Local heights of toric varieties over non-archimedean fields
aa r X i v : . [ m a t h . N T ] J a n LOCAL HEIGHTS OF TORIC VARIETIES OVERNON-ARCHIMEDEAN FIELDS
WALTER GUBLER AND JULIUS HERTEL
Abstract.
We generalize results about local heights previously proved in thecase of discrete absolute values to arbitrary non-archimedean absolute values.First, this is done for the induction formula of Chambert-Loir and Thuillier.Then we prove the formula of Burgos–Philippon–Sombra for the toric localheight of a proper normal toric variety in this more general setting. We applythe corresponding formula for Moriwaki’s global heights over a finitely gener-ated field to a fibration which is generically toric. We illustrate the last resultin a natural example where non-discrete non-archimedean absolute values re-ally matter.
MSC2010: 14M25 , 14G40, 14G22.
Contents
Introduction 1Acknowledgements 6Terminology 61. Local heights 61.1. Analytic and formal geometry 61.2. Metrics, local heights and measures 81.3. Semipositivity 111.4. Induction formula for DSP local heights 132. Metrics and local heights of toric varieties 182.1. Toric varieties 182.2. Toric schemes over valuation rings of rank one 222.3. Toric Cartier divisors on toric schemes 252.4. Metrized line bundles on toric varieties 292.5. Semipositive metrics and measures on toric varieties 312.6. Local heights of toric varieties 353. Global heights of varieties over finitely generated fields 363.1. Global heights of varieties over an M -field 363.2. Relative varieties over a global field 413.3. Global heights of toric varieties over finitely generated fields 513.4. Heights of projectively embedded toric varieties over the function fieldof an elliptic curve 53Appendix A. Convex geometry 55Appendix B. Strictly semistable models 58References 65 Introduction
The height of an algebraic point of a proper variety X over a number field F measures the arithmetic complexity of its coordinates. It is a tool to control the number and distribution of these points which is essential for proving finitenessresults in diophantine geometry like the theorems of Mordell–Weil and Faltings (see,for example, [5]). More generally, there is a height of (sub-)varieties which mightbe seen as an arithmetic analogue of the degree used in algebraic geometry. In[19], Faltings made this precise writing the height of X with respect to a hermitianline bunde L as an arithmetic degree using the arithmetic intersection theory ofGillet–Soulé [22].In the adelic language introduced by Zhang [44], a hermitian line bundle can beseen as a line bundle L over X endowed with a smooth metric at every archimedeanplace and with a metric induced by a global model of the line bundle for everynon-archimedean place of F . This flexible point of view allows to consider moregenerally semipositive (continuous) metrics which are obtained from uniform limitsof semipositive hermitian metrics or even DSP metrics which are differences of thesesemipositive continuous metrics. A remarkable application of Zhang’s heights is hisproof of the Bogomolov conjecture for abelian varieties over a number field in [45].Following Weil and Néron, it is more convenient to define the height as a sumof local heights. Here, “local” means that we consider the contribution of a fixedplace and work over the corresponding completion. Local heights of subvarietiescan be studied for any field with a given absolute value which was systematicallydone in [24], [25], [27]. By base change, we may assume that our base field is analgebraically closed field K endowed with a non-trivial complete absolute value.The local height λ ( L ,s ) ,..., ( L n ,s n ) ( X ) of the n -dimensional proper variety X over K with respect to DSP-metrized line bundle L , . . . , L n depends also on the choiceof non-zero meromorphic sections s j of L j for j = 1 , . . . , n and is a well-defined realnumber under the assumption that | div( s ) | ∩ · · · ∩ | div( s n ) | = ∅ . (0.1)In [14], Chambert-Loir introduced a measure c ( L ) ∧ . . . ∧ c ( L n ) on the analyti-fication X an . It plays an important role for equidistribution theorems. For detailsabout the theory of local heights and Chambert-Loir measures, we refer to Section1. The main result in Section 1 is the following induction formula which generalizesa result of Chambert-Loir and Thuillier [15, Théorème 4.1]. Theorem I (Induction formula) . Under the hypotheses above, the function log k s k is integrable with respect to c ( L ) ∧ . . . ∧ c ( L n ) and we have λ ( L ,s ) ,..., ( L n ,s n ) ( X ) = λ ( L ,s ) ,..., ( L n ,s n ) (cyc( s )) − Z X an log k s k c ( L ) ∧ . . . ∧ c ( L n ) where cyc( s ) is the Weil divisor associated to s . In fact, we will show in Theorem 1.4.3 a more general result involving pseudo-divisors. Chambert-Loir and Thuillier proved the formula under the additionalassumptions that K is a completion of a number field and s , . . . , s n are globalsections such that their Cartier divisors intersect properly. The heart of the proofis an approximation theorem saying that log k s k can be approximated by suitablefunctions log k k n , where k · k n are formal metrics on the trivial bundle O X . Toshow this over any non-archimedean field K , we use techniques from analytic andformal geometry.In Section 2, we deal with local heights of toric varieties. Toric varieties are a spe-cial class of varieties that have a nice description through combinatorial data fromconvex geometry. So they are well-suited for testing conjectures and for computa-tions in algebraic geometry. Let K be any field, then a complete fan Σ of stronglyconvex rational polyhedral cones in a vector space N R ≃ R n corresponds to a proper OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 3 toric variety X Σ over K with torus T ≃ Spec K [ x ± , . . . , x ± n ]. The torus T actson X Σ and hence every toric object should have a certain invariance property withrespect to this action in order to describe it in terms of convex geometry. We recallthe classical theory of toric varieties in § 2.1.A support function on Σ, i. e. a concave function Ψ : N R → R which is linear oneach cone of Σ and has integral slopes, corresponds to a base-point-free toric linebundle L on X Σ together with a toric section s . Moreover, one can associate to Ψa polytope ∆ Ψ = { m ∈ M R | m ≥ Ψ } in the dual space M R of N R . Then a famousresult in classical toric geometry is the degree formula:deg L ( X Σ ) = n ! vol M (∆ Ψ ) , where vol M is the Haar measure on M R such that the underlying lattice M ≃ Z n has covolume one. As mentioned above, the arithmetic analogue of the degree of avariety with respect to a line bundle is the height of a variety. Burgos, Philipponand Sombra proved in the monograph [11] a similar formula for the toric localheight. In the non-archimedean case, they assume that the valuation is discrete. Itis the goal of Section 2 to remove this hypothesis. The problem is that the valuationring is not noetherian and hence the usual methods from algebraic geometry do notapply.Let K be a field endowed with a non-trivial non-archimedean complete absolutevalue and value group Γ as a subgroup of R . Let X Σ be the proper toric varietyover K associated to the complete fan Σ and let L be a toric line bundle on X Σ together with any toric section s . Let Ψ be the corresponding support functionand let ∆ Ψ be the dual polytope. In § 2.2, we recall the theory of toric schemesover the valuation ring K ◦ given in [32]. In particular, a strongly convex Γ-rationalpolyhedral complex Π induces a normal toric scheme X Π over K ◦ . Assuming thatthe recession cones in Π give the fan Σ, the toric scheme X Π is a K ◦ -model of X Σ .Generalizing the program from [11], we describe toric Cartier divisors on X Π interms of piecewise affine functions on Π (see § 2.3). To describe local heights, wemay additionally assume that K is algebraically closed. A continuous metric k · k on L is toric if the function p
7→ k s ( p ) k is invariant under the action of the formaltorus in T an . We will give the following classification of toric metrics: Theorem II.
Let Ψ be a support function on Σ . Then there is a bijective corre-spondence between the sets of(i) semipositive toric metrics on L ;(ii) concave functions ψ on N R such that the function | ψ − Ψ | is bounded;(iii) continuous concave functions ϑ on ∆ Ψ . For the first bijection, one associates to the toric metric k · k the function ψ on N R given by ψ ( u ) = log k s ◦ trop − ( u ) k , where trop: N R → T an is the tropicaliza-tion map from tropical geometry (see 2.4.5). The second bijection is given by theLegendre-Fenchel dual from convex analysis (see A.7).This theorem was proven by Burgos, Philippon and Sombra in [11] if the absolutevalue on K is discrete or archimedean. We will prove our generalization in Theorem2.5.8. Essential for the proof are characterizations of semipositive formal metricsdeveloped in [33] and some new results for strictly semistable formal schemes shownin Appendix B. Note that the concave function ψ = Ψ defines a canonical metricon L which will be used later.In Theorem 2.5.10, we will show that the measure c ( L ) ∧ n , induced by a semi-positive toric metrized line bundle L = ( L, k · k ) on the n -dimensional proper toricvariety X Σ , satisfies the following formulatrop ∗ (cid:0) c ( L ) ∧ n (cid:1) = n ! M M ( ψ ) , WALTER GUBLER AND JULIUS HERTEL where ψ is the concave function given by k · k and M M ( ψ ) is the Monge-Ampèremeasure of ψ (see A.17).Now all ingredients from the program in [11] are generalized to show the formulafor the local toric height. Let X Σ be an n -dimensional projective toric variety over K and L a semipositive toric metrized line bundle. Denote by L can the same linebundle equipped with the canonical metric. The toric local height of X Σ withrespect to L is defined as λ tor L ( X Σ ) = λ ( L,s ) ,..., ( L,s n ) ( X Σ ) − λ ( L can ,s ) ,..., ( L can ,s n ) ( X Σ ) , where s , . . . , s n are any invertible meromorphic sections of L satisfying the inter-section condition (0.1). Theorem III.
Using the above notation, we have λ tor L ( X Σ ) = ( n + 1)! Z ∆ Ψ ϑ dvol M , where ϑ : ∆ Ψ → R is the concave function associated to ( L, s ) given by Theorem II. A slightly more general version for a proper toric variety X Σ will be shown inTheorem 2.6.6. The proof is analogous to [11]. It is based on induction relative to n and uses the induction formula (Theorem I) in an essential way.In Section 3, we return to global heights. In [24], the notion of an M -field wasintroduced to capture all situations where global heights occur. This is a field K together with a measure space ( M, µ ) of absolute values on K . The standardexamples are number fields and function fields, but the notion of M -field may bealso used to study the characteristic function in Nevanlinna theory which is ananalogue of a global height according to Osgood and Vojta.Let us consider a projective n -dimensional variety X over an M -field K satisfyingthe product formula and a line bundle L on X . For v ∈ M , we write X v and L v for the base change to the completion C v of the algebraic closure of K . A DSP M -metric on L is a family of DSP metrics k · k v on L v , v ∈ M . Write L = ( L, ( k · k v ) v )and L v = ( L v , k · k v ) for each v ∈ M . We consider now DSP M -metrized linebundles L , . . . , L n on X assuming that the local heights M −→ R , v λ ( L ,v ,s ) ,..., ( L n,v ,s n ) ( X v )are µ -integrable for any choice of non-zero meromorphic sections s , . . . , s n of L which satisfy condition (0.1). Then the (global) height of X is defined ash L ,...,L n ( X ) = Z M λ ( L ,v ,s ) ,..., ( L n,v ,s n ) ( X v ) d µ ( v ) . By the product formula, this definition is independent of the choice of sections.Using integration over all places, we give in Theorem 3.1.13 a global version ofthe induction formula. In § 3.1, the theory of global height is presented moregenerally for proper varieties over an M -field. In the classical case of number fieldsor function fields, the above integrability condition is always satisfied if the metrizedline bundles are quasi-algebraic. The latter means for L that up to finitely manyplaces v ∈ M , the metrics k · k v are induced by a global model of L . In Theorem3.1.13, we give the induction formula for global heights.In [40], Moriwaki defined the global height of a variety over a finitely generatedfield K over Q as an arithmetic intersection number and generalized the Bogomolovconjecture to such fields. As observed in [27, Example 11.22], this finitely generatedextension has a M -field structure for a natural set of places M related to the normalvariety B with K = Q ( B ). Burgos–Philippon–Sombra proved in [12, Theorem 2.4]that the height of Moriwaki can be written as an integral of local heights over M .In Section 3, we will generalize their result as follows. OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 5
Let B be a b -dimensional normal projective variety over a global field F . Wedenote by K the function field of B which is a finitely generated extension of F .We define M := B (1) ⊔ G v ∈ M F B gen v , where B (1) is the set of discrete absolute values corresponding to the orders inthe prime divisors of B and where B gen v is the set of absolute values induced byevaluating at generic points of the analytification of B with respect to the place v of F . Note that the elements in B gen v may lead to non-discrete valuations.Choosing quasi-algebraic metrized line bundles H , . . . , H b on B , we can equip K with a natural structure ( M , µ ) of an M -field satisfying the product formula.Here, we assume that the line bundles H j are all nef which means that all themetrics are semipositive and that the height of every algebraic point of B is non-negative. The measure µ on M is given by the counting measure on B (1) andby µ ( v ) c ( H , k · k v ) ∧ . . . ∧ c ( H b , k · k v ) on B gen v where µ ( v ) is the weight of theproduct formula for F in v . For more details, we refer to § 3.2.Let π : X → B be a dominant morphism of projective varieties over F of relativedimension n and denote by X the generic fiber of π . Let L , . . . , L n be semipositivequasi-algebraic line bundles on X and choose any invertible meromorphic sections s , . . . , s n of L , . . . , L n respectively, which satisfy (0.1). These line bundles induce M -metrized line bundles L , . . . , L n on X . We prove in Theorem 3.2.6: Theorem IV.
The function M → R , w λ ( L ,w ,s ) ,..., ( L n,w ,s n ) ( X ) , is µ -integrableand we have h π ∗ H ,...,π ∗ H b , L ,..., L n ( X ) = Z M λ ( L ,w ,s ) ,..., ( L n,w ,s n ) ( X ) d µ ( w ) . We will prove a more general version of this result in Theorem 3.2.6 where wealso allow proper varieties. Burgos–Philippon–Sombra have shown this formulain the case when F = Q and the varieties X , B and the occurring metrized linebundles are induced by models. Then the measure µ has support in the subset of M given by the archimedean and discrete absolute values. The main difficulty intheir proof appears at the archimedean place, where well-known techniques fromanalysis as Ehresmann’s fibration theorem are used. In our proof, the archimedeanpart and the part on B (1) follow from their arguments, but the contribution of B gen v for non-archimedean v is much more complicated as the support of µ can alsocontain non-discrete absolute values.In § 3.3, we will give the following application of the formula in Theorem III.This was suggested to us by José Burgos Gil. In the setting of Theorem IV, let π : X → B be a dominant morphism of projective varieties over a global field F suchthat its generic fiber X is an n -dimensional normal toric variety over the functionfield K = F ( B ). This field is equipped with the M -field structure induced by themetrized line bundles H , . . . , H b . Assume that L = · · · = L n = L and that theinduced semipositive M -metrized line bundle L is toric. Let s be any toric sectionof L and Ψ the associated support function. Then L defines, for each w ∈ M , aconcave function ϑ w : ∆ Ψ → R . Combining theorems III and IV, we will obtain inCorollary 3.3.4 the formulah π ∗ H ,...,π ∗ H b , L ,..., L ( X ) = ( n + 1)! Z M Z ∆ Ψ ϑ w ( x ) dvol( x ) d µ ( w ) . By means of this formula we can compute the height of a non-toric variety comingfrom a fibration with toric generic fiber. It generalizes Corollary 3.1 in [12] wherethe global field is Q and the metrized line bundles are induced by models over Z and hence only archimedean and discrete non-archimedean places occur. WALTER GUBLER AND JULIUS HERTEL
In § 3.4, we will illustrate the formula in a special case with B an elliptic curveover the global field F . Then the canonical metric on an ample line bundle H = H of B leads to a natural example where non-discrete non-archimedean absolute valuesreally matter. Acknowledgements.
This is an extended version of the second author’s thesis.We are very grateful to José Burgos Gil for suggesting us the application in § 3.3and for his comments. We thank Jascha Smacka for proofreading and the refereefor his helpful comments. This research was supported by the DFG grant: SFB1085 “Higher invariants”.
Terminology.
For the inclusion A ⊂ B of sets, A may be equal to B . Thecomplement is denoted by B \ A . A disjoint union is denoted by A ⊔ B . A measureis a signed measure, i. e. it is not necessarily a positive measure.The set N of natural numbers contains zero. All occuring rings and algebras arecommutative with unity. For a ring R , the group of units is denoted by R × .A variety over a field k is an irreducible and reduced scheme which is separatedand of finite type over k . The function field of a variety X over k is denoted by k ( X ). For a proper scheme Y over a field, we denote by Y ( n ) the set of subvarietiesof codimension n . A prime cycle on Y is just a closed subvariety of Y .By a line bundle we mean a locally free sheaf of rank one. For an invertiblemeromorphic section s of a line bundle, we denote by div( s ) the associated Cartierdivisor and by cyc( s ) the associated Weil divisor. The support of div( s ) is denotedby | div( s ) | .A non-archimedean field is a field K which is complete with respect to a non-trivial non-archimedean absolute value |·| . Its valuation ring is denoted by K ◦ withvaluation val := − log | · | , value group Γ := val( K × ) and residue field ˜ K := K ◦ /K ◦◦ ,where K ◦◦ is the maximal ideal of K ◦ .For the notations used from convex geometry, we refer to Appendix A.1. Local heights
In this section, we recall foundational notions and results for this work. In thefirst subsection, we collect results about Berkovich spaces and admissible formalschemes in the sense of Raynaud. In the next subsection, we will introduce formaland algebraic models of proper algebraic varieties and their line bundles. The asso-ciated formal and algebraic metrics in the sense of Zhang lead to local heights andChambert–Loir measures. This is generalized in § 1.3 to semipositive continuousmetrics on line bundles. The new results in this section are in § 1.4 where the in-duction formula for local heights of Chambert-Loir and Thuillier is generalized toarbitrary non-archimedean fields.1.1.
Analytic and formal geometry.
