · ·L_n₋₁→M ·L1· · ·Ln−1=0, A·L₁· · ·L_n₋₁→ A·L1· · ·Ln−1>0.

Here we explain the last inequality, which uses the assumption thatLi is big and nef for eachi. We refer to [25] for bigness and nefness in the geometric setting. In particular, the bigness ofLi implies that there are ampleQ-line bundlesAi such that Li −Ai is effective. It follows that

A·L1·L2· · ·Ln−1≥ A·A1·L2· · ·Ln−1≥ · · · ≥ A·A1· · ·An−1>0.

Therefore, the inequalityM²·L1· · ·Ln−1≤0 is the limit of the inequalityM²· L₁· · ·L_n₋₁ ≤ 0. Here every L_i is arithmetically positive. This shows that we may assume (2).

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By approximation, it suffices to prove

M²·L1· · ·Ln−1≤0

for a model (X,M,L1, . . . ,Ln−1)of (X,M,L1, . . . ,Ln−1)under the following assumptions:

(3) Xis smooth andXis normal;

(4) Liarithmetically ample onX in the sense of Lemma5.4.

Then the inequality was proved by Moriwaki [29] in the case that allLiare equal. The current case is similar, but we still sketch a proof as follows.

Lemma 3.6 There is a smooth metric · 0onMat archimedean places, unique up to scalars, such that the curvature form ofM=(M, · 0)on X(C)pointwise satisfies

c1(M)c1(L1)· · ·c1(Ln−1)=0,

where we omit the symbol for the wedge product of forms of degree2. Moreover, with this metric, one has pointwise

c1(M)²c1(L1)· · ·c1(Ln−2)≤0.

Proof For the first equality, writeM=(M,·)and·0=e^φ·for a real-valued C^∞-functionφonX(C). Then the above equation becomes

1

πi∂∂φ·c1(L1)· · ·c1(Ln−1)= −c1(M)c1(L1)· · ·c1(Ln−1).

Set = c1(L1)· · ·c1(Ln−1). Then the left-hand side is a scalar multiple of the Laplacianφwith respect to. The right-hand side is an exact form on every connected component X_σ(C)of X(C). In fact, its cohomology class is represented by

−M·L1· · ·Ln−1, which is zero in H²ⁿ(X_σ(C),C)=C. The solutionφexists by Gromov [14, Corollary 2.2 A2].

The inequality of the lemma is implied by the equality by Aleksandrov’s lemma. See Gromov [14, Lemma 2.1 A] by setting:=c1(L1)· · ·c1(Ln−2)andω0:=c1(Ln−1).

Write the original metric · ofMase^−φ · 0. Then by the above lemma, we have

M²·L1· · ·Ln−1=M²·L1· · ·Ln−1−

X(C)φ 1

πi∂∂φ∧c1(L1)· · ·c1(Ln−1).

By integration by parts, the second term on the right

−

X(C)φ 1

πi∂∂φ∧c1(L1)· · ·c1(Ln−1)=

X(C)

1

πi∂φ∧∂φ∧c1(L1)· · ·c1(Ln−1).

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It is non-positive since the volume form 1

πi∂φ∧∂φ∧c1(L1)· · ·c1(Ln−1)≤0. Hence, it suffices to prove

M²·L1· · ·Ln−1≤0.

By Moriwaki’s arithmetic Bertini theorem (cf. [28], Theorem 4.2 and Theorem 5.3), which is also stated at the beginning to [29, §1], replacingLn−1by a multiple if necessary, there is a nonzero section s ∈ H⁰(X,Ln−1)satisfying the following conditions:

• The supremum normssup=sup_x_∈_X_(C)s(x)<1;

• The horizontal part of div(s)onX is a generically smooth arithmetic varietyY;

• The vertical part of div(s)onX is a positive linear combination

℘a_℘X℘ of smooth fibersX℘ofX above (good) prime ideals℘ofOK.

Then

M²·L1· · ·Ln−1=M|²_Y ·L1|_Y· · ·Ln−2|_Y +

℘

a_℘M|²_X_℘·L1|_X_℘· · ·Ln−2|_X_℘

−

X(C)logsc1(M)²c1(L1)· · ·c1(Ln−2).

It suffices to prove each term on the right-hand side is non-positive. The non positivity of the “main term”

M|²_Y·L1|_Y· · ·Ln−2|_Y ≤0 follows from the induction hypothesis. By flatness ofX/OK,

M|²_X_℘·L1|_X_℘· · ·Ln−2|_X_℘ =(M²·L1· · ·Ln−2)log(#OK/℘)≤0.

Here we have used the Hodge index theorem in the geometric case as stated in Theorem 5.20. By the above lemma,

−

X(C)logsc1(M)²c1(L1)· · ·c1(Ln−2)≤0.

3.4 Equality

In this section, we prove the equality part of Theorem3.2and Theorem3.5. We have already treated the casen=1, so we assumen≥2 in the following discussion.