Let K be a non-archimedean field , i. e. afield which is complete with respect to a non-trivial non-archimedean absolute value | · | . Its valuation ring is denoted by K ◦ , the associated maximal ideal by K ◦◦ andthe residue field by ˜ K = K ◦ /K ◦◦ .In this subsection we recall some facts about the (Berkovich) analytification ofa scheme X of finite type over K and of an admissible formal K ◦ -scheme X in thesense of Raynaud. The
Tate algebra K h x , . . . , x n i consists of the formal power series f = P ν a ν x ν in K [[ x , . . . , x n ]] such that | a ν | → | ν | → ∞ . This K -algebra is thecompletion of K [ x , . . . , x n ] with respect to the Gauß norm k f k = max ν | a ν | .A K -affinoid algebra is a K -algebra A which is isomorphic to K h x , . . . , x n i /I for an ideal I . We may use the quotient norm from K h x , . . . , x n i to define a OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 7 K -Banach algebra ( A , k · k ). The presentation and hence the induced norm of anaffinoid algebra is not unique but two norms on A are equivalent and thus theydefine the same concept of boundedness. The
Berkovich spectrum M ( A ) of a K -affinoid algebra A is defined as theset of multiplicative seminorms p on A satisfying p ( f ) ≤ k f k for all f ∈ A . It onlydepends on the algebraic structure on A . As above we endow it with the coarsesttopology such that the maps p p ( f ) are continuous for all f ∈ A and we obtaina nonempty compact space.Roughly speaking, a Berkovich analytic space is given by an atlas of affinoidBerkovich spectra. For the precise definition, we refer to [2]. Note that we hereonly consider analytic spaces which are called strict in [2]. We need mainly thefollowing construction related to algebraic schemes: First let X = Spec( A ) be affine. The analytification X an is the set ofmultiplicative seminorms on A extending the absolute value | · | on K . We endowit with the coarsest topology such that the functions X an → R , p p ( f ) arecontinuous for every f ∈ A . The sheaf of analytic functions O X an on X an gives X an the structure of a Berkovich analytic space (see [1, §3.4] and [11, § 1.2]). For any scheme X of finite type over K we define the analytification X an as a Berkovich analytic space by gluing the affine analytic spaces obtained from anopen affine cover of X . For a morphism ϕ : X → Y of schemes of finite type over K we have a canonical map ϕ an : X an → Y an defined by ϕ an ( p ) := p ◦ ϕ ♯ on suitableaffine open subsets.The analytification functor preserves many properties of schemes and their mor-phisms. An analytic space X an is Hausdorff (resp. compact) if and only if X isseparated (resp. proper). For more details, we refer to [1, § 3.4].The analytification of a formal scheme is more difficult because at first we needarbitrary analytic spaces. We call a K ◦ -algebra A admissible if it is isomorphic to K ◦ h x , . . . , x n i /I for an ideal I and A has no K ◦ -torsion (or equivalently A is K ◦ -flat). If A isadmissible, then I is finitely generated. A formal scheme X over K ◦ is called admissible if there is a locally finite covering of open subsets isomorphic to formalaffine schemes Spf( A ) for admissible K ◦ -algebras A .Then the generic fiber X an of X is the analytic space locally defined by theBerkovich spectrum of the K -affinoid algebra A = A ⊗ K ◦ K . Moreover we definethe special fiber e X of X as the ˜ K -scheme locally given by Spec( A/K ◦◦ A ), i. e. e X isa scheme of locally finite type over ˜ K with the same topological space as X and thestructure sheaf O e X := O X ⊗ K ◦ ˜ K .There is a reduction map red : X an → e X assigning each seminorm p in a neigh-borhood M ( A ⊗ K ◦ K ) to the prime ideal { a ∈ A | p ( a ⊗ < } /K ◦◦ A . Thismap is surjective and anti-continuous. If e X is reduced, then red coincides with thereduction map in [1, 2.4]. In this case, for every irreducible component V of e X ,there is a unique point ξ V ∈ X an such that red( ξ V ) is the generic point of V (see[1, Proposition 2.4.4]). Assume that K is algebraically closed and let X = Spf( A ) be an admissibleformal affine scheme over K ◦ with reduced generic fiber X an , but not necessarilywith reduced special fiber. Let A = A ⊗ K ◦ K be the associated K -affinoid al-gebra and let A ◦ be the K ◦ -subalgebra of power bounded elements in A . Then X ′ := Spf( A ◦ ) is an admissible formal scheme over K ◦ with X ′ an = X an and withreduced special fiber e X ′ . The identity on the generic fiber extends to a canonical WALTER GUBLER AND JULIUS HERTEL morphism X ′ → X whose restriction to the special fibers is finite and surjective.By gluing, these assertions also hold for non-necessarily affine formal schemes. Fordetails, we refer to [25, Proposition 1.11 and 8.1]. Let X be a flat scheme of finite type over K ◦ with generic fiber X and π some non-zero element in K ◦◦ . Locally we can replace the coordinate ring A bythe π -adic completion of A and get an admissible formal scheme ˆ X over K ◦ withspecial fiber equal to the special fiber f X of X . The generic fiber ˆ X an , denotedby X ◦ , is an analytic subdomain of X an locally given by { p ∈ (Spec A ⊗ K ◦ K ) an | p ( a ) ≤ ∀ a ∈ A } . If X is proper over K ◦ , then X ◦ = X an and the reduction map is defined on thewhole of X an . If f X is reduced, then each maximal point of f X has a unique inverseimage in X ◦ . We refer to [32, 4.9–4.13] for details.If K is algebraically closed and X is reduced, then the construction in 1.1.6 givesus a formal admissible scheme X over K ◦ with generic fiber X an = X ◦ and withreduced special fiber e X such that the canonical morphism e X → f X is finite andsurjective.1.2. Metrics, local heights and measures.
From now on, we assume that thenon-archimedean field K is algebraically closed. This is no serious restriction be-cause we can always perform base change to the completion of the algebraic closureof any non-archimedean field and local heights and measures do not depend on thechoice of the base field.Let X be a reduced proper scheme over K and L a line bundle on X . Thisdefines a line bundle L an on the compact space X an .In this subsection, we introduce algebraic (resp. formal) models of X and L and their associated algebraic (resp. formal) metrics on L an . After introducingmetrized pseudo-divisors, we can study local heights of subvarieties and Chambert-Loir measures. Definition 1.2.1. A metric k · k on L is the datum, for any section s of L an on aopen subset U ⊆ X an , of a continuous function k s ( · ) k : U → R ≥ such that(i) it is compatible with the restriction to smaller open subsets;(ii) for all p ∈ U , k s ( p ) k = 0 if and only if s ( p ) = 0;(iii) for any λ ∈ O X an ( U ) and for all p ∈ U , k ( λs )( p ) k = | λ ( p ) | · k s ( p ) k .On the set of metrics on L we define the distance functiond( k · k , k · k ′ ) := sup p ∈ X an | log( k s p ( p ) k / k s p ( p ) k ′ ) | , where s p is any local section of L an not vanishing at p . Clearly, this definition isindependent of the choices of the s p . The pair L :=( L, k · k ) is called a metrized linebundle . Operations on line bundles like tensor product, dual and pullback extendto metrized line bundles. Definition 1.2.2. A formal ( K ◦ -)model of X is an admissible formal scheme X over K ◦ with a fixed isomorphism X an ≃ X an . Note that we identify X an with X an via this isomorphism.A formal ( K ◦ -)model of ( X, L ) is a triple ( X , L , e ) consisting of a formal model X of X , a line bundle L on X and an integer e ≥
1, together with an isomorphism L an ≃ ( L ⊗ e ) an . When e = 1, we write ( X , L ) instead of ( X , L , Definition 1.2.3.
To a formal K ◦ -model ( X , L , e ) of ( X, L ) we associate a metric k · k on L in the following way: If U is a formal trivialization of L and if s is a OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 9 section of L an on U an such that s ⊗ e corresponds to λ ∈ O X an ( U an ) with respect tothis trivialization, then k s ( p ) k = | λ ( p ) | /e for all p ∈ U an . A metric on L obtained in this way is called a Q -formal metric and,if e = 1, a formal metric .Such a Q -formal metric is said to be semipositive if the reduction e L of L on thespecial fiber e X is nef, i. e. we have deg e L ( C ) ≥ C in e X . The dual, the tensor product and the pullback of ( Q -)formal metrics areagain ( Q -)formal metrics. Furthermore, the tensor product and the pullback ofsemipositive Q -formal metrics are semipositive. Every line bundle L on X has a formal K ◦ -model ( X , L ) and hence a formalmetric k · k . For proofs of this and the following statements we refer to [25, § 7].Since K is algebraically closed and X is reduced, we may always assume that X hasreduced special fiber (see 1.1.6). Then the formal metric determines the K ◦ -model L on X up to isomorphisms, more precisely we have canonically L ( U ) ∼ = { s ∈ L an ( U an ) | k s ( p ) k ≤ ∀ p ∈ U an } (1.1)for each formal open subset U of X . Definition 1.2.6. An algebraic K ◦ -model X of X is a flat and proper scheme over K ◦ together with an isomorphism of the generic fiber of X onto X . An algebraic K ◦ -model ( X , L , e ) of ( X, L ) consists of a line bundle L on an algebraic K ◦ -model X of X and a fixed isomorphism L | X ∼ = L e .As in Definition 1.2.3, an algebraic model ( X , L , e ) of ( X, L ) induces a metric k · k on L , called algebraic metric . Such a metric is said to be semipositive if, forevery closed integral curve C in the special fiber f X , we have deg L ( C ) ≥ Q -formal metrics. Proposition 1.2.7.
Let L be a line bundle on a proper variety X over K and let k · k be a metric on L . Then, k · k is Q -formal if and only if k · k is algebraic. A metrized pseudo-divisor ˆ D on X is a triple ˆ D :=( L, Y, s ) where L is ametrized line bundle, Y is a closed subset of X and s is a nowhere vanishing sectionof L on X \ Y . Then ( O ( D ) , | D | , s D ) :=( L, Y, s ) is a pseudo-divisor in the sense of[21, 2.2]. In contrast to Cartier divisors, we can always define the pullback of ametrized pseudo-divisor ˆ D on X by a proper morphism ϕ : X ′ → X , namely ϕ ∗ ˆ D :=( ϕ ∗ O ( D ) , ϕ − | D | , ϕ ∗ s D ) . Example 1.2.9.
Let L be a metrized line bundle on X and s an invertible me-romorphic section of L , i. e. there is an open dense subset U of X such that s restricts to a non-vanishing section of L on U . Then the pair ( L, s ) determines apseudo-divisor c div( s ) := (cid:0) L, | div( s ) | , s | X \| div( s ) | (cid:1) , where | div( s ) | is the support of the Cartier divisor div( s ).Every real-valued continuous function ϕ on X an defines a metric on the trivialline bundle O X given by k k = e − ϕ . We denote this metrized line bundle by O ( ϕ ).Then we get a metrized pseudo-divisor b O ( ϕ ) :=( O ( ϕ ) , ∅ , Let ˆ D , . . . , ˆ D t be metrized pseudo-divisors with Q -formal metrics and let Z be a t -dimensional cycle on X with | D | ∩ · · · ∩ | D t | ∩ | Z | = ∅ . (1.2)Note that condition (1.2) is much weaker than the usual assumption that ˆ D , . . . , ˆ D t intersect properly on Z , that is, for all I ⊆ { , . . . , t } , each irreducible componentof Z ∩ T i ∈ I | D i | has dimension t − | I | .For Q -formal metrized pseudo-divisors there is a refined intersection productwith cycles on X developed in [25, §8] and [27, §5]. By means of this product, onecan define the local height λ ˆ D ,..., ˆ D t ( Z ) as the real intersection number of ˆ D , . . . , ˆ D t and Z on a joint formal K ◦ -model. For details, we refer to [25, §9] and [27, §9]. If K ◦ is a discrete valuation ring with value group Γ = Z and all the K ◦ -models arealgebraic, then we can use the usual intersection product. Proposition 1.2.11.
The local height λ ( Z ) := λ ˆ D ,..., ˆ D t ( Z ) is characterized by thefollowing properties:(i) It is multilinear and symmetric in ˆ D , . . . , ˆ D t and linear in Z .(ii) For a proper morphism ϕ : X ′ → X and a t -dimensional cycle Z ′ on X ′ satisfying | D | ∩ · · · ∩ | D t | ∩ | ϕ ( Z ′ ) | = ∅ , we have λ ϕ ∗ ˆ D ,...,ϕ ∗ ˆ D t ( Z ′ ) = λ ˆ D ,..., ˆ D t ( ϕ ∗ Z ′ ) . (iii) Let λ ′ ( Z ) be the local height obtained by replacing the metric k · k of ˆ D byanother Q -formal metric k · k ′ . If the Q -formal metrics of ˆ D , . . . , ˆ D t aresemipositive and if Z is effective, then | λ ( Z ) − λ ′ ( Z ) | ≤ d( k · k , k · k ′ ) · deg O ( D ) ,..., O ( D t ) ( Z ) . (1.3) Proof.
The properties (i) and (ii) follow from [27, Theorem 10.6] and the last prop-erty follows from the metric change formula in [27, Remark 9.5]. (cid:3) If X is an algebraic K ◦ -model of X , then there is a K ◦ -model Y of X withreduced special fiber and a proper K ◦ -morphism Y → X which is the identity on X . This follows from [8, Theorem 2.1’].Moreover, let L , L ′ be algebraic metrized line bundles on X induced by algebraic K ◦ -models ( X , L , e ) and ( X ′ , L ′ , e ′ ) respectively. Taking the closure X ′′ of X in X × K ◦ X ′ and pulling back L , L ′ to X ′′ , we obtain models inducing the samemetrics on L and L ′ but living on the same model X ′′ .Hence, we can always assume that L and L ′ live on a common model withreduced special fiber. The same holds for formal models, see 1.1.6.For global heights and archimedean local heights of subvarieties there is an in-duction formula which can be taken as definition for the heights (see [10, (3.2.2)]and [27, Proposition 3.5]). A. Chambert-Loir has introduced a measure on X an such that an analogous induction formula holds for non-archimedean local heights(cf. [14, 2.3]). Definition 1.2.13.
Let L i , i = 1 , . . . , d be Q -formal metrized line bundles on thereduced proper scheme X over K of dimension d . By 1.2.12, there is a formal K ◦ -model X of X with reduced special fiber and, for each i , a formal K ◦ -model( X , L i , e i ) of ( X, L i ) inducing the metric of L i . We denote by e X (0) the set of ir-reducible components of the special fiber e X . Then we define a discrete (signed)measure on X an byc ( L ) ∧ · · · ∧ c ( L d ) = 1 e · · · e d X V ∈ e X (0) deg e L ,..., e L d ( V ) · δ ξ V , OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 11 where δ ξ V is the Dirac measure in the unique point ξ V ∈ X an such that red( ξ V ) isthe generic point of V (see 1.1.5).More generally, let Y be a t -dimensional subvariety of X , then we definec ( L ) ∧ · · · ∧ c ( L t ) ∧ δ Y = i ∗ (cid:0) c ( L | Y ) ∧ · · · ∧ c ( L t | Y ) (cid:1) , where i : Y an → X an is the induced immersion. We also write c ( L ) · · · c ( L t ) δ Y .This measure extends by linearity to t -dimensional cycles. This measure is multilinear and symmetric in metrized line bundles. More-over, the total mass of c ( L ) ∧ · · · ∧ c ( L t ) ∧ δ Y equals the degree deg L ,...,L t ( Y ),and it is a positive measure if the metrics of the L i are semipositive. Proposition 1.2.15 (Induction formula) . Let ˆ D , . . . , ˆ D t be Q -formal metrizedpseudo-divisors and let Z be a t -dimensional prime cycle with | D | ∩ · · · ∩ | D t | ∩| Z | = ∅ . If | Z | * | D t | , then let s D t ,Z := s D t | Z , otherwise we choose any non-zero meromorphic section s D t ,Z of O ( D t ) | Z . Let Y be the Weil divisor of s D t ,Z considered as a cycle on X . Then we have λ ˆ D ,..., ˆ D t ( Z ) = λ ˆ D ,..., ˆ D t − ( Y ) − Z X an log k s D t ,Z k · c ( O ( D )) ∧ · · · ∧ c ( O ( D t − )) ∧ δ Z . Proof.
This follows from [27, Remark 9.5] and Definition 1.2.13. (cid:3)
Remark 1.2.16. If | Z | ⊆ | D t | , one might wonder why the left hand side dependson the metrized pseudo-divisor ˆ D t , which does not play a role on the right hand side,where however an arbitrarily chosen meromorphic section of O ( D t ) | Z occurs. Thisis closely related to the refined intersection products of pseudo-divisors on formal K ◦ -models given in [27, §5] and the dependence of the choice of the meromorphicsection fizzles out by the condition | D | ∩ · · · ∩ | D t − | ∩ | Z | = ∅ .1.3. Semipositivity.
It would be nice if we could extend local heights to all contin-uous metrics. Although the Q -formal metrics are dense in the space of continuousmetrics, this is not possible because the continuity property (1.3) in Proposition1.2.11 only holds for semipositive Q -formal metrics. Following Zhang, we extend thetheory of local heights to limits of semipositive Q -formal metrics which is importantfor canonical metrics and equidistribution theorems.Let X be a proper variety over an algebraically closed non-archimedean field K . Definition 1.3.1.
Let L = ( L, k·k ) be a metrized line bundle on X . The metric k·k is called semipositive if there exists a sequence ( k · k n ) n ∈ N of semipositive Q -formalmetrics on L such that lim n →∞ d( k · k n , k · k ) = 0 . In this case we say that L = ( L, k · k ) is a semipositive (metrized) line bundle . Themetric is said to be DSP (for “difference of semipositive”) if there are semipositivemetrized line bundles M , N on X such that L = M ⊗ N − . Then L is called a DSP (metrized) line bundle as well.
Remark 1.3.2. If k · k is a Q -formal metric, then [33, Proposition 7.2] says that k · k is semipositive in the sense of Definition 1.2.3 if and only if k · k is semipositiveas defined in Definition 1.3.1. So there is no ambiguity in the use of the termsemipositive metric. This answers the question raised in [11, Remark 1.4.2]. The tensor product and the pullback (with respect to a proper morphism)of semipositive metrics are again semipositive. The tensor product, the dual andthe pullback of DSP metrics are also DSP.
By means of Proposition 1.2.11, we can easily extend the local heights toDSP metrics. Concretely, let Y be a t -dimensional prime cycle and ˆ D i = ( L i , k ·k i , | D i | , s i ), i = 0 , . . . , t , a collection of semipositive metrized pseudo-divisors on X with | D | ∩ · · · ∩ | D t | ∩ Y = ∅ . By Definition 1.3.1, there is, for each i , an associatedsequence of semipositive Q -formal metrics k·k i,n on L i such that d( k·k i,n , k·k i ) → n → ∞ . Then we define the local height of Y with respect to ˆ D , . . . , ˆ D t as λ ˆ D ,..., ˆ D t ( Y ) := lim n →∞ λ ( L , k·k ,n , | D | ,s ) ,..., ( L t , k·k t,n , | D t | ,s t ) ( Y ) . (1.4)This does not depend on the choice of the approximating semipositive formal met-rics and extends to cylces. For details, see [24, § 1] or [26, Theorem 5.1.8].Let Z be a t -dimensional cycle of X and ( L i , s i ), i = 0 , . . . , t , DSP metrizedline bundles on X with invertible meromorphic sections such that | div( s ) | ∩ · · · ∩| div( s t ) | ∩ | Z | = ∅ . By Example 1.2.9, we obtain DSP metrized pseudo-divisors c div( s i ), i = 0 , . . . , t . Then, we denote the local height by λ ( L ,s ) ,..., ( L t ,s t ) ( Z ) := λ c div( s ) ,..., c div( s t ) ( Z ) . (1.5) Proposition 1.3.5.
Let Z be a t -dimensional cycle of X and ˆ D , . . . , ˆ D t DSPmetrized pseudo-divisors on X with | D | ∩ · · · ∩ | D t | ∩ | Z | = ∅ . Then there is aunique local height λ ( Z ) := λ ˆ D ,..., ˆ D t ( Z ) ∈ R satisfying the following properties:(i) If ˆ D , . . . , ˆ D t are Q -formal metrized, then λ ( Z ) is the local height of 1.2.10.(ii) λ ( Z ) is multilinear and symmetric in ˆ D , . . . , ˆ D t and linear in Z .(iii) For a proper morphism ϕ : X ′ → X and a t -dimensional cycle Z ′ on X ′ satisfying | D | ∩ · · · ∩ | D t | ∩ | ϕ ( Z ′ ) | = ∅ , we have λ ϕ ∗ ˆ D ,...,ϕ ∗ ˆ D t ( Z ′ ) = λ ˆ D ,..., ˆ D t ( ϕ ∗ Z ′ ) . In particular, λ ˆ D ,..., ˆ D t ( Z ) does not change when restricting the metrizedpseudo-divisors to the prime cycle Z .(iv) Let λ ′ ( Z ) be the local height obtained by replacing the metric k · k of ˆ D by another DSP metric k · k ′ . If the metrics of ˆ D , . . . , ˆ D t are semipositiveand if Z is effective, then | λ ( Z ) − λ ′ ( Z ) | ≤ d( k · k , k · k ′ ) · deg O ( D ) ,..., O ( D t ) ( Z ) . (v) Let f be a rational function on X and let ˆ D = c div( f ) be endowed with thetrivial metric on O ( D ) = O X . If Y = P P n P P is a cycle representing D . · · · .D t .Z ∈ CH ( | D | ∩ · · · ∩ | D t | ∩ | Z | ) , then λ ( Z ) = X P n P · log | f ( P ) | . Proof.
This follows immediately from Proposition 1.2.11 and the construction in1.3.4 and is established in [27, Theorem 10.6]. (cid:3)
Similarly, there is a generalization of Chambert Loir’s measures to semipositiveand DSP line bundles:
Proposition 1.3.6.
Let Y be a t -dimensional subvariety of X and L i = ( L i , k ·k i ) , i = 1 , . . . , t , semipositive line bundles. For each i , let ( k · k i,n ) n ∈ N be thecorresponding sequence of Q -formal semipositive metrics on L i converging to k · k i .Then the measures c ( L , k · k ,n ) ∧ · · · ∧ c ( L t | , k · k t,n ) ∧ δ Y converge weakly to a regular Borel measure on X an . This measure is independentof the choice of the sequences.Proof. This follows from [29, Proposition 3.12]. (cid:3)
OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 13
Definition 1.3.7.
Let Y be a t -dimensional subvariety of X and L i = ( L i , k · k i ), i = 1 , . . . , t , semipositive line bundles. We denote the limit measure in 1.3.6 byc ( L ) ∧ · · · ∧ c ( L t ) ∧ δ Y or shortly by c ( L ) . . . c ( L t ) δ Y . By multilinearity thisnotion extends to a t -dimensional cycle Y of X and DSP line bundles L , . . . , L t .Chambert Loir’s measure is uniquely determined by the following property whichis taken as definition in [29, 3.8]. Proposition 1.3.8.
Let L , . . . , L t be DSP line bundles on X and let Z be a t -dimensional cycle. For j = 1 , . . . , t we choose any metrized pseudo-divisor ˆ D j with O ( D j ) = L j , for example ˆ D j = ( L j , X, .If g is any continuous function on X an , then there is a sequence of Q -formalmetrics ( k · k n ) n ∈ N on O X such that log k k − n tends uniformly to g for n → ∞ and Z X an g · c ( L ) ∧ · · · ∧ c ( L t ) ∧ δ Z = lim n →∞ λ ( O X , k·k n , ∅ , , ˆ D ,..., ˆ D t ( Z ) . Proof.
By [25, Theorem 7.12], the Q -formal metrics are dense in the space of con-tinuous metrics on O X . This implies the existence of the sequence ( k · k n ) n ∈ N . Thesecond part follows from [29, Proposition 3.8]. (cid:3) Corollary 1.3.9.
Let Z be a cycle on X of dimension t and let ˆ D , . . . , ˆ D t be DSPmetrized pseudo-divisors with | D | ∩ · · · ∩ | D t | ∩ | Z | = ∅ . Replacing the metric k · k on O ( D ) by another DSP metric k · k ′ , we obtain a metrized pseudo-divisor ˆ E .Then g := log( k s D k / k s D k ′ ) extends to a continuous function on X and λ ˆ D ,..., ˆ D t ( Z ) − λ ˆ E, ˆ D ,..., ˆ D t ( Z ) = Z X an g · c ( O ( D )) ∧ · · · ∧ c ( O ( D t )) ∧ δ Z . Proof.
Clearly g defines a continuous function on X and the claim follows easilyfrom Proposition 1.3.8. (cid:3) Proposition 1.3.10.
Let Z be a t -dimensional cycle of X and L , . . . , L t DSP linebundles. Then the measure c ( L ) ∧ · · · ∧ c ( L t ) ∧ δ Z has the following properties:(i) It is multilinear and symmetric in L , . . . , L t and linear in Z .(ii) Let ϕ : X ′ → X be a morphism of proper schemes over K and Z ′ a t -dimensional cycle of X ′ , then ϕ ∗ (cid:0) c ( ϕ ∗ L ) ∧ · · · ∧ c ( ϕ ∗ L t ) ∧ δ Z ′ (cid:1) = c ( L ) ∧ · · · ∧ c ( L t ) ∧ δ ϕ ∗ Z ′ . (iii) If the metrics of L , . . . , L t are semipositive, then c ( L ) ∧ · · · ∧ c ( L t ) ∧ δ Z is a positive measure with total mass deg L ,...,L t ( Z ) .Proof. We refer to Corollary 3.9 and Proposition 3.12 in [29]. (cid:3)
Remark 1.3.11.
In the archimedean case, i.e. for K = C , there is a similartheory of local heights and Chambert-Loir measures as presented above. Formalmetrics are replaced by smooth metrics, semipositivity means positive curvature.Then uniform limits of semipositive smooth metrics lead to semipositive continuousmetrics on the complex analytification of the line bundle. The DSP metrics aredefined as above and we get Chambert-Loir measures as before. All of the aboveproperties remain valid. For details, we refer to [31].1.4. Induction formula for DSP local heights.
It is quite difficult to generalizethe induction formula from Proposition 1.2.15 to DSP line bundles. In the case ofa discrete valuation (or an archimedean place) and properly intersecting Cartierdivisors on a projective variety, this was done by Chambert-Loir and Thuillier in[15, Théorème 4.1]. The goal of this subsection is to generalize their result to anyproper variety X over an algebraically closed non-archimedean field K . Note that, once the algebraically closed case is settled, invariance of the local heights by basechange gives the induction formula for any non-archimedean field. Theorem 1.4.1 (Approximation theorem) . Let L be a line bundle on X endowedwith a semipositive formal metric k · k and let s be a global section of L on X whichdoes not vanish identically. Then there is a sequence ( k · k n ) n ∈ N of formal metricson the trivial bundle O X with the following properties: (i) The sequence (cid:0) log k k − n (cid:1) n ∈ N converges pointwise to log k s k − and it ismonotonically increasing. (ii) For each n ∈ N , the formal metric k · k / k · k n on L is semipositive. Over a complete discrete valuation ring and for a projective variety X , this wasproven by Chambert-Loir and Thuillier [15, Théorème 3.1]. We will use a similar,but more analytic approach. Proof.
We fix some non-zero element π in K ◦◦ and define, for each n ∈ N , thestrictly analytic domains A n := { x ∈ X an | k s ( x ) k ≥ | π n |} and B n := { x ∈ X an | k s ( x ) k ≤ | π n |} . (1.6)By 1.2.5, the formal metric k·k on L is given by a (finite) G-covering { U i } i ∈ I of X an by strictly affinoid domains and non-vanishing regular sections t i ∈ L an ( U i ) with k t i k ≡
1. We refer to [6, § 9] for the G-topology on rigid analytic varieties and to [2,§1.6] for the transition to Berkovich analytic spaces. Let g ij = t j /t i ∈ O ( U i ∩ U j ) × be the transition functions. Then the nowhere vanishing restrictions s | U i ∩ A n maybe identified with regular functions f i ∈ O ( U i ∩ A n ) × satisfying f i = g ij f j on U i ∩ U j ∩ A n . There is a unique continuous metric k · k n on O X with k k n := max {k s k , | π | n } . Since the functions f − i ∈ O ( U i ∩ A n ), π − n ∈ O ( U i ∩ B n ) are local frames of O X an on a G-covering of X an by strictly affinoid domains and since these frames havenorm 1 with respect to k · k n , it follows from 1.2.5 that k · k n is a formal metric on O X . By construction, we have k k n = ( k s k on A n , | π | n on B n . (1.7)Clearly, the sequence (cid:0) log k k − n (cid:1) n ∈ N is monotonically increasing and convergespointwise to log k s k − . This proves (i).To prove (ii), we show that, for each n ∈ N , the formal metric k · k ′ n := k · k / k · k n is semipositive on L ⊗ O − X = L . Note that k s k ′ n = k s kk k n = ( A n , k s k · | π − n | on B n . (1.8)For the G-covering { U i ∩ A n , U i ∩ B n } i ∈ I of X an by strictly affinoid domains, thereexists a formal K ◦ -model X n of X an and a formal open covering { U i,n , V i,n } i ∈ I of X n such that U an i,n = U i ∩ A n and V an i,n = U i ∩ B n (see [7, Theorem 5.5]). We mayassume that X n has reduced special fiber (cf. 1.2.12). Then, by 1.2.5, the formalmetric k · k ′ n is associated to the formal K ◦ -model ( L ′ n , X n ) of ( X, L ) given by L ′ n ( U ) = { r ∈ L an ( U an ) | k r ( x ) k ′ n ≤ ∀ x ∈ U an } (1.9)on a formal open subset U of X n . Therefore, we can consider s as a global sectionof L ′ n as we have k s k ′ n = k s k n / k k n ≤ C ⊆ e X n be a closed integral curve. If s doesn’t vanish identically on C , thendeg e L ′ n ( C ) = deg(c ( e L ′ n ) .C ) = deg(div( s | C )) ≥ . OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 15 If s vanishes identically on C , let B n be the union of the formal open subsets( V i,n ) i ∈ I . Then it follows from (1.6) and (1.8) that e B n = red( B n ) contains C . Bypassing from the G-covering { U i } i ∈ I to the refinement { U i ∩ A n , U i ∩ B n } i ∈ I andusing the frame t i with k t i k = 1 on U i ∩ A n and on U i ∩ B n , we see again from1.2.5 that the formal metric k · k is given by a K ◦ -model L n on X n . Moreover, L n satisfies a similar formula as in (1.9) with k · k replacing k · k ′ n . We get anisomorphism L n | B n ∼ = L ′ n | B n given by r π n · r . By assumption, the formalmetric k · k is semipositive and hence e L n is nef. Since ˜ B n is a neighborhood of C ,we obtain deg e L ′ n ( C ) = deg e L n ( C ) ≥ . This implies the semipositivity of k · k / k · k n proving (ii). (cid:3) Corollary 1.4.2.
We use the notations from the approximation theorem 1.4.1 andin addition, we consider semipositive line bundles L , . . . , L t − on the t -dimensionalproper variety X . Let µ be a (signed) Radon measure on X an such that, for everyformal metric k · k ′ on O X , lim n →∞ Z X an log k k ′ · c ( O X , k · k n ) c ( L ) . . . c ( L t − ) = Z X an log k k ′ · µ . (1.10) Then the sequence (cid:0) c ( O X , k · k n ) c ( L ) . . . c ( L t − ) (cid:1) n ∈ N of measures on X an con-verges weakly to µ .Proof. We define ν := c ( L, k · k ) c ( L ) . . . c ( L t − ) and, for each n ∈ N , we set µ n := c ( O X , k · k n ) c ( L ) . . . c ( L t − ). Then, by the approximation theorem 1.4.1and Proposition 1.3.10 (iii), the measures ν − µ n = c (cid:16) L, k·kk·k n (cid:17) c ( L ) . . . c ( L t − )are positive with finite total mass deg L,L ,...,L t − ( X ) independent of n . By linearity,the equation (1.10) also holds for any Q -formal metric on O X . By [25, Theorem7.12], the space {− log k k ′ | k · k ′ Q -formal metric on O X } is dense in C ( X an ). Aneasy application of the triangle inequality shows thatlim n →∞ Z X an f · ( ν − µ n ) = Z X an f · ( ν − µ )for any f ∈ C ( X an ). This proves the claim. (cid:3) Theorem 1.4.3 (Induction formula) . Let Z be a t -dimensional prime cycle on X and let ˆ D = ( L , | D | , s ) , . . . , ˆ D t = ( L t , | D t | , s t ) be DSP pseudo-divisors with | D | ∩ · · · ∩ | D t | ∩ | Z | = ∅ . If | Z | * | D t | , then let s t,Z := s t | Z , otherwise we chooseany non-zero meromorphic section s t,Z of L t | Z . Let cyc( s t,Z ) be the Weil divisorof s t,Z considered as a cycle on X .Then the function log k s t,Z k is integrable with respect to c ( L ) ∧· · ·∧ c ( L t − ) ∧ δ Z and we have λ ˆ D ,..., ˆ D t ( Z ) = λ ˆ D ,..., ˆ D t − (cyc( s t,Z )) − Z X an log k s t,Z k · c ( L ) ∧ · · · ∧ c ( L t − ) ∧ δ Z . (1.11) Remark 1.4.4. If L , . . . , L t have Q -formal metrics, then this result is just Propo-sition 1.2.15. It is also evident if L t is the trivial bundle and hence, log k s t,Z k is acontinuous function on Z . The difficulties of the general case arise from the relationbetween the limit process defining the measure, and the singularities of the functionlog k s t,Z k . Our proof is based on [15, Théorème 4.1] where the formula is demonstratedunder the additional assumptions that X is projective over a complete discretevaluation ring and s , . . . , s t are global sections with associated Cartier divisorsintersecting Z properly. As explained at the beginning of our proof, our moregeneral setting of pseudo-divisors satisfying | D |∩· · ·∩| D t |∩| Z | = ∅ can be reducedto the projective situation with properly intersecting global sections. So the maindifficulty is to deal with a non-discrete valuation where the main ingredient is nowour generalization of the approximation theorem.The induction formula 1.4.3 also holds in the archimedean case, i.e. for K = C .Indeed, this was proven by Chambert-Loir and Thuillier in the projective situationdescribed above and extends to our setting as explained above. Proof of the induction formula 1.4.3.
By Proposition 1.3.5 (iii), we may assume that X = Z , especially s t = s t,Z . Furthermore, we can suppose that X is projective byChow’s lemma and functoriality of the height (Proposition 1.3.5). We denote thecrucial difference of local heights by∆( ˆ D , . . . , ˆ D t ) := λ ˆ D ,..., ˆ D t ( X ) − λ ˆ D ,..., ˆ D t − (cyc( s t )) . (1.12)First note that ∆( ˆ D , . . . , ˆ D t ) depends continuously on the metrics ( k · k , . . . , k · k t )by the metric change formula 1.3.9. If all the metrics are formal, then the inductionformula 1.2.15 holds and hence a continuity argument shows that ∆( ˆ D , . . . , ˆ D t )only depends on L , . . . , L t − and ˆ D t . In fact, it is easy to see that∆ L ,...,L t − ( ˆ D t ) := ∆( ˆ D , . . . , ˆ D t )makes sense for any DSP metrized line bundles L , . . . , L t − and any metrized DSPpseudo-divisor ˆ D t by choosing (generic) pseudo-divisors ˆ D , . . . , ˆ D t − for L , . . . , L t − with | D | ∩ · · · ∩ | D t | = ∅ . By Proposition 1.3.5(i), ∆ L ,...,L t − ( ˆ D t ) is multilinear in( L , . . . , L t − , ˆ D t ). Using this and projectivity, we may assume that L , . . . , L t arevery ample, s t is a global section and the metrics are semipositive.Note that − log k s t k is a measurable function on X an . In the following, we willintegrate it with respect to positive Radon measures on X an allowing the value ∞ for the integral, the value −∞ is out of question as the function is bounded frombelow and X an is compact. To prove the theorem, it is enough to show that∆ L ,...,L t − ( ˆ D t ) = − Z X an log k s t k · c ( L ) ∧ · · · ∧ c ( L t − ) (1.13)which implies also integrability of − log k s t k . The metric change formula 1.3.9 showsthat the last metric k · k t may be assumed to be a semipositive formal metric. Usinggeneric hyperplane sections of L , . . . , L t − , we may assume that the local heights inthe theorem and in (1.12) are with respect to very ample pseudo-divisors D , . . . , D t which intersect properly.We prove by induction on k ∈ { , . . . , t } that the theorem holds if L i is a formallymetrized line bundle for i ≥ k . The case k = 0 is just the induction formula forformal metrics (see Proposition 1.2.15). We assume that the statement holds for k and demonstrate it for k + 1.In the following we fix a semipositive formal metric k · k ′ on L k . We denotethe corresponding metrized line bundle by M k and the metrized pseudo-divisor( M k , | D k | , s k ) by ˆ E k . Then we can extend ϕ k := log k s k k ′ − log k s k k to a continuousfunction on X an and O X ( ϕ k ) := L k ⊗ M − k is a DSP line bundle whose underlyingline bundle is trivial. To emphasize that the following line bundles are equippedwith formal metrics, we will use M i := L i and ˆ E i := ˆ D i for i = k + 1 , . . . , t . Since OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 17 M k , . . . , M t − are formally metrized line bundles, our induction hypothesis implies Z X an log k s t k · c ( L ) . . . c ( L k − ) c ( M k ) . . . c ( M t − )= λ ˆ D ,..., ˆ D k − , ˆ E k ,..., ˆ E t − (cyc( s t )) − λ ˆ D ,..., ˆ D k − , ˆ E k ,..., ˆ E t ( X ) . (1.14)If we apply the change of metrics formula 1.3.9 twice in (1.14) to replace ˆ E k by ˆ D k ,then the same computation as in the proof of [15, Théorème 4.1] shows that (1.13)is equivalent to the claim Z X an ϕ k · c ( L ) . . . c ( L k − ) c ( M k +1 ) . . . c ( M t )= Z X an ϕ k · c ( L ) . . . c ( L k − ) c ( M k +1 ) . . . c ( M t − ) δ cyc( s t ) − Z X an log k s t k · c ( O ( ϕ k )) c ( L ) . . . c ( L k − ) c ( M k +1 ) . . . c ( M t − ) . Using our reduction steps at the beginning, we can apply the approximation theorem1.4.1 : Let ( k ·k n ) n ∈ N be a sequence of formal metrics on O X such that the functions g n := log k k − n tend pointwise to log k s t k − , the sequence ( g n ) n ∈ N is monotonicallyincreasing and ( O X , k · k n ) is a DSP line bundle. Applying Lebesgue’s monotoneconvergence theorem and using Proposition 1.3.8 and 1.3.10 (i), we obtain Z X an log k s t k − · c ( O ( ϕ k )) c ( L ) . . . c ( L k − ) c ( M k +1 ) . . . c ( M t − )= lim n →∞ Z X an g n · c ( O ( ϕ k )) c ( L ) . . . c ( L k − ) c ( M k +1 ) . . . c ( M t − )= lim n →∞ λ b O ( g n ) , b O ( ϕ k ) , ˆ D ,..., ˆ D k − , ˆ E k +1 ,..., ˆ E t − ( X )= lim n →∞ λ b O ( ϕ k ) , b O ( g n ) , ˆ D ,..., ˆ D k − , ˆ E k +1 ,..., ˆ E t − ( X )= lim n →∞ Z X an ϕ k · c ( O X , k · k n ) c ( L ) . . . c ( L k − ) c ( M k +1 ) . . . c ( M t − ) . For ϕ k = log( k · k ′ k / k · k k ), this shows that (1.13) is equivalent to the claimlim n →∞ Z X an ϕ k · c ( O X , k · k n ) c ( L ) . . . c ( L k − ) c ( M k +1 ) . . . c ( M t − )= Z X an ϕ k · (cid:0) c ( L ) . . . c ( L k − ) c ( M k +1 ) . . . c ( M t ) (1.15) − c ( L ) . . . c ( L k − ) c ( M k +1 ) . . . c ( M t − ) δ cyc( s t ) (cid:1) . The induction hypothesis implies that equation (1.15) always holds if k · k ′ k / k · k k is a formal metric. But then Corollary 1.4.2 says that this equation is also true if ϕ k is only continuous. This shows the induction formula (1.11) and therefore theintegrability of log k s t k with respect to c ( L ) · · · c ( L t − ). (cid:3) Corollary 1.4.5.
Let L , . . . , L n be DSP-metrized line bundles on the n -dimensionalproper variety X over K and let Y be any proper closed subset of X endowed withthe induced reduced structure. Then Y an has measure zero with respect to the mea-sure c ( L ) ∧ . . . ∧ c ( L n ) on X an .Proof. By Chow’s lemma and Proposition 1.3.10 (ii), we may assume that X isprojective. Then Y is contained in the support of an effective pseudo-divisor( L, | div( s ) | , s ). By the induction formula, the function log k s k − is integrablewith respect to c ( L ) ∧ . . . ∧ c ( L n − ), but it takes the value + ∞ on | div( s ) | an . Thus | div( s ) | an and the subset Y an have measure zero with respect to the measurec ( L ) ∧ . . . ∧ c ( L n ) on X an . (cid:3) Corollary 1.4.6.
Let L , . . . , L n be semipositive metrized line bundles on the n -dimensional proper variety X over K . For i = 1 , . . . , m , let s i be a non-trivial mero-morphic section of a DSP-metrized line bundle M i with DSP-metric k · k i . Then φ := max i log k s i k i is integrable with respect to the measure µ := c ( L ) ∧ . . . ∧ c ( L n ) and the function ( L , . . . , L n ) R X an φ dµ is continuous with respect to uniformconvergence of the semipositive metrics on the fixed line bundles L , . . . , L n .Proof. Using Chow’s lemma, we may assume that X is projective. We first handlethe case m = 1. Then it follows from Theorem 1.2.15 that log k s k is µ -integrable andthat we may write R X an φ dµ as a difference of two local heights. Then continuitywith respect to uniform convergence of the metrics follows from the similar propertyof local heights (see Proposition 1.3.5(iv)).Now we deal with the case m ≥
2. Let H be an ample line bundle on X .Using that M i is the difference of M i ⊗ H k and H k and both are very ample for k sufficiently large, we deduce easily that s i = t i /t for t i ∈ Γ( X, M i ⊗ H k ) and t ∈ Γ( X, H k ). Moreover, we may assume that the same k and that the samedenominator t work for all i = 1 , . . . , m . We endow H with any DSP-metric and M i ⊗ H k with the tensor metric. Using the case m = 1 handled above, it is enoughto show the claim for the function max i log k t i k i . Hence we may assume that all s i are global sections.Using the case m = 1, we deduce that φ is a maximum of µ -integrable functionsand hence φ is also µ -integrable. Since all the s i are now global sections, there is C ∈ R such that log k s i k i ≤ φ ≤ C for any i = 1 , . . . , m . A sandwich argument andthe case m = 1 yield that φ is continuous with respect to uniform convergence ofthe semipositive metrics L , . . . , L n . (cid:3) Metrics and local heights of toric varieties
In this section, we study local heights of proper toric varieties with respect toline bundles endowed with toric metrics. We generalize the program of Burgos–Philippon–Sombra [11, Chapter 4], where they assumed to a large extend thatvaluations are discrete, to arbitrary non-archimedean fields. In the first two sections,we recall the classical theory of toric varieties and the theory of toric schemes from[32]. The first new results come in § 2.3 where we describe Cartier divisors ontoric schemes in terms of piecewise affine functions. In § 2.4, we recall toric metrics,tropicalizations and the Kajiwara–Payne compactification. In § 2.5, the main resultis the characterization of semipositive toric metrics in terms of concave functions.Moreover, we give a formula for the Chambert–Loir measure of a semipositive toricmetric in terms of the associated Monge–Ampère measure. Finally, we prove in § 2.6the formula for toric local heights explained in Theorem III of the introduction.2.1.
Toric varieties.
We give a short overview of the theory of (normal) toricvarieties over a field K following closely the notation in [11, 3.1-3.4]. For details,we refer to [38], [20] and [16].Let M be a free abelian group of rank n and N := M ∨ := Hom( M, Z ) its dualgroup. The natural pairing between m ∈ M and u ∈ N is denoted by h m, u i := u ( m ).We have the split torus T := Spec( K [ M ]) over a field K of rank n . Then M canbe considered as the character lattice of T and N as the lattice of one-parametersubgroups. For m ∈ M we will write χ m for the corresponding character. If G is an abelian group, we set N G = N ⊗ Z G . In particular, N R = N ⊗ Z R is an n -dimensional real vector space with dual space M R . OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 19
Definition 2.1.1.
Let K be a field and T a split torus over K . A ( T -) toric variety is an irreducible variety X over K containing T as a (Zariski) open subset such thatthe translation action of T on itself extends to an algebraic action µ : T × X → X . There is a nice description of normal toric varieties in combinatorial data.At first we have a bijection between the sets of(i) strongly convex rational polyhedral cones σ in N R ,(ii) isomorphism classes of affine normal T -toric varieties X over K .This correspondence is given by σ U σ = Spec( K [ M σ ]), where K [ M σ ] is thesemigroup algebra of M σ = σ ∨ ∩ M = { m ∈ M | h m, u i ≥ ∀ u ∈ σ } . The action of T on U σ is induced by K [ M σ ] → K [ M ] ⊗ K [ M σ ] , χ m χ m ⊗ χ m . More generally, we consider a fan Σ in N R (Definition A.4). If σ, σ ′ ∈ Σ, then U σ and U σ ′ glue together along the open subset U σ ∩ σ ′ . this leads to a normal T -toricvariety X Σ = [ σ ∈ Σ U σ . This construction induces a bijection between the set of fans Σ in N R and the setof isomorphism classes of normal toric varieties X Σ with torus T . Many properties of normal toric varieties are encoded in their fans, forexample:(i) A normal toric variety X Σ is proper if and only if the fan is complete, i. e. | Σ | := S σ ∈ Σ σ = N R .(ii) A normal toric variety X Σ is smooth if and only if the minimal generatorsof each cone σ ∈ Σ are part of a Z -basis of N . Let X Σ be the normal toric variety of the fan Σ in N R . Then there isa bijective correspondence between the cones in Σ and the T -orbits in X Σ . Theclosures of the orbits in X Σ have the structure of normal toric varieties what wedescribe in the following: For σ ∈ Σ we set N ( σ ) = N/ h N ∩ σ i , M ( σ ) = N ( σ ) ∨ = M ∩ σ ⊥ , O ( σ ) = Spec( K [ M ( σ )]) , where σ ⊥ denotes the orthogonal space to σ . Then O ( σ ) is a torus over K ofdimension n − dim( σ ) which can be identified with a T -orbit in X Σ via the surjection K [ M σ ] −→ K [ M ( σ )] , χ m ( χ m if m ∈ σ ⊥ , . We denote by V ( σ ) the closure of O ( σ ) in X Σ . Then V ( σ ) can be identified withthe normal O ( σ )-toric variety X Σ( σ ) which is given by the fanΣ( σ ) = { τ + h N ∩ σ i R | τ ∈ Σ , τ ⊇ σ } (2.1)in N ( σ ) R = N R / h N ∩ σ i R . Definition 2.1.5.
Let X i , i = 1 ,
2, be toric varieties with dense open torus T i . Wesay that a morphism ϕ : X → X is toric if ϕ maps T into T and ϕ | T : T → T is a group morphism. Any toric morphism ϕ : X → X is equivariant , i. e. we have a commutativediagram T × X µ / / ϕ | T × ϕ (cid:15) (cid:15) X ϕ (cid:15) (cid:15) T × X µ / / X , where µ , µ denote the torus actions.Toric morphisms can be desribed in combinatorial terms. For i = 1 ,
2, let N i be a lattice with associated torus T i = Spec K [ N ∨ i ] andlet Σ i be a fan in N i, R . Let H : N → N be a linear map which is compatible with Σ and Σ . That is, for each cone σ ∈ Σ , there exists a cone σ ∈ Σ with H ( σ ) ⊆ σ .Then H induces a group morphism T → T of tori and, by the compatibility of H , this group morphism extends to a toric morphism ϕ H : X Σ → X Σ .We fix N i , T i and Σ i , i = 1 ,
2, as above. Then the assignment H ϕ H inducesa bijection between the sets of(i) linear maps H : N → N which are compatible with Σ and Σ ;(ii) toric morphisms ϕ : X Σ → X Σ .A toric morphism ϕ H : X Σ → X Σ is proper if and only if H − ( | Σ | ) = | Σ | . Definition 2.1.8. A T -Cartier divisor on a T -toric variety X is a Cartier divisor D on X which is invariant under the action of T on X , i. e. we have µ ∗ D = p ∗ D denoting by µ : T × X → X the toric action and by p : T × X → X the secondprojection.Torus-invariant Cartier divisors on a normal T -toric variety X = X Σ can bedescribed in terms of support functions: Definition 2.1.9.
A function Ψ : | Σ | −→ R is called a virtual support function onΣ, if there exists a set { m σ } σ ∈ Σ of elements in M such that, for each cone σ ∈ Σ, wehave Ψ( u ) = h m σ , u i for all u ∈ σ . It is said to be strictly concave if, for maximalcones σ, τ ∈ Σ, we have m σ = m τ if and only if σ = τ . A support function is aconcave virtual support function on a fan. Let Ψ be a virtual support function given by the data { m σ } σ ∈ Σ . Then Ψdetermines a T -Cartier divisor D Ψ := (cid:8)(cid:0) U σ , χ − m σ (cid:1)(cid:9) σ ∈ Σ on X Σ . The map Ψ D Ψ is an isomorphism between the group of virtual supportfunctions on Σ and the group of T -Cartier divisors on X Σ . The divisors D Ψ and D Ψ are rationally equivalent if and only if Ψ − Ψ is linear. Definition 2.1.11.
Let X be a toric variety. A toric line bundle on X is a pair( L, z ) consisting of a line bundle L on X and a non-zero element z in the fiber L x of the unit point x of U = T . A toric section is a rational section s of a toric linebundle which is regular and non-vanishing on the torus U and such that s ( x ) = z . Let D be a T -Cartier divisor on a normal toric variety X Σ . Then there isan associated line bundle O ( D ) and a rational section s D such that div( s D ) = D .Since the support of D lies in the complement of T , the section s D is regular andnon-vanishing on T . Thus, D corresponds to a toric line bundle ( O ( D ) , s D ( x ))with toric section s D . This assignment determines an isomorphism between thegroup of T -Cartier divisors on X Σ and the group of isomorphism classes of toricline bundles with toric sections. OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 21
Let Ψ be a virtual support function on Σ. This function corresponds bijectively tothe isomorphism class of the toric line bundle with toric section (( O ( D Ψ ) , s D Ψ ( x )) , s D Ψ ),which we simply denote by ( L Ψ , s Ψ ).Note that a line bundle with a toric section admits a unique structure of a T -equivariant line bundle such that the toric section becomes T -invariant. Conversely,any T -equivariant toric line bundle has a unique toric section which is T -equivariant(see [11, Remark 3.3.6]). Let X Σ be a T -toric variety. We denote by Pic( X Σ ) the Picard group of X Σ and by Div T ( X Σ ) the group of T -Cartier divisors. Then we have a short exactsequence of abelian groups M −→ Div T ( X Σ ) −→ Pic( X Σ ) −→ , where the first morphism is given by m div( χ m ). In particular, every toric linebundle admits a toric section and, if s and s ′ are two toric sections, then there isan m ∈ M such that s ′ = χ m s . Let D Ψ be a T -Cartier divisor on a normal toric variety X Σ . Then theassociated Weil divisor cyc( s Ψ ) is invariant under the torus action. Indeed, letΣ (1) be the set of one-dimensional cones in Σ. Each ray τ ∈ Σ (1) gives a minimalgenerator v τ ∈ τ ∩ N and a corresponding T -invariant prime divisor V ( τ ) on X Σ (see 2.1.4). Then we have cyc( s Ψ ) = X τ ∈ Σ (1) − Ψ( v τ ) V ( τ ) . (2.2) We describe the intersection of a T -Cartier divisor with the closure of anorbit. Let Σ be a fan in N R and Ψ a virtual support function on Σ given by thedefining vectors { m τ } τ ∈ Σ . Let σ be a cone of Σ and V ( σ ) the corresponding orbitclosure. Each cone τ (cid:23) σ corresponds to a cone τ of the fan Σ( σ ) defined in(2.1). Since m τ − m σ | σ = 0, we have m τ − m σ ∈ M ( σ ) = M ∩ σ ⊥ . Thus, thedefining vectors { m τ − m σ } τ ∈ Σ( σ ) gives us a well-defined virtual support function(Ψ − m σ )( σ ) on Σ( σ ).When Ψ | σ = 0, then D Ψ and V ( σ ) do not intersect properly. But D Ψ is rationallyequivalent to D Ψ − m σ and the latter divisor properly intersects V ( σ ). Moreover, wehave D Ψ − m σ | V ( σ ) = D (Ψ − m σ )( σ ) . For details, we refer to [11, Proposition 3.3.14].We end this subsection with some positivity statements about T -Cartier divisors.For this, we consider a complete fan Σ in N R and a virtual support function Ψ onΣ given by the defining vectors { m σ } σ ∈ Σ . Many properties of the associated toric line bundle O ( D Ψ ) are encoded inits (virtual) support function.(i) O ( D Ψ ) is generated by global sections if and only if Ψ is concave;(ii) O ( D Ψ ) is ample if and only if Ψ is strictly concave.If Ψ is concave, then the stability set ∆ Ψ from A.7 is a lattice polytope and { χ m } m ∈ M ∩ ∆ Ψ is a basis of the K -vector space Γ( X Σ , O ( D Ψ )). Moreover, we havein this case deg O ( D Ψ ) ( X Σ ) = n ! vol M (∆ Ψ ) . (2.3) Assume that Ψ is strictly concave or equivalently that D Ψ is ample. Weuse the notations and statements from A.20. Then the stability set ∆ := ∆ Ψ is afull dimensional lattice polytope and Σ coincides with the normal fan Σ ∆ of ∆.Thus, a facet F of ∆ correspond to a ray σ F of Σ and we can reformulate (2.2),cyc( s Ψ ) = X F − h F, v F i V ( σ F ) , where the sum is over the facets F of ∆ and where v F is the primitive inner normalvector to F (see A.21). Assume that Ψ is concave, i.e. D Ψ is generated by global sections. Then∆ = ∆ Ψ is a (not necessarily full dimensional) lattice polytope. We set M (∆) = M ∩ L ∆ , N (∆) = M (∆) ∨ = N (cid:14)(cid:0) N ∩ L ⊥ ∆ (cid:1) , where L ∆ denotes the linear subspace of M R associated to the affine hull aff(∆)of ∆. We choose any m ∈ aff(∆) ∩ M . Then, the lattice polytope ∆ − m is fulldimensional in L ∆ = M (∆) R . Let Σ ∆ be the normal fan of ∆ − m in N (∆) R (seeA.20). The projection H : N → N (∆) is compatible with Σ and Σ ∆ and so, by2.1.7, it induces a proper toric morphism ϕ : X Σ → X Σ ∆ . We set ∆ ′ = ∆ − m andconsider the functionΨ ∆ ′ : N (∆) R −→ R , u min m ′ ∈ ∆ ′ h m ′ , u i . This is a strictly concave support function on Σ ∆ . By 2.1.16, the divisor D Ψ ∆ ′ isample and, D Ψ = ϕ ∗ D Ψ ∆ ′ + div( χ − m ) . (2.4)2.2. Toric schemes over valuation rings of rank one.
In this subsection wesummarize some facts from the theory of toric schemes over valuation rings of rankone developed in [32] and [35].Let K be a field equipped with a non-archimedean absolute value | · | . Then wehave a valuation ring K ◦ with valuation val := − log | · | of rank one and a valuegroup Γ := val( K × ). As usual, we fix a free abelian group M of rank n with dual N . Then we denote by T S the split torus T S = Spec ( K ◦ [ M ]) over S = Spec( K ◦ )with generic fiber T = Spec( K [ M ]) and special fiber T ˜ K = Spec( ˜ K [ M ]) over theresidue field ˜ K . Definition 2.2.1.
A ( T S -) toric scheme is an integral separated S -scheme X suchthat the generic fiber X η contains T as an open subset and the translation actionof T on itself extends to an algebraic action T S × S X → X over S .Note that by [32, Lemma 4.2] a toric scheme X is flat over S and the genericfiber X η is a T -toric variety over K . Definition 2.2.2.
Let X be a T -toric variety and X a T S -toric scheme. Then X is called a ( T S -) toric model of X if it comes with a fixed isomorphism X η ≃ X which is the identity on T . If X and X ′ are toric models of X and α : X → X ′ is an S -morphism, we say that α is a morphism of toric models if its restriction to T is the identity. A Γ -admissible cone σ in N R × R ≥ is a strongly convex cone which is ofthe form σ = k \ i =1 { ( u, r ) ∈ N R × R ≥ | h m i , u i + l i · r ≥ } with m i ∈ M, l i ∈ Γ , i = 1 , . . . , k. For such a cone σ , we define K [ M ] σ := n X m ∈ M α m χ m ∈ K [ M ] | h m, u i + val( α m ) · r ≥ ∀ ( u, r ) ∈ σ o . This is an M -graded K ◦ -subalgebra of K [ M ] which is an integrally closed domain.Hence, we get an affine normal T S -toric scheme U σ := Spec( K [ M ] σ ) over S . Itis finitely generated as a K ◦ -algebra if and only if the following condition (F) isfulfilled: OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 23 (F) The value group Γ is discrete or the vertices of σ ∩ ( N R ×{ } ) are containedin N Γ × { } .Hence U σ is of finite type if and only if (F) holds. If Γ is discrete or divisible, then(F) is always satisfied. A fan in N R × R ≥ is called Γ -admissible if it consists of Γ-admissible cones.Given such a fan e Σ, the affine T S -toric schemes U σ , σ ∈ e Σ, glue together along theopen subschemes corresponding to the common faces as in the case of toric varieties.This leads to a normal T S -toric scheme X e Σ = [ σ ∈ e Σ U σ (2.5)over S . It is universally closed if and only if e Σ is complete, i. e. | e Σ | = N R × R ≥ (see [32, Proposition 11.8]). We have the following generalization of the classification of toric varietiesover a field: By [35, Theorem 3], e Σ X e Σ defines a bijection between the sets of(i) Γ-admissible fans in N R × R ≥ whose cones satisfy condition (F),(ii) isomorphism classes of normal T S -toric schemes of finite type over S .In this case, X e Σ is proper over S if and only if e Σ is complete.
It is also possible to describe toric schemes in terms of polyhedra in N R . Let σ be a cone in N R × R ≥ . For r ∈ R ≥ , we set σ r := { u ∈ N R | ( u, r ) ∈ σ } . Then σ σ defines a bijection between the set of Γ-admissible cones in N R × R ≥ ,which are not contained in N R × { } , and the set of strongly convex Γ-rationalpolyhedra in N R . The inverse map is given by Λ c(Λ), where c(Λ) is the closureof R > (Λ × { } ) in N R × R ≥ . Let e Σ be a Γ-admissible fan. Then we have two kinds of cones σ in e Σ:(i) If σ is contained in N R × { } , then K [ M ] σ = K [ M σ ]. Hence, U σ is equalto the toric variety U σ associated to the cone σ (see 2.1.2) and it iscontained in the generic fiber of X e Σ .(ii) If σ is not contained in N R × { } , then Λ := σ is a strongly convex Γ-rational polyhedron in N R . It easily follows that K [ M ] σ is equal to K [ M ] Λ := n X m ∈ M α m χ m ∈ K [ M ] | h m, u i + val( α m ) ≥ ∀ u ∈ Λ o . Thus, U σ equals the T S -toric scheme U Λ := Spec( K [ M ] Λ ). The genericfiber of U Λ = U σ is identified with the T -toric variety U σ = U rec(Λ) .We set Σ := { σ | σ ∈ e Σ } and Π := { σ | σ ∈ e Σ } . Then Σ is a fan in N R and Π isa Γ-rational strongly convex polyhedral complex in N R . Now we can rewrite theopen cover (2.5) as X e Σ = [ σ ∈ Σ U σ ∪ [ Λ ∈ Π U Λ (2.6)using the same gluing data. The generic fiber of this toric scheme is the T -toricvariety X Σ associated to Σ, i. e. X e Σ is a toric model of X Σ . If the value group Γ is discrete, then the special fiber X s is reduced for X := X e Σ if and only if the vertices of all Λ ∈ Π are contained in N Γ . If thevaluation is not discrete, then X s is always reduced (see [32, Proposition 7.11 and7.12]). Conversely, if we start with an arbitrary Γ-rational strongly convex polyhe-dral complex Π, we can’t expect thatc(Π) := { c(Λ) | Λ ∈ Π } ∪ { rec(Λ) × { } | Λ ∈ Π } is a fan in N R × R ≥ . Burgos and Sombra have shown that the correspondenceΠ c(Π) gives a bijection between complete Γ-rational strongly convex polyhedralcomplexes in N R and complete Γ-admissible fans in N R × R ≥ (see [13, Corollary3.11]). We set X Π := X c(Π) and we identify the generic fiber X Π ,η with the toricvariety X rec(Π) .We end this subsection with a description of the orbits of a toric scheme. As in2.2.7, we consider a Γ-admissible fan e Σ with associated fan Σ := { σ | σ ∈ e Σ } in N R and with associated polyhedral complex Π := { σ | σ ∈ e Σ } in N R . Notation 2.2.10.
For Λ ∈ Π, let L Λ be the R -linear subspace of N R associated tothe affine space aff(Λ). We set N (Λ) = N/ ( N ∩ L Λ ) , M (Λ) = N (Λ) ∨ = M ∩ L ⊥ Λ , generalizing the notation in 2.1.4. Furthermore, we define f M (Λ) = { m ∈ M (Λ) | h m, u i ∈ Γ ∀ u ∈ Λ } , e N (Λ) = f M (Λ) ∨ . Because of the Γ-rationality of Λ, the lattice f M (Λ) is of finite index in M (Λ). Wedefine the multiplicity of a polyhedron Λ ∈ Π bymult(Λ) = (cid:2) M (Λ) : f M (Λ) (cid:3) . (2.7)Let Λ ′ ∈ Π and Λ a face of Λ ′ . The local cone (or angle ) of Λ ′ at Λ is defined as ∠ (Λ , Λ ′ ) := { t ( u − v ) | u ∈ Λ ′ , v ∈ Λ , t ≥ } . This is a polyhedral cone.There is a bijection between torus orbits of X e Σ and the two kinds of cones in e Σcorresponding to cones in Σ as well as polyhedra in Π.First, the cones in Σ correspond to the T -orbits on the generic fiber X Σ via σ O ( σ ) as in 2.1.4. We denote by V ( σ ) the Zariski closure of O ( σ ) in X e Σ . Then V ( σ ) is a scheme of relative dimension n − dim( σ ) over S . Moreover, we have τ (cid:22) σ if and only if O ( σ ) ⊆ V ( τ ). Proposition 2.2.11.
Using the notation above, there is a canonical isomorphismfrom V ( σ ) to the Spec( K ◦ [ M ( σ )]) -toric scheme over K ◦ associated to the Γ -admissiblefan e Σ( σ ) := { ( π σ × id R ≥ )( ν ) | ν ∈ Σ , ν ⊃ σ } in N ( σ ) R × R ≥ . The associatedpolyhedral complex Π( σ ) in N ( σ ) R = N R / h N ∩ σ i R is given by Π( σ ) = { Λ + h N ∩ σ i R | Λ ∈ Π , rec(Λ) ⊇ σ } . Proof.
This follows from [32, Proposition 7.14]. (cid:3)
The polyhedra of Π correspond to the T ˜ K -orbits on the special fiber of X e Σ . Thisbijective correspondence is given by O : Λ red(trop − (ri Λ)) , where red is the reduction map from 1.1.7, trop is the tropicalization map from2.4.5 and ri(Λ) is the relative interior of Λ from A.1. For details, we refer to [32,Proposition 6.22 and 7.9]. For Λ ∈ Π, we denote by V (Λ) the Zariski closure of O (Λ) in X e Σ . Then V (Λ) is contained in the special fiber of X e Σ and has dimension n − dim(Λ). For Λ , Λ ′ ∈ Π and σ ∈ Σ, we haveΛ (cid:22) Λ ′ ⇐⇒ O (Λ ′ ) ⊆ V (Λ) and σ (cid:22) rec(Λ) ⇐⇒ O (Λ) ⊆ V ( σ ) . (2.8) OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 25
Proposition 2.2.12.
The variety V (Λ) is equivariantly (but non-canonically) iso-morphic to the Spec( ˜ K [ f M (Λ)]) -toric variety over ˜ K associated to the fan Π(Λ) = { ∠ (Λ , Λ ′ ) + L Λ | Λ ′ ∈ Π , Λ ′ ⊇ Λ } (2.9) in e N (Λ) R = N (Λ) R = N R / h N ∩ L Λ i R .Proof. This is [32, Proposition 7.15]. (cid:3)
In particular, there is a bijection between vertices of Π and the irreduciblecomponents of the special fiber of X e Σ . For each v ∈ Π , the component V ( v ) isa toric variety over e K with torus associated to the character lattice { m ∈ M |h m, v i ∈ Γ } and given by the fan Π( v ) = { R ≥ (Λ ′ − v ) | Λ ′ ∈ Π , Λ ′ ∋ v } in N R .2.3. Toric Cartier divisors on toric schemes.
We extend the theory of T -Cartier divisors to the case of toric schemes over a valuation ring of rank one. Thisgeneralizes [11, § 3.6] where the case of discrete valuation is handled and which weuse as a guideline. We keep the notations of the previous subsection. Definition 2.3.1. A T S -Cartier divisor on a T S -toric scheme X is a Cartier divisor D on X which is invariant under the action of T S on X , i. e. we have µ ∗ D = p ∗ D denoting by µ : T S × X → X the toric action and by p : T S × X → X the secondprojection.In the following, we consider the T S -toric scheme X given by a Γ-admissiblefan e Σ in N R × R ≥ with associated fan Σ := { σ | σ ∈ e Σ } in N R and with associatedpolyhedral complex Π := { σ | σ ∈ e Σ } in N R as in 2.2.7. A function ψ : | e Σ | → R is called a Γ -admissible virtual support function on e Σ if for every σ ∈ e Σ, there is m σ ∈ M and l σ ∈ Γ such that ψ ( u, r ) = h m σ , u i + l σ r (2.10)for all ( u, r ) ∈ σ . By restriction to e Σ = Σ (resp. e Σ = Π), we get a virtual supportfunction ψ on Σ (resp. a Γ-rational piecewise affine function ψ on Π).Conversely, suppose that we have ( m σ , l σ ) ∈ M × R for every σ ∈ e Σ satisfyingthe condition h m σ , u i + l σ r = h m σ ′ , u i + l σ ′ r for all ( u, r ) ∈ σ ∩ σ ′ and all σ, σ ′ ∈ e Σ . (2.11)Then it is clear that (2.10) defines a Γ-admissible virtual support function on e Σ.On each open subset U σ , the vector ( m σ , l σ ) determines a rational function α − σ χ − m σ , where α σ ∈ K × is any element with val( α σ ) = l σ . For σ, σ ′ ∈ e Σ,condition (2.11) implies that this function is regular and non-vanishing on U σ ∩ U σ ′ = U σ ∩ σ ′ . By construction { U σ } σ ∈ e Σ is an open covering of X . Thus, ψ definesa Cartier divisor D ψ = (cid:8)(cid:0) U σ , α − σ χ − m σ (cid:1)(cid:9) σ ∈ e Σ , (2.12)where α σ ∈ K × is any element with val( α σ ) = l σ . The divisor D ψ only dependson ψ and not on the particular choice of the defining vectors and elements α σ . Itis easy to see that D ψ is T S -invariant. Theorem 2.3.3.
Let e Σ be a Γ -admissible fan in N R × R ≥ with corresponding T S -toric scheme X = X e Σ .(i) The assignment ψ D ψ is an isomorphism between the group of Γ -admissible virtual support functions on e Σ and the group of T S -Cartierdivisors on X . (ii) The divisors D ψ and D ψ ′ are rationally equivalent if and only if ψ ′ − ψ isaffine. For the proof, we need the following helpful lemma.
Lemma 2.3.4.
Let σ a Γ -admissible cone in N R × R ≥ . Then for each T S -Cartierdivisor D on U σ we have D = div( αχ m ) for some m ∈ M and α ∈ K × .Proof. Using the decomposition (2.6), we may assume that U σ = U Λ for Λ ∈ Π with c(Λ) = σ as the case σ ∈ Σ is well-known. Let us consider the K ◦ -algebra A := O U Λ ( U Λ ) = K [ M ] Λ and the fractional ideal I := Γ( U Λ , O U Λ ( − D ))of A . Since D is T S -invariant, the K ◦ -module I is graded by M , i. e. we can write I = L m ∈ M I m , where I m is a K ◦ -submodule contained in Kχ m . Because K ◦ is avaluation ring of rank one, either I m = (0) or I m = K ◦◦ α m χ m or I m = K ◦ α m χ m or I m = Kχ m for some m ∈ M , α m ∈ K × . Since I is finitely generated as an A -module, we deduce I = M α m χ m ∈ I K ◦ α m χ m . (2.13)Now we fix a point p ∈ O (Λ). Then D is principal on an open neighborhood U of p in U Λ . We may assume that U = Spec( A h ) for some h ∈ A with h ( p ) = 0.Hence, D | U = div( f ) | U for some f ∈ K ( M ) × = Quot( A ) × . This implies I h = O U Λ ( − D )( U ) = f · O U Λ ( U ) = f · A h . In particular, f ∈ I h , and by (2.13), we can write f = X i c i h k α m i χ m i with c i ∈ K ◦ \ { } , k ∈ N . Since α m i χ m i /f ∈ O U Λ ( U ) and p ∈ U , we deduce ( α m i χ m i /f ) ( p ) = 0 for some i . There exists an open neighborhood W ⊆ U of p on which α m i χ m i /f is non-vanishing and thus, div( α m i χ m i ) | W = div( f ) | W = D | W . (2.14)By [35, Corollary 2.12 (c)], we have an injective homomorphism D cyc( D )from the group of Cartier divisors on U Λ to the group of Weil divisors on U Λ ,which restricts to a homomorphism of the corresponding groups of T S -invariantdivisors. The T S -invariant prime (Weil) divisors are exactly the T S -orbit closuresof codimension one. By (2.8), p ∈ O (Λ) ⊆ \ v ∈ Π ,v (cid:22) Λ V ( v ) ∩ \ τ ∈ rec(Π) ,τ (cid:22) rec(Λ) V ( τ ) , and therefore, W meets each T S -invariant prime divisor of U Λ . Thus, equation(2.14) implies cyc( D ) = cyc (div( α m i χ m i )) and hence D = div( α m i χ m i ). (cid:3) Proof of Theorem 2.3.3. (i) Let σ be a Γ-admissible virtual support function on e Σgiven by defining vectors { ( m σ , val( α σ )) } σ ∈ e Σ . Then, by the construction in 2.3.2, D ψ is a well-defined T S -Cartier divisor on X . It is easy to see that this assignmentdefines a group homomorphism.To prove injectivity, we assume that ψ maps to the zero divisor ( X , σ ∈ e Σ, the function α − σ χ − m σ is invertible on U σ or equivalently, ψ ( u, r ) = h m σ , u i + val( α σ ) r = 0 for all ( u, r ) ∈ σ. Therefore, ψ is identically zero and we proved the injectivity. OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 27
For surjectivity, let D be an arbitrary T S -Cartier divisor on X . By Lemma2.3.4, there exist, for each σ ∈ e Σ, elements α σ ∈ K × and m σ ∈ M , such that D | U σ = div( α σ χ m σ ) | U σ . Since D is a Cartier divisor, we have, for σ, σ ′ ∈ e Σ,div( α σ χ m σ ) | U σ ∩ σ ′ = div( α σ ′ χ m σ ′ ) | U σ ′∩ σ , which implies thatval( α σ ) r + h m σ , u i = val( α σ ′ ) r + h m σ ′ , u i for all ( u, r ) ∈ σ ∩ σ ′ . (2.15)For each σ ∈ e Σ, we set ψ ( u, r ) := h− m σ , u i − val( α σ ) r for all ( u, r ) ∈ σ . By (2.15),this determines a well-defined Γ-admissible virtual support function ψ : N R → R and, by (2.12), ψ maps to D .(ii) We claim that a T S -Cartier divisor on X is principal if and only if it hasthe form div( αχ m ) for α ∈ K × , m ∈ M . Indeed, let D be any principal T S -Cartierdivisor on X , i. e. D = div( f ) for some f ∈ K ( X ) × . The support of D is disjointfrom the torus T . Therefore, when regarded as an element of K ( T ) × , the divisorof f | T is zero. This implies f ∈ K [ M ] × and thus, f = αχ m for some α ∈ K × and m ∈ M .Using this equivalence, statement (ii) follows easily from (i). (cid:3) Let X be a toric scheme over S . A toric line bundle on X is a pair ( L , z )consisting of a line bundle L on X and a non-zero element z in the fiber L x ofthe unit point x ∈ X η . A toric section is a rational section s of a toric line bundlewhich is regular and non-vanishing on the torus T ⊆ X η and such that s ( x ) = z .As in 2.1.12, each T -Cartier divisor D on X gives us a toric line bundle ( O ( D ) , s D ( x ))with toric section s D . Usually, we are concerned with toric schemes X := X e Σ as-sociated to a Γ-admissible fan e Σ in N R × R . Then each Γ-admissible virtual supportfunction ψ on e Σ defines a toric line bundle with toric section (( O ( D ψ ) , s D ψ ( x )) , s D ψ ),which we simply denote by ( L ψ , s ψ ).Let ( X Σ , D Ψ ) be a toric variety with a T -Cartier divisor. A toric model of( X Σ , D Ψ ) is a triple ( X , D, e ) consisting of a T S -toric model X of X Σ associatedto a Γ-admissible fan e Σ in N R × R ≥ , a T S -Cartier divisor D on X and an integer e > D | X Σ = eD Ψ . Remark 2.3.6.
In our applications, we consider a complete Γ-rational polyhedralcomplex Π or equivalently a complete Γ-admissible cone e Σ = c(Π). As in 2.2.9,we get a universally closed T S -toric scheme X Π := X c(Π) . In this case, the map ψ → ψ from 2.3.2 gives a bijective correspondence between Γ-admissible virtualsupport functions on e Σ and Γ-lattice functions on Π.For a Γ-lattice function φ with corresponding Γ-admissible virtual support func-tion ψ , 2.3.2 gives an associated Cartier divisor D φ := D φ . We conclude fromTheorem 2.3.3 that φ D ψ is an isomorphism from the group of Γ-lattice func-tions on Π onto the group of T -Cartier divisors on X Π . Moreover, the Cartierdivisors D φ and D φ ′ are linearly equivalent if and only if φ ′ − φ is affine. Theorem 2.3.7.
Suppose that the value group Γ is discrete or that Γ is divisible.Let Σ be a complete fan in N R and Ψ a virtual support function on Σ . Then theassignment (Π , φ ) ( X Π , D φ ) gives a bijection between the sets of(i) pairs (Π , φ ) , where Π is a complete Γ -rational polyhedral complex in N R and rec(Π) = Σ , and φ is a Γ -lattice function on Π with rec( φ ) = Ψ ;(ii) isomorphism classes of proper normal toric models ( X , D, of ( X Σ , D Ψ ) .Proof. Let (Π , φ ) be a pair as in (i) and let { ( m Λ , val( α Λ )) } Λ ∈ Π be defining vectorsof φ . Then D φ | X Σ = (cid:8)(cid:0) U Λ , α − χ − m Λ (cid:1)(cid:9)(cid:12)(cid:12) X rec(Π) = (cid:8)(cid:0) U rec(Λ) , χ − m Λ (cid:1)(cid:9) = D rec( φ ) = D Ψ . Hence, ( X Π , D φ ,
1) is a toric model of ( X Σ , D Ψ ). The statement follows from 2.2.5and Theorem 2.3.3. (cid:3) Now we describe the restriction of T S -Cartier divisors to closures of orbits. Butwe are only interested in the case of orbits lying in the special fiber. The other casecan be handled analogously to [11, Proposition 3.6.12].Let e Σ be a Γ-admissible fan in N R × R ≥ with associated fan Σ := e Σ in N R and associated polyhedral complex Π := e Σ in N R . Let ψ be a Γ-admissible virtualsupport function on e Σ and let D ψ be the associated T S -Cartier divisor. We alsoconsider the associated Γ-lattice function φ := ψ on Π.Let Λ ∈ Π be a polyhedron. Then we have φ ( u ) = h m ∆ , u i + l ∆ on ∆ for some m ∆ ∈ M and l ∆ ∈ Γ. We assume φ | Λ = 0. Using Notation 2.2.10 and (2.9), wedefine a virtual support function φ (Λ) on the rational fan Π(Λ) in N (Λ) R given bythe following defining vectors { m π } π ∈ Π(Λ) . For each cone π ∈ Π(Λ), let Λ π ∈ Πbe the unique polyhedron with Λ (cid:22) Λ π and ∠ (Λ , Λ π ) + L Λ = π . The condition φ | Λ = 0 implies that m Λ π ∈ L ⊥ Λ and h m Λ π , u i = − l Λ π ∈ Γ for all u ∈ Λ. Hence, m Λ π lies in f M (Λ). We set m π := m Λ π . Proposition 2.3.8.
We use the above notation. If φ | Λ = 0 , then D ψ properlyintersects the orbit closure V (Λ) . Moreover, the restriction of D ψ to V (Λ) is thedivisor D φ (Λ) .Proof. The T S -Cartier divisor D ψ is given on U Λ by α − χ − m Λ , where α Λ ∈ K × isany element of K × with val( α Λ ) = l Λ . If φ | Λ = 0, then val( α Λ )+ h m Λ , u i = 0 for all u ∈ Λ. Thus, the local equation α − χ − m Λ of D ψ in U Λ is a unit in O ( U Λ ) = K [ M ] Λ .Hence, the orbit O (Λ) ⊆ U Λ does not meet the support of D ψ and hence V (Λ) and D ψ intersect properly. Furthermore, D ψ | V (Λ) = (cid:8)(cid:0) U Λ π ∩ V (Λ) , α − π χ − m Λ π | U Λ π ∩ V (Λ) (cid:1)(cid:9) π ∈ Π(Λ) . Using the non-canonical isomorphism ˜ K [ U π ] ≃ ˜ K [ U Λ π ∩ V (Λ)], we get D ψ | V (Λ) = (cid:8)(cid:0) U π , χ − m π (cid:1)(cid:9) π ∈ Π(Λ) = D φ (Λ) proving the claim. (cid:3) Proposition 2.3.9.
Let Π be a complete Γ -rational polyhedral complex in N R withassociated T S -toric scheme X Π and let φ be a concave Γ -lattice function on Π withassociated T S -Cartier divisor D φ on X Π . Let Λ ∈ Π be a k -dimensional polyhedronand v ∈ ri(Λ) . Then we have mult(Λ) deg D φ ( V (Λ)) = ( n − k )! vol M (Λ) ( ∂φ ( v )) , (2.16) where mult(Λ) is the multiplicity of Λ (see (2.7)) and ∂φ ( v ) is the sup-differentialof φ at v (see A.15) which is in fact a polytope contained in a translate of M (Λ) R .Proof. It follows from Proposition 2.2.12 that V (Λ) is a toric variety over ˜ K andthat the associated fan Π(Λ) is complete. We conclude from 2.1.3 that V (Λ) is aproper variety and hence the degree deg D φ ( V (Λ)) is well-defined. We have φ ( u ) = h m Λ , u i + l Λ on Λ for some m Λ ∈ M and l Λ ∈ Γ. Then D φ is rationally equivalent to D φ − m Λ − l Λ and ∂ ( φ − m Λ − l Λ )( v ) = ∂φ ( v ) − m Λ . Thus, replacing φ by φ − m Λ − l Λ does not change both sides of equation (2.16) and we may assume that φ | Λ = 0.By Proposition 2.3.8 and (2.3),deg D φ ( V (Λ)) = deg D φ (Λ) (cid:0) X Π(Λ) (cid:1) = ( n − k )! vol e M (Λ) (∆ φ (Λ) ) . OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 29
Using Remark A.16, we have ∆ φ (Λ) = ( ∂φ (Λ))(0) = ∂φ ( v ). We havevol e M (Λ) (∆ φ (Λ) ) = vol e M (Λ) ( ∂ ( φ (Λ))( v )) = 1 (cid:2) M (Λ) : f M (Λ) (cid:3) vol M (Λ) ( ∂ ( φ (Λ))( v )) , and hence we get the claim. (cid:3) Metrized line bundles on toric varieties.
In this subsection, we recallresults about toric metrics on a toric line bundle from [11, Chapter 4]. Note that in[11, § 4.1–4.3] the non-archimedean fields are not assumed to be discrete, in contrastto the rest of their Chapter 4.We fix the following notation. Let K be field which is complete with respectto a non-trivial non-archimedean absolute value | · | . Then we have a valuationval := − log | · | and a value group Γ := val( K × ) of rank one. We fix a free abeliangroup M of rank n with dual N and denote by T = Spec( K [ M ]) the n -dimensionalsplit torus over K .Let Σ be a complete fan in N R and X Σ the corresponding proper toric variety.Furthermore, let Ψ be a virtual support function on Σ and ( L, s ) the associatedtoric line bundle with toric section.
Definition 2.4.1.
A metric k · k on L is called toric if, for all p, q ∈ T an satisfying | χ m ( p ) | = | χ m ( q ) | for each m ∈ M , we have k s ( p ) k = k s ( q ) k .It easily follows from 2.1.13 that this definition is independent of the choice ofthe toric section s . Remark 2.4.2.
In [11, 4.2], the authors study the action of the analytic group T an on X anΣ and in particular, the action of the compact analytic subgroup S = { p ∈ T an | | χ m ( p ) | = 1 for all m ∈ M } . By [11, Proposition 4.2.15], we have for p ∈ T an , S · p = { q ∈ T an | | χ m ( p ) | = | χ m ( q ) | for all m ∈ M } . Hence, a metric k · k is toric if and only if the function p
7→ k s ( p ) k is invariant underthe action of S . Given an arbitrary metric k · k on L , we can associate to it a toric metricin the following way: For σ ∈ Σ, let s σ be a toric section of L which is regular andnon-vanishing in U σ . Then we set, for p ∈ U an σ , k s σ ( p ) k S := k s σ (˜ p ) k , where ˜ p ∈ U an σ is given by X m ∈ M σ α m χ m max m | α m || χ m ( p ) | . It is easy to check that this defines a toric metric k · k S on L . This process is called torification of k · k . Proposition 2.4.4.
Toric metrics are invariant under torification. Moreover, tori-fication is multiplicative with respect to products of metrized line bundles and con-tinuous with respect to uniform convergence of metrics.Proof.
This is established in [11, Proposition 4.3.4] and follows easily from thedefinition. (cid:3)
We have the tropicalization map trop : T an → N R , p trop( p ), wheretrop( p ) is the element of N R = Hom( M, R ) given by h m, trop( p ) i := − log | χ m ( p ) | . This defines a proper surjective continuous map. For details, we refer to [41, § 3].Let k · k be a toric metric on L . Then consider the following diagram T an log k s ( · ) k / / trop " " ❉❉❉❉❉❉❉❉ R N R ; ; ①①①①① . Since k · k is toric, log k s ( · ) k is constant along the fibers of trop. Moreover, trop issurjective and closed, and hence, there exists a unique continuous function on N R making the above diagram commutative. This leads to the following definition. Definition 2.4.6.
Let L = ( L, k · k ) be a metrized toric line bundle on X Σ and s a toric section of L . We define the function ψ L,s : N R −→ R , u log k s ( p ) k S , where p ∈ T an is any element with trop( p ) = u . The line bundle and the toricsection are usually clear from the context and we alternatively denote this functionby ψ k·k For an alternative description of ψ L,s , we consider the continuous map ρ : N R → T an defined, for each u ∈ N R , by the multiplicative norm ρ ( u ) : K [ M ] −→ R ≥ , X m ∈ M α m χ m max m ∈ M | α m | exp( − h m, u i ) . Then it is easy to see that ψ L,s ( u ) = log k s ( ρ ( u )) k for all u ∈ N R .We note that ρ is a homeomorphism of N R onto a canonical closed subset S ( T ) := ρ ( N R ) of T an called the skeleton of T an . Berkovich showed in [1, §6.3] that τ :=trop ◦ ρ is a strong proper deformation retraction from T an onto S ( T ). Proposition 2.4.8.
Let L = ( L, k · k ) and L ′ be metrized toric line bundles on X Σ with toric sections s and s ′ , respectively. Let ϕ : X Σ ′ → X Σ be a toric morphismwith corresponding linear map H as in 2.1.7. Then ψ L ⊗ L ′ ,s ⊗ s ′ = ψ L,s + ψ L ′ ,s ′ , ψ L − ,s − = − ψ L and ψ ϕ ∗ L,ϕ ∗ s = ϕ L,s ◦ H .
Moreover, if ( k · k n ) n ∈ N is a sequence of metrics on L that converges uniformly to k · k , then (cid:0) ψ k·k n (cid:1) n ∈ N converges uniformly to ψ k·k .Proof. This is established in the propositions 4.3.14 and 4.3.19 in [11] and followseasily from the definitions. (cid:3)
In order to characterize toric metrics by functions on N R , we need the Kajiwara-Payne tropicalization of X Σ introduced by [36] and [41]. This is a topo-logical space N Σ together with a tropicalization map X anΣ → N Σ . As a set, N Σ isa disjoint union of linear spaces N Σ = a σ ∈ Σ N ( σ ) R , where N ( σ ) = N/ h N ∩ σ i is the quotient lattice as in 2.1.4. We refer to [41] for adescription of the topology.The toric variety X Σ is the disjoint union of tori T N ( σ ) = Spec K [ M ( σ )] , σ ∈ Σ.Hence, we can define the tropicalization map trop : X anΣ −→ N Σ as the disjoint union of tropicalization maps trop : T an N ( σ ) → N ( σ ) R as defined in2.4.5. This is also a proper surjective continuous map. Especially, N Σ = trop( X anΣ )is a compact space. OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 31
By [11, 4.2.12], the canonical section ρ : N R → T an from 2.4.7 extends uniquelyto a continuous proper section ρ Σ : N Σ → X anΣ . Proposition 2.4.10.
Let Σ be a complete fan in N R and Ψ a virtual supportfunction on Σ . We set L = L Ψ . Then, for any metric k · k on L , the function ψ k·k − Ψ extends to a continuous function on N Σ . In particular, the function | ψ k·k − Ψ | is bounded.Moreover, the assignment k · k 7→ ψ k·k is a bijection between the sets of(i) toric metrics on L ;(ii) continuous functions ψ : N R → R sucht that ψ − Ψ can be extended to acontinuous function on N Σ .Proof. This is proved in Proposition 4.3.10 and Corollary 4.3.13 in [11]. The inversemap is given as follows: Let ψ be a function as in (ii) and { m σ } a set of definingvectors of Ψ. For each cone σ ∈ Σ, the section s σ = χ m σ s is a non-vanishing regularsection on U σ . Then we obtain a toric metric k · k ψ on L characterized by k s σ ( p ) k ψ := exp (cid:0) ( ψ − m σ )(trop( p )) (cid:1) (2.17)on U σ . (cid:3) Definition 2.4.11.
Let L be a toric line bundle on X Σ with toric section s andlet Ψ be the associated virtual support function on Σ. By Proposition 2.4.10, thefunction ψ := Ψ defines a toric metric on L . This metric is called the canonicalmetric of L . We denoted it by k · k can and write L can = ( L, k · k can ). Remark 2.4.12.
By [11, Proposition 4.3.15], the canonical metric only dependson the structure of toric line bundle of L and not on the choice of s . Proposition 2.4.13.
Let L , L ′ be toric line bundles on X Σ and let ϕ : X ′ Σ → X Σ be a toric morphism. Let σ ∈ Σ and ι : V ( σ ) → X Σ the closed immersion of 2.1.4.Then L ⊗ L ′ can = L can ⊗ L ′ can , L − = ( L can ) − , ϕ ∗ L can = ϕ ∗ L can , ι ∗ L can = ι ∗ L can . Proof.
The first two statements are established in [11, Proposition 4.3.16]. The lasttwo statements are the corollaries 4.3.20 and 4.3.18 in [11]. (cid:3)
Semipositive metrics and measures on toric varieties.
We continue ourstudy of metrized line bundles on a toric variety. We assume that the reader isfamiliar with the notation introduced in § 2.4. We give a complete characterizationof semipositive toric metrics in terms of concave functions. Moreover, we describethe Chambert-Loir measure of a semipositive toric metric as the Monge–Ampèremeasure of the associated concave function.In this subsection, we will use the following setup. Let K be an algebraicallyclosed field which is complete with respect to a non-trivial non-archimedean ab-solute value | · | . Then we have a valuation val := − log | · | and a value groupΓ := val( K × ) of rank one. We fix a free abelian group M of rank n with dual N and denote by T = Spec( K [ M ]) the n -dimensional split torus over K . We considera complete rational fan Σ in N R with associated T -toric variety X Σ . For a virtualsupport function Ψ on Σ, we denote by D Ψ the associated T -toric Cartier divisoron X Σ and by L := O ( D Ψ ) the associated toric line bundle. Let Π be a complete Γ-rational polyhedral complex in N R with rec(Π) = Σ,and let φ be a Γ-rational piecewise affine function on Π with rec( φ ) = Ψ. Let e > eφ is a Γ-lattice function given by the defining vectors { ( m Λ , l Λ ) } Λ ∈ Π in M × Γ. Then eφ defines a T S -Cartier divisor D eφ = (cid:8)(cid:0) U Λ , α − χ − m Λ (cid:1)(cid:9) Λ ∈ Π , where α Λ ∈ K × with val( α Λ ) = l Λ , and the pair (Π , eφ ) defines a toric model( X Π , D eφ , e ) of ( X Σ , D Ψ ) (see Theorem 2.3.7). We write L = O ( D eφ ) for thecorresponding toric line bundles. The model ( X Π , L , e ) induces an algebraic metric k · k L on L (see 1.2.6). Proposition 2.5.2.
In the above notation, the metric k · k L is toric. Moreover,the equalities ψ k·k L = ψ and k · k L = k · k ψ hold.Proof. Let Λ ∈ Π. Recall that U Λ := Spec( K [ M ] Λ ) is an algebraic K ◦ -model of U rec(Λ) . The associated formal scheme has generic fiber U ◦ rec(Λ) := { p ∈ U anrec(Λ) | p ( f ) ≤ ∀ f ∈ K [ M ] Λ } . Then U Λ is a trivialization of L on which s ⊗ e Ψ , considered as a rational section of L , corresponds to the rational function α − χ − m Λ (see [32, 4.9]). Hence we have k s Ψ ( p ) k L = | α − χ − m Λ ( p ) | /e for all p ∈ U ◦ rec(Λ) . Let u ∈ Λ and p ∈ T an with trop( p ) = u . Lemma 2.5.3 belowimplies that p ∈ U ◦ rec(Λ) and we obtainlog k s Ψ ( p ) k L = log | α − χ − m Λ ( p ) | /e = 1 e ( h m Λ , u i + l Λ ) = ψ ( u ) . This shows that the metric k · k L is toric. We deduce, by Definition 2.4.6, that ψ k·k L = ψ and, by Proposition 2.4.10, that k · k L = k · k ψ . (cid:3) Lemma 2.5.3.
Let Π be a complete Γ -rational polyhedral complex in N R and rec(Π) = Σ , and let red: X anΣ → X Π ,s be the reduction map from 1.1.7. Let Λ ∈ Π and p ∈ T an . Then trop( p ) ∈ Λ ⇐⇒ p ∈ U ◦ rec(Λ) ⇐⇒ red( p ) ∈ U Λ ,s . Proof.
By [32, Lemma 6.21], we have trop( p ) ∈ Λ if and only if | p ( f ) | ≤ f ∈ K [ M ] Λ or in other words p ∈ U ◦ rec(Λ) . By the description of the reduction mapin 1.1.7, this is equivalent to red( p ) ∈ U Λ ,s . (cid:3) Corollary 2.5.4.
Let ψ be a Γ -rational piecewise affine concave function on N R with rec( ψ ) = Ψ . Then the metric k · k ψ on L is induced by a toric model.Proof. As in the proof of [11, Theorem 3.7.3], we can show that there exists acomplete Γ-rational polyhedral complex Π in N R such that rec(Π) = Σ and ψ ispiecewise affine on Π. Since Γ is discrete or divisible, the complex Π induces aproper toric scheme X Π (see 2.2.5). Then Proposition 2.5.2 says that k · k ψ isinduced by a toric model ( X Π , D eψ , e ) of ( X Σ , D Ψ ). (cid:3) Proposition 2.5.5.
Let k · k be an algebraic metric on L . Then the function ψ k·k is Γ -rational piecewise affine.Proof. There exists a proper K ◦ -model ( X , L , e ) of ( X Σ , L ) inducing the metric k · k . Let { U i } i ∈ I be a trivialization of L . Then the subsets U ◦ i = red − (cid:0) U i ∩ f X (cid:1) form a finite closed cover of X anΣ . On U i the rational section s ⊗ e corresponds to arational function λ i ∈ K ( M ) × such that on U ◦ i we have k s ( p ) k = | λ i ( p ) | /e . OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 33
We write λ i = P m ∈ M α m χ m P m ∈ M β m χ m . Using the continuous map ρ : N R → T an from 2.4.7,we have on the closed subset Λ i := ρ − ( U ◦ i ∩ T an ) ⊆ N R , ψ k·k ( u ) = log k s ( ρ ( u )) k = log | λ i ( ρ ( u )) | /e = 1 e log (cid:18) max m ∈ M | α m | exp( − h m, u i ) (cid:19) − e log (cid:18) max m ∈ M | β m | exp( − h m, u i ) (cid:19) = 1 e min m ∈ M ( h m, u i + val( β m )) − e min m ∈ M ( h m, u i + val( α m )) . We see that ψ k·k | Λ i is the difference of two Γ-rational piecewise affine concavefunctions. Since { Λ i } i ∈ I is a finite closed cover of N R , we deduce that ψ k·k isΓ-rational piecewise affine (see A.10 and A.11). (cid:3) Next we study semipositive toric metrics on L . Proposition 2.5.6.
Let k · k be an algebraic metric on L .(i) If k · k is semipositive, then ψ k·k is concave.(ii) We assume that k · k is toric. Then k · k is semipositive if and only if ψ k·k is concave.Proof. (ii) Because each algebraic metric is formal (see 1.1.7), this follows fromCorollary 8.12 in [33].(i) For k · k semipositive, we have to show that ψ k·k is concave along any affineline. By a density argument, we may assume that the line is Γ-rational. Similarlyas in [11, proof of Proposition 4.7.1], we use pull-back with respect to a suitableequivariant morphism to reduce the concavity on the affine line to the case of P K and hence the claim follows from Corollary B.18 and (ii). (cid:3) Corollary 2.5.7.
Let k · k be a semipositive algebraic metric on L . Then the toricmetric k · k S is also algebraic and semipositive.Proof. By the propositions 2.5.6 (i), 2.5.5 and 2.4.10, the function ψ = ψ k·k is aconcave Γ-rational piecewise affine function with rec( ψ ) = Ψ. Then Corollary 2.5.4says that the metric k · k S = k · k ψ is algebraic and Proposition 2.5.6 (ii) impliessemipositivity. (cid:3) Theorem 2.5.8.
Let Ψ be a support function on the complete fan Σ in N R and set L = L Ψ . Then there is a bijection between the sets of(i) semipositive toric metrics on L ;(ii) concave functions ψ on N R such that the function | ψ − Ψ | is bounded;(iii) continuous concave functions on ∆ Ψ .The bijections are given by k · k 7→ ψ k·k and by ψ ψ ∨ . This theorem was proven by Burgos–Philippon–Sombra [11, Theorem 4.8.1] inthe case of a discrete or an archimedean absolute value. Note that the bijectionbetween (i) and (ii) holds also in case of a non-concave virtual support function Ψas then both sets are empty. This follows from the arguments in the proof below.
Proof.
The bijection between (ii) and (iii) follows from Proposition A.9. To provethe bijection between (i) and (ii), let k · k be a semipositive toric metric on L . ByProposition 2.4.10, the function | ψ k·k − Ψ | is bounded. Furthermore, there exists asequence ( k · k n ) n ∈ N of semipositive algebraic metrics converging to the toric metric k · k . Proposition 2.5.6 (i) says that the functions ψ k·k n are concave. By Proposition ψ k·k n ) n ∈ N converges uniformly to ψ k·k and hence the latter isalso concave.Conversely, let ψ be a concave function on N R such that | ψ − Ψ | is bounded.Then by Proposition A.14, there is a sequence of Γ-rational piecewise affine concavefunctions ( ψ k ) k ∈ N with rec( ψ k ) = Ψ, that uniformly converges to ψ . Because ψ k is apiecewise affine concave function with rec( ψ k ) = Ψ, the function ψ k − Ψ continuouslyextends to N Σ . We conclude that ψ − Ψ continuously to N Σ . By Proposition 2.4.10,we obtain toric metrics k·k ψ and k·k ψ k , k ∈ N , given as in (2.17). Then the sequenceof metrics ( k · k ψ k ) k ∈ N converges to k · k ψ . By Proposition 2.5.2, the metric k · k ψ k isalgebraic and therefore, by Proposition 2.5.6 (ii), semipositive. It follows that themetric k · k ψ is also semipositive. (cid:3) Remark 2.5.9.
Theorem 2.5.8 induces a bijective correspondence between semi-positive algebraic toric metrics k · k on L and concave Γ-rational piecewise affinefunctions ψ on N R with rec( ψ ) = Ψ. Moreover, such metrics always have a toricmodel. One direction follows from the propositions 2.5.5, 2.5.6 (i) and 2.4.10. Theconverse and the last claim are a consequence of Corollary 2.5.7.Now we characterize the Chambert-Loir measure associated to a semipositivetoric metric. Let ψ : N R → R be a concave function. We extend the Monge-Ampèremeasure M M ( ψ ) on N R (Definition A.17) to a measure M M ( ψ ) on N Σ by setting M M ( ψ )( E ) = M M ( ψ ) ( E ∩ N R )for any Borel subset E of N Σ . Theorem 2.5.10.
Let k · k be a semipositive toric metric on L and ψ = ψ k·k theassociated concave function on N R . Then trop ∗ (c ( L, k · k ) n ) = n ! M M ( ψ ) . Moreover, we have c ( L, k · k ) n = ( ρ Σ ) ∗ (cid:0) n ! M M ( ψ ) (cid:1) . Proof.
In the case of an algebraic metric, the claim follows from the same argumentsas in [11, Theorem 4.7.4] based in our case on Remark 2.5.9, 2.2.13 and Proposition2.3.9. Note that our assumption K algebraically closed yields that Γ is divisibleand hence the special fiber of a toric model X Π is reduced (see 2.2.8) and themultiplicity of a vertex of Π is one as well. This simplifies this argument a littlebit.The general case is based on the algebraic case as in [11, Corollary 4.7.5]. It isused here that the boundary X anΣ \ T an is a set of measure zero with respect toc ( L, k · k ) n by Corollary 1.4.5. (cid:3) At the end of this subsection, we quote a result about the restriction of semi-positive metrics to toric orbits which will be useful in the proof of the local heightformula. Recall that Ψ is a support function on Σ with associated toric line bun-dle (
L, s ) with toric section. Let σ be a cone of Σ and V ( σ ) the correspond-ing orbit closure with the structure of a toric variety (cf. 2.1.4). We denote by ι : V ( σ ) → X Σ the closed immersion. Let m σ ∈ M be a defining vector of Ψ at σ and set s σ = χ m σ s . By 2.1.15, the divisor D Ψ − m σ = div( s σ ) intersects V ( σ )properly and we can restrict s σ to V ( σ ) to obtain a toric section ι ∗ s σ of the toricline bundle O (cid:0) D (Ψ − m σ )( σ ) (cid:1) ≃ ι ∗ L . Proposition 2.5.11.
Let notation be as above and denote by F σ the face of ∆ Ψ associated to σ (see A.20). Let k · k be a semipositive toric metric on L . Then, forall m ∈ F σ − m σ , ψ ∨ ι ∗ L,ι ∗ s σ ( m ) = ψ ∨ L,s ( m + m σ ) . OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 35
Proof.
We can prove the statement as in [11, Proposition 4.8.8] since the discrete-ness of the valuation doesn’t play a role in the argument. (cid:3)
Local heights of toric varieties.
We prove a formula to compute the localheight of a normal toric variety over an algebraically closed non-archimedean field.Note that this is no restriction of generality as we can always achieve that by abase extension and as local heights are invariant under such a base extension. Thisgeneralizes work by Burgos, Philippon and Sombra who showed this formula overfields with a discrete valuation (cf. [11, Theorem 5.1.6]).Let K be an algebraically closed field which is complete with respect to a non-trivial absolute value | · | and denote by Γ = − log | K × | the associated value group.We fix a lattice M ≃ Z n with dual M ∨ = N and denote by T = Spec( K [ M ]) the n -dimensional split torus over K . Let Σ be a complete fan on N R and X Σ theassociated proper T -toric variety.Following [11, § 5.1] we define a local height for toric metrized line bundles thatdoes not depend on the choice of sections. Definition 2.6.1.
Let L i , i = 0 , . . . , t , be toric line bundles on X Σ equipped withDSP toric metrics. We denote by L can i the same line bundle endowed with thecanonical metric. Let Y be a t -dimensional prime cycle of X Σ and let ϕ : Y ′ → Y be a birational morphism such that Y ′ is projective. Recall the definition of localheights in 1.3.4. Then the toric local height of Y with respect to L , . . . , L t isdefined as λ tor L ,...,L t ( Y ) = λ ( ϕ ∗ L ,s ) ,..., ( ϕ ∗ L t ,s t ) ( Y ′ ) − λ ( ϕ ∗ L can0 ,s ) ,..., ( ϕ ∗ L can t ,s t ) ( Y ′ ) , where s . . . , s t are invertible meromorphic sections of L with | div( s ) | ∩ · · · ∩ | div( s t ) | ∩ Y = ∅ . (2.18)This definition extends to cycles by linearity. When L = · · · = L t = L , we writeshortly λ tor L ( Y ) = λ tor L ,...,L t ( Y ). Remark 2.6.2.
Proposition 1.3.5 (iii, v) implies that the toric local height doesnot depend on the choice of ϕ and Y ′ nor on the choice of the meromorphic sec-tions. When | div( s ) | , . . . , | div( s t ) | intersect properly on Y , then condition (2.18)is fullfilled. Proposition 2.6.3.
The toric local height is symmetric and multilinear in themetrized line bundles.Proof.
This follows easily from Proposition 1.3.5 (ii). (cid:3)
Definition 2.6.4.
Let L = ( L, k · k ) be a semipositive metrized toric line bundlewith a toric section s . Let Ψ be the corresponding support function on Σ and ψ L,s the associated concave function on N R . The roof function associated to ( L, s ) isthe concave function ϑ L,s : ∆ Ψ → R given by ϑ L,s = ψ ∨ L,s , where ψ ∨ L,s denotes the Legendre-Fenchel dual (see A.7). We will denote ϑ L,s by ϑ k·k if the line bundle and section are clear from the context. Let notation be as above. If k · k is an algebraic metric, then, by Proposition2.5.5 and A.12, the roof function ϑ k·k is piecewise affine concave. Theorem 2.6.6.
Let Σ be a complete fan on N R . Let L = ( L, k · k ) be a toricline bundle on X Σ equipped with a semipositive toric metric. We choose any toric section s of L and denote by Ψ the corresponding support function on Σ . Then, thetoric local height of X Σ with respect to L is given by λ tor L ( X Σ ) = ( n + 1)! Z ∆ Ψ ϑ L,s d vol M (2.19) where ∆ Ψ is the stability set of Ψ and vol M is the Haar measure on M R such that M has covolume one.Proof. The proof is completely analogous to [11, Theorem 5.1.6]. It is based oninduction relative to n and uses the induction formula 1.4.3 in an essential way. (cid:3) Remark 2.6.7.
The above theorem also holds in the archimedean case (see [11,Theorem 5.1.6]). In [11, § 5.1], the formula in (2.19) is extended to toric localheights with respect to distinct line bundles in the archimedean case or in case of adiscrete valuation. Moreover, the toric local height of a translated toric subvarietyand its behavior with respect to equivariant morphisms is studied. For arbitrarynon-archimedean fields, all these results remain valid and the arguments are thesame.3.
Global heights of varieties over finitely generated fields
In this section, we apply the results about toric local heights from the previoussection to global heights. First, we recall the theory of global heights over an M -field and we give an induction formula for global heights. In § 3.2, we considerMoriwaki’s M -field structure on a finitely generated field over a global field andwe prove Theorem IV from the introduction. In § 3.3, we apply Theorem III andTheorem IV to a fibration with toric generic fiber leading to a combinatorial formulafor a suitable global height of the fibration. In the last subsection, we illustrate thisformula for projective toric varieties over the function field of an elliptic curve.3.1. Global heights of varieties over an M -field. Diophantine geometry isusually considered over a number field or a function field. Osgood and Vojta re-alized a stunning similarity to Nevanlinna theory where the base field is the fieldof meromorphic function on C . The characteristic function in Nevanlinna theorycorresponds to the height in diophantine geometry. Inspired by the analogy and theproof of the second main theorem of Nevanlinna theory, Vojta found a new proofof the Mordell conjecture which was later generalized by Faltings to prove Lang’sconjecture for subvarieties of abelian varieties.In [24, Definition 2.1], the notion of an M -field was introduced to capture allthree situations simultaneously. In this subsection, we will recall the definition andthe theory of global heights over an M -field. This will be later applied to heightsover finitely generated fields introduced by Moriwaki. Definition 3.1.1.
Let K be a field and ( M, µ ) be a measure space endowed with apositive measure µ . We assume that for µ -almost every v ∈ M , we fix a non-trivialabsolute value | | v on K . Then K is called an M -field if log | f | v ∈ L ( M, µ ) for all f ∈ K × . We say that K satisfies the product formula if R M log | f | v dµ ( v ) = 0 for all f ∈ K × . Remark 3.1.2.
In [24, Definition 2.1], a more general definition of an M -field isgiven which is more suitable to uncountable fields as occurring in Nevanlinna theory.For countable fields, the definitions are the same. For our purposes in this paper,the above definition will suffice. We refer to [24] for the theory of global heightsover the more general M -fields noticing that this much more technical and that onereduces always to the above special case by passing to a sufficiently large finitelygenerated subfield. OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 37
Example 3.1.3.
Let F be a number field and M F be the set of places endowed withthe discrete measure µ given by µ ( v ) = [ F v : Q v ][ F : Q ] for v ∈ M F with completions F v and Q v . We denote by | | v the absolute value for the place v ∈ M F which extends thestandard euclidean or p -adic absolute value on Q . Then F is an M F -field satisfyingthe product formula. Example 3.1.4.
Let us fix the function field F := k ( C ) of a regular projectivecurve C over an arbitrary field k and let M F be the set of places of F correspondingto the closed points of C . We fix q ∈ R with q >
1. For v ∈ M F , we choose thestandard absolute value given by | α | v = q − ord v ( α ) (3.1)for α ∈ F . We endow M F with the discrete measure given by µ ( v ) = [ k ( v ) : k ].Then F is an M F -field satisfying the product formula.More generally, we consider a finite extension F of F . Then F is the functionfield of a regular projective curve Z over C . Let M F be the set of places correspond-ing to the closed points of Z . For v ∈ M F , we use the representative | | v whichextends | | v to K , where v := v | F . We endow F with the discrete measure givenby µ ( v ) = [ F v : F ,v ][ F : F ] µ ( v ) . Then F is an M F -field satisfying the product formula. A global field F is either a number field or a finite extension of the functionfield of a fixed regular projective curve over a finite field k . We endow F alwayswith the M F -field structure given in Examples 3.1.3 and 3.1.4. In the latter case,we choose q as the number of elements of k . In fact, we never use that k is finite andwe could choose any constant q >
1. For some results of the next subsection, it isrequired that k is countable (otherwise one has to state them differently restrictingto a countable subfield of k ). Definition 3.1.6.
Let K be a field with an absolute value | | v . We denote by C v the completion of an algebraic closure of the completion of K with respect to v ∈ M . Note that C v is a minimal algebraically closed complete field extending( K, | | v ) with residue field equal to an algebraic closure of ˜ K (see [6, Proposition3.4.1/3, Lemma 3.4.1/4]). By abuse of notation, we denote the absolute value of C v also by | | v .Let X be a variety over K . We set X v := X × K Spec( C v ). If v is archimedean,then C v = C and we denote by X an v = X v ( C v ) the complex analytic space associatedto X . If v is non-archimedean, then X an v is the Berkovich analytic space associatedto X v over C v as defined in 1.1.4. We call X an v the analytification of X with respectto v (or | · | v ). In the following, K is always an M -field. Our goal is to define an M -metricon a line bundle L over the proper variety X over K . For µ -almost every v ∈ M , wehave an associated absolute value | | v . An ( M -)metric on L is a family of metrics k · k v on L an v for µ -almost every v ∈ M as above. The corresponding metrized linebundle is denoted by L = ( L, ( k · k v ) v ).An M -metric on L is said to be semipositive if k · k v is semipositive for µ -almostall v ∈ M (cf. Definition 1.3.1 and Remark 1.3.11). Moreover, a metrized linebundle L is DSP if there are semipositive metrized line bundles M , N on X suchthat L = M ⊗ N − . Let Z be a t -dimensional cycle on X and ( L i , s i ), i = 0 , . . . , t , DSP metrizedline bundles on X with invertible meromorphic sections such that | div( s ) | ∩ · · · ∩ | div( s t ) | ∩ | Z | = ∅ . For v ∈ M , we set for the local height at v , λ ( L ,s ) ,..., ( L t ,s t ) ( Z, v ) := λ c div( s ) v ,..., c div( s t ) v ( Z v ) , where c div( s i ) v is the pseudo-divisor on X v induced by c div( s i ) (cf. Example 1.2.9). A t -dimensional prime cycle Y of X is called integrable with respect toDSP metrized line bundles L i , i = 0 , . . . , t , on X if there is a birational map ϕ : Y ′ → Y from a projective variety Y ′ and invertible meromorphic sections s i of ϕ ∗ L i , i = 0 , . . . , t , with div( s ) , . . . , div( s t ) intersecting properly, such that thefunction M −→ R , v λ ( ϕ ∗ L ,s ) ,..., ( ϕ ∗ L t ,s t ) ( Y ′ , v ) (3.2)is µ -integrable on M . A t -dimensional cycle is integrable if its components areintegrable. Let Y be a prime cycle. Then there exists a generically finite surjectivemorphism ϕ : Y ′ → Y from a proper variety Y ′ and invertible meromorphic sections s i of ϕ ∗ L i , i = 0 , . . . , t , satisfying | div( s ) | ∩ · · · ∩ | div( s t ) | = ∅ . By Chow’s lemma, we may even assume that Y is projective, ϕ is birational and thatdiv( s ) , . . . , div( s t ) intersect properly, but we don’t want to make these additionalassumptions here. Then Y is µ -integrable with respect to L , . . . , L t if and onlyif the µ -integrability of (3.2) holds. Moreover, the notion of integrability of cyclesis closed under tensor product and pullback of DSP metrized line bundles. Thisfollows from [27, 11.4, 11.5]. Definition 3.1.11.
Let X be a proper variety over an M -field K and Y a t -dimensional prime cycle on X which is integrable with respect to DSP metrizedline bundles L , . . . , L t on X . Let Y ′ and s , . . . , s t be as in 3.1.9. Then the globalheight of Y with respect to L, . . . , L t is defined ash L ,...,L t ( Y ) = Z M λ ( ϕ ∗ L ,s ) ,..., ( ϕ ∗ L t ,s t ) ( Y ′ , v ) d µ ( v ) . By linearity, we extend this definition to all t -dimensional cycles on X .Using Corollary 1.3.5 (iii), the archimedean analogon mentioned in Remark 1.3.11and the product formula of K , we see that this definition is independent of the choiceof the sections. Proposition 3.1.12.
The global height of integrable cycles has the following basicproperties:(i) It is multilinear and symmetric with respect to tensor products of DSPmetrized line bundles.(ii) Let ϕ : X ′ → X be a morphism of proper varieties over K and let Z ′ be a t -dimensional cycle such that ϕ ∗ Z ′ is integrable with respect to DSPmetrized line bundles L , . . . , L t on X . Then we have h ϕ ∗ L ,...,ϕ ∗ L t ( Z ′ ) = h L ,...,L t ( ϕ ∗ Z ′ ) . Proof.
Using 3.1.10, we get the results by integrating the corresponding formulasstated in Proposition 1.3.5. (cid:3)
Theorem 3.1.13 (Global induction formula) . Let X be a d -dimensional propervariety over the M -field K . Let L , . . . , L d be DSP metrized line bundles on X and OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 39 let s d be any invertible meromorphic section of L d . If X is integrable with respectto L , . . . , L d and if cyc( s d ) is integrable with respect to L , . . . , L d − , then φ ( v ) := Z X an v log k s d k d,v c ( L ,v ) ∧ · · · ∧ c ( L d − ,v ) is in L ( M, µ ) and h L ,...,L d ( X ) = h L ,...,L d − (cyc( s d )) − Z M F φ ( v ) dµ ( v ) . Proof.
We may assume that K is infinite as otherwise M has measure zero andall heights are zero as well. By Proposition 3.1.12 and Chow’s lemma, we mayassume that X is projective and there are invertible meromorphic sections s j of L j for j = 0 , . . . , d with | div( s ) | ∩ . . . ∩ | div( s d ) | = ∅ . Then the claim follows from Theorem 1.4.3. (cid:3)
Definition 3.1.14.
Let F be a global field with the structure ( M F , µ ) of an M F -field as in Example 3.1.5. Let X be a proper variety over a global field F and L aline bundle on X . We call an M F -metric on L quasi-algebraic if there exist a finitesubset S ⊆ M F containing the archimedean places and a proper algebraic model( X , L , e ) of ( X , L ) over the ring F ◦ S = { α ∈ F | | α | v ≤ ∀ v / ∈ S } , such that, for each v / ∈ S , the metric k · k v is induced by the localization( X × F ◦ S Spec C ◦ v , L ⊗ F ◦ S C ◦ v , e ) . Proposition 3.1.15.
Let X be a proper variety over a global field F . Then every d -dimensional cycle of X is µ -integrable with respect to DSP quasi-algebraic M F -metrized line bundles L , . . . , L d on X .Proof. This is [11, Proposition 1.5.14]. (cid:3)
Remark 3.1.16.
In the situation of Proposition 3.1.15 and with dim( X ) = d ,the hypotheses of the global induction formula 3.1.13 are always satisfied for anyinvertible meromorphic section s d of L d and hence there is only a finite number of v ∈ M F such that Z X an v log k s d k d,v c ( L ,v ) ∧ · · · ∧ c ( L d − ,v ) = 0and the global induction formula holds. Proposition 3.1.17.
Let F be a global field and F ′ a finite extension of F with theinduced structure of an M F ′ -field (see Example 3.1.5). Let X be an F -variety, L i , i = 0 , . . . , t , quasi-algebraic DSP metrized line bundles on X and Z a t -dimensionalcycle on X . We denote by π : X ′ → X the morphism, by Z ′ the cycle and by π ∗ L i the M F ′ -metrized line bundles obtained by base change to F ′ . Then h π ∗ L ,...,π ∗ L t ( Z ′ ) = h L ,..., L t ( Z ) . Proof.
This follows from [11, Proposition 1.5.10]. (cid:3)
Definition 3.1.18.
Let F be a global field and let L be a quasi-algebraic M F -metrized line bundle on the proper variety X over F . We say that L is nef if k · k issemipositive and, for each point p ∈ X ( F ), the global height h L ( p ) is non-negative. Example 3.1.19.
Let L = ( L, ( k · k v ) v ) be a semipositive quasi-algebraic metrizedline bundle. We assume that L is generated by small global sections, i. e. foreach point p ∈ X ( F ), there exists a global section s such that p / ∈ | div( s ) | andsup x ∈X an v k s ( x ) k v ≤ v ∈ M F . Then L is nef.The idea of the following proof was suggested to us by José Burgos Gil. Lemma 3.1.20.
Let V be a d -dimensional subvariety of X and let L , . . . , L d benef quasi-algebraic M F -metrized line bundles on X . Then, h L ,..., L d ( V ) ≥ . Proof.
We may assume that V = X and, by Chow’s Lemma and Proposition3.1.12 (ii), that there is a closed immersion ϕ : X ֒ → P mF . Consider the line bundle ϕ ∗ O P mF (1) on X , equipped with the metric ϕ ∗ k · k can ,v at one place v ∈ M F and with the metric ϕ ∗ k · k can ,v at all other places v = v . This M F -metrized linebundle is denoted by L . For each point p ∈ X ( F ) with function field F ( p ), thereexists a homogeneous coordinate x j , considered as a global section of O P mF (1), suchthat p / ∈ | div( ϕ ∗ x j ) | and hence,h L ( p ) = − X w ∈ M F ( p ) µ ( w ) log k x j ◦ ϕ ( p ) k can ,w + X w ∈ M F ( p ) w | v µ ( w ) log 2 ≥ log 2 > . (3.3)We extend the group of isomorphism classes of M F -metrized line bundles on X by Q -coefficients and write its group structure additively. For i = 1 , . . . , d , and apositive rational number ε , we set L i,ε := L i + ε L . Since L i is nef, we obtain, by(3.3) and the multilinearity of the heights, for each point p ∈ X ( F ),h L i ,ε ( p ) = h L i ( p ) + ε h L ( p ) ≥ ε log 2 > . (3.4)Now, we distinguish between number fields and function fields. First, let F bea number field. Since L i,ε is semipositive quasi-algebraic, there exists a sequence( L i,ε,k ) k ∈ N that converges to L i,ε and that consists of M F -metrized line bundleswhich are induced by vertically nef smooth hermitian Q -line bundles L i,ε,k , k ∈ N ,on a common model X ε,k over the ring of integers O F . By Proposition 1.3.5 (iv),we have, for all k ∈ N and all p ∈ X ( F ), (cid:12)(cid:12)(cid:12) h L i,ε,k ( p ) − h L i,ε ( p ) (cid:12)(cid:12)(cid:12) ≤ X v ∈ M F µ ( v ) d ( k · k i,ε,k,v , k · k i,ε,v ) . Note that the sum is finite and does not depend on p . Hence, by (3.4), there is a k ∈ N such that for all k ≥ k and all p ∈ X ( F ),h L i,ε,k ( p ) = h L i,ε,k ( p ) ≥ . Thus, for all k ≥ k , we have nef smooth hermitian Q -line bundles L ,ε,k , . . . , L d,ε,k .By a result of Zhang (see Moriwaki’s paper [40, Proposition 2.3 (1)]), the proposi-tion holds for such line bundles and hence we geth L ,ε,k ,..., L d,ε,k ( X ) = h L ,ε,k ,..., L d,ε,k ( X ) ≥ . (3.5)Next, let F be the function field of a smooth projective curve C over any field.Since L i,ε is semipositive quasi-algebraic, there exists a sequence ( L i,ε,k ) k ∈ N thatconverges to L i,ε and that consists of M F -metrized line bundles which are inducedby vertically nef Q -line bundles L i,ε,k , k ∈ N , on a common model π ε,k : X ε,k → C .As in the number field case, we can deduce, for sufficiently large k ’s and for all p ∈ X ( F ), h L i,ε,k ( p ) ≥ . (3.6) OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 41
By [30, Theorem 3.5 (d)], the height with respect to such algebraic metrized linebundles is given as an algebraic intersection number of the associated models. So,the inequality (3.6) just says that the line bundles L ,ε,k , . . . , L d,ε,k on the model X ε,k are horizontally nef. Using that they are also vertically nef, it follows fromKleiman’s Theorem [39, Theorem III.2.1] thath L ,ε,k ,..., L d,ε,k ( X ) = deg C (( π ε,k ) ∗ (c ( L ,ε,k ) . . . c ( L d,ε,k ))) ≥ . (3.7)Finally, by (3.5) for number fields and by (3.7) for function fields, continuity ofheights in the metrized line bundles yieldsh L ,..., L n ( X ) = lim ε → h L ,ε ,..., L d,ε ( X ) = lim ε → lim k →∞ h L ,ε,k ,..., L d,ε,k ( X ) ≥ , proving the lemma. (cid:3) Relative varieties over a global field.
Let F be a global field with thecanonical M F -field structure from Example 3.1.5. Let B be a b -dimensional normalproper variety over F with function field K = F ( B ).We first endow the field K with the structure of an M -field where M is a naturalset of places and where the positive measure is induced by fixed nef quasi-algebraic M F -metrized line bundles H , . . . , H b on B . This generalizes the M -fields obtainedby Moriwaki’s construction in [40, § 3] where the function field of an arithmeticvariety and a family of nef hermitian line bundles on B are considered (see also [27,Example 11.22] and [12, §2]).We consider a dominant morphism π : X → B of proper varieties over F ofrelative dimension n . The generic fiber X = X × B Spec( K ) of π is a proper varietyover K . Then we prove the main result of this section (Theorem 3.2.6) showingthat the intersection number h π ∗ H ,...,π ∗ H b , L ,..., L n ( X ) with respect to DSP quasi-algebraic M F -metrized line bundles L i is equal to the height h L ,...,L n ( X ) withrespect to induced M -metrized line bundles L i . Note that the first height is a sumof local heights over M F whereas the second is an integral over M . This generalizesTheorem 2.4 in [11] where the global field is a number field and only hermitianline bundles are considered. Our more general assumptions above on the metricsof the polarizations H , . . . , H b lead to the problem that also non-discrete non-archimedean places in M have to be considered leading to considerable difficultiesin the proof of the theorem. Let H , . . . , H b be nef quasi-algebraic line bundles on B . By Lemma 3.1.20,we deduce, for every one-codimensional prime cycle V on B ,h H ,...,H b ( V ) ≥ . (3.8)Let B (1) be the set of prime cycles of B of codimension 1. By (3.8), the cycle V ∈ B (1) induces a non-archimedean absolute value on K given, for f ∈ K , by | f | V = e − h H ,...,Hb ( V ) ord V ( f ) , (3.9)where ord V is the discrete valuation associated to the regular local ring O B,V . Weequip B (1) with the counting measure µ fin .Let us fix a place v ∈ M F . Then we define the generic points of B an v as B gen v = B an v \ [ V ∈ B (1) V an v . Since each V ∈ B (1) is contained in the support of the divisor of a rational function,a point p ∈ B an v lies in B gen v if and only if, for each f ∈ K × , p does not lie in the analytification (with respect to v ) of the support of div( f ). Thus every p ∈ B gen v defines a well-defined absolute value on K given by | f | v,p = | f ( p ) | . (3.10)If v is non-archimedean, then this absolute value is just p . Let µ ( v ) be the weightof the product formula for F in v as given in Example 3.1.3 and Example 3.1.4. On B an v we have the positive measure µ v = µ ( v ) · c ( H ,v ) ∧ · · · ∧ c ( H b,v ) , which is a weak limit of smooth volume forms in the archimedean case (cf. [11,Definition 1.4.6]) and defined as in Definition 1.3.7 in the non-archimedean case.Each V an v , V ∈ B (1) , has measure zero with respect to µ v by Corollary 1.4.5 (non-archimedean case) and by [15, Corollaire 4.2] (archimedean case). Since F is count-able, the set B (1) is also countable and therefore B an v \ B gen v has measure zero withrespect to µ v . So we get a positive measure on B gen v , which we also denote by µ v .In conclusion, we obtain a measure space( M , µ ) = ( B (1) , µ fin ) ⊔ ( G v ∈ M F B gen v , G v ∈ M F µ v ) , (3.11)which is in bijection with a set of absolute values on K .The following shows that ( K, M , µ ) is an M -field: Proposition 3.2.2.
Let f ∈ K × , then the function M → R , w log | f | w isintegrable with respect to µ and we have the product formula Z M log | f | w d µ ( w ) = 0 . Proof.
Let f ∈ K × be a non-zero rational function on B . Then for almost every V ∈ B (1) , we have f ∈ O × B,V . Hence, the function on B (1) given by V log | f | V is µ fin -integrable.For every v ∈ M F , the function on B gen v given by p log | f ( p ) | is µ v -integrable(see Theorem 1.4.3). Since the trivially metrized line bundle O B and H , . . . , H b are quasi-algebraic, there is, by Remark 3.1.16, only a finite number of v ∈ M F such that Z B gen v log | f ( p ) | d µ v ( p ) = 0 . Summing up, the function M → R , w log | f | w , is µ -integrable.By the global induction formula 3.1.16, Z M log | f | w d µ ( w ) = X V ∈ B (1) − ord V ( f ) h H ,...,H b ( V ) + X v ∈ M F Z B an v log | f ( p ) | d µ v ( p )= − h H ,...,H b (cyc( f )) + X v ∈ M F Z B an v log | f ( p ) | d µ v ( p )= − h O B ,H ,...,H b ( B )= 0 , which concludes the proof. (cid:3) Let L = ( L , ( k · k v ) v ) be an M F -metrized line bundle on X . Then L inducesan M -metric on the line bundle L = L ⊗ K on X given as follows:For each V ∈ B (1) , consider the non-archimedean absolute value | · | V on K from(3.9) and let C V be the completion of an algebraic closure of the completion of K with respect to |·| V . We get a proper C ◦ V -model ( X V , L V ) := ( X × B Spec C ◦ V , L ⊗ C ◦ V ) OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 43 of (
X, L ). By Definition 1.2.6, the model ( X V , L V ) induces a metric k · k V on theanalytification L an V over X an V with respect to | · | V .Let us fix a place v ∈ M F . By (3.10), a generic point p ∈ B gen v induces anabsolute value | · | v,p on K . We denote by C v,p the completion of an algebraicclosure of the completion of K with respect to | · | v,p and by X an v,p the analytificationof X with respect to | · | v,p . Then the projection X v × B v Spec C v,p → X v induces amorphism i p : X an v,p → X an v . (3.12)Note that i p is injective if v is an archimedean place (cf. [12, (2.1)]), but notnecessarily in the non-archimedean case. The analytification L an v,p of L with respectto |·| v,p can be identified with the line bundle i ∗ p L an v and we equip it with the metric k · k v,p := i ∗ p k · k v . This leads to an M -metrized line bundle L = ( L, ( k · k w ) w ∈ M ) (3.13)on X . Lemma 3.2.4.
Let π v : X an v → B an v be the morphism of C v -analytic spaces inducedby π : X → B and let i p : X an v,p → X an v be the morphism from (3.12). Then we have i p ( X an v,p ) = π − v ( p ) . Proof.
We only show this for a non-archimedean place v , the archimedean case isestablished at the beginning of [12, § 2]. We may assume that B = Spec( A ) resp. X = Spec( C ) for finitely generated F -algebras A and C . Then π corresponds to aninjective F -algebra homomorphism A ֒ → C and we have X = Spec( C ⊗ A K ) with K = F ( B ) = Quot( A ).Let q ∈ X an v , that means q is a multiplicative seminorm on C ⊗ F C v satisfying q | C v = | · | v . Then q lies in i p ( X an v,p ) if and only if it extends to a multiplicativeseminorm ˜ q on C ⊗ A C v,p with ˜ q | C v,p = | · | v,p . This is illustrated in the followingdiagram, A ⊗ F C v (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) C v,p (cid:127) _ (cid:15) (cid:15) |·| v,p (cid:23) (cid:23) C ⊗ F C v (cid:31) (cid:127) / / q - - C ⊗ A C v,p ˜ q % % ❑❑❑❑❑ R ≥ . On the one hand, if we have such a commutative diagram, then π v ( q ) = q | A ⊗ C v = | · | v,p | A ⊗ C v = p. On the other hand, if π v ( q ) = p , then we have a multiplicative seminorm ˜ q givenby C ⊗ A C v,p = ( C ⊗ F C v ) ⊗ ( A ⊗ C v ) C v,p −→ H ( q ) b ⊗ H ( p ) C v,p y −→ R ≥ , where y is some element of the non-empty Berkovich spectrum M (cid:0) H ( q ) b ⊗ H ( p ) C v,p (cid:1) (cf. [18, 0.3.2]). It follows easily that we obtain a commutative diagram as above.This proves the result. (cid:3) We need the following projection formula for heights. This formula is possiblebecause we have one more line bundle H j on the base B than usual (compare withTheorem 3.2.6). Proposition 3.2.5.
Let π : W → V be a morphism of proper varieties over aglobal field F of dimensions n + b − and b − respectively, with b, n ≥ . Let H i , i = 1 , . . . , b , and L j , j = 1 , . . . , n , be DSP quasi-algebraic line bundles on V and W respectively. Then h π ∗ H ,...,π ∗ H b , L ,..., L n ( W ) = deg L ,..., L n ( W η ) h H ,...,H b ( V ) , where W η denotes the generic fiber of π . In particular, if dim( π ( W )) ≤ b − , then h π ∗ H ,...,π ∗ H b , L ,..., L n ( W ) = 0 .Proof. The proof is similar as for [12, Proposition 2.3] and we only sketch the addi-tional contributions. By continuity of the height, we may assume that the metricsin H i and L j are smooth or algebraic for all i, j . We prove this proposition by in-duction on n . The case n = 0 follows from functoriality of the height (Proposition3.1.12). Let n ≥
1. We choose any invertible meromorphic section s n of L n anddenote by k · k n = ( k · k n,v ) v the metric of L n . Then the global induction formula3.1.16 impliesh π ∗ H ,...,π ∗ H b , L ,..., L n ( W ) = h π ∗ H ,...,π ∗ H b , L ,..., L n − (cyc( s n )) − X v ∈ M F µ ( v ) Z W an v log k s n k n,v b ^ i =1 c ( π ∗ H i,v ) ∧ n − ^ j =1 c ( L j,v ) . If v is archimedean, then V bi =1 c ( H i,v ) is the zero measure on V an v since dim( V an v ) = b −
1. Thus, the measure in the above integral vanishes and so the integral is zero.If v is non-archimedean, then the metrics in H i,v , i = 1 , . . . , b , are induced bymodels H i of H e i i,v on a common model V of V v over Spec C ◦ v . By linearity, we mayassume that e i = 1 for all i . Analogously, the metrics in L j,v , j = 1 , . . . , n , areinduced by models L j of L j,v on a common model W of W v . Moreover, we mayassume that the morphism π v : W v → V v extends to a morphism τ : W → V overSpec C ◦ v . Since the special fiber e V of V has dimension b −
1, the degree with respectto H , . . . , H b of a cycle of e V is zero. Hence, for every irreducible component Y ofthe special fiber of W , we have by means of the projection formula,deg τ ∗ H ,...,τ ∗ H b , L ,..., L n − ( Y ) = deg H ,..., H b ( τ ∗ (c ( L ) . . . c ( L n − ) .Y )) = 0 . Therefore, for each v ∈ M F , the measure in the above integral vanishes and so theintegral is zero.Finally, we obtain by the induction hypothesis,h π ∗ H ,...,π ∗ H b , L ,..., L n ( W ) = h π ∗ H ,...,π ∗ H b , L ,..., L n − (cyc( s n ))= deg L ,..., L n − (cyc( s n ) η ) h H ,...,H b ( V )= deg L ,..., L n ( W η ) h H ,...,H b ( V ) , proving the result. (cid:3) Theorem 3.2.6.
Let B be a b -dimensional normal proper variety over a global field F and let H , . . . , H b be nef quasi-algebraic line bundles on B . Let K = F ( B ) bethe function field of B and ( M , µ ) the associated structure of an M -field on K asin (3.11).Let π : X → B be a dominant morphism of proper varieties over F and X thegeneric fiber of π . Let Y be an n -dimensional prime cycle of X and Y its closurein X . For j = 0 , . . . , n, let L j be an M -metrized line bundle on X which is inducedby a DSP quasi-algebraic line bundle L j on X as in (3.13).Then Y is integrable with respect to L , . . . , L n and we have h L ,...,L n ( Y ) = h π ∗ H ,...,π ∗ H b , L ,..., L n ( Y ) . (3.14) OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 45
Proof.
By Chow’s lemma (e. g. [23, Theorem 13.100]) and functoriality of theheight (Proposition 3.1.12 (ii)), we reduce to the case when the proper varieties areprojective over F . Then π is also projective. By (multi-)linearity of the height(Proposition 3.1.12 (i)), we may assume that the line bundles L j are very ampleand their M F -metrics are semipositive. Making a finite base change and usingProposition 3.1.17, we may suppose that B and X are geometrically integral.We prove this theorem by induction on the dimension of Y . If dim( Y ) = − Y = ∅ , then Y is integrable since the local heights of Y are zero. Equation(3.14) holds in this case because Y is empty as well.From now on we suppose that dim( Y ) = n ≥
0. Then the restriction π | Y : Y → B is dominant. By Proposition 3.1.12 (ii), the height does not change if we restrict thecorresponding metrized line bundles to Y . So we may assume that Y = X , Y = X and n = dim( Y ) = dim( X ).Let s , . . . , s n be global sections of L , . . . , L n respectively, whose divisors inter-sect properly on X , and let ρ : M → R be the function given by w λ ( L ,s | X ) ,..., ( L n ,s n | X ) ( X, w ) . We must show that ρ is µ -integrable and that Z M ρ ( w ) d µ ( w ) = h π ∗ H ,...,π ∗ H b , L ,..., L n ( X ) . By the induction formula of local heights (Theorem 1.4.3), there is a decomposi-tion ρ = ρ + ρ into well-defined functions ρ , ρ : M → R given by ρ ( w ) = λ ( L ,s | X ) ,..., ( L n − ,s n − | X ) (cyc( s n | X ) , w )and ρ ( w ) = Z X an w log k s n | X w k − n,w c ( L ,w ) ∧ · · · ∧ c ( L n − ,w ) . Moreover, we can write the cycle cyc( s n ) in X ascyc( s n ) = cyc( s n ) hor /B + cyc( s n ) vert /B , where cyc( s n ) hor /B contains the components which are dominant over B and cyc( s n ) vert /B contains the components not meeting X .By the induction hypothesis, the function ρ is µ -integrable and Z M ρ ( w ) d µ ( w ) = h L ,...,L n − (cyc( s n | X ))= h π ∗ H ,...,π ∗ H b , L ,..., L n − (cyc( s n ) hor /B ) . (3.15)If w = V ∈ B (1) , then we deduce as in the corresponding part of the proof of[12, Theorem 2.4] that ρ ( V ) = X W∈X (1) π ( W )= V h H ,...,H b ( V ) ord W ( s n ) deg L ,..., L n − ( W V ) , (3.16)where W V denotes the generic fiber of π | W : W → V . This formula implies theintegrability of ρ on B (1) with respect to the counting measure µ fin because thereare only finitely many W ∈ X (1) such that ord W ( s n ) = 0. The same arguments asin the corresponding part of the proof of [12, Theorem 2.4] show that Z B (1) ρ ( w ) d µ fin ( w ) = h π ∗ H ,...,π ∗ H b , L ,..., L n − (cyc( s n ) vert /B ) . (3.17) Now, let v be a place of M F and p a generic point of B an v . We claim that thefunction ρ ( p ) = Z X an v,p log i ∗ p k s n k − n,v n − ^ j =0 c ( i ∗ p L j,v ) (3.18)is integrable with respect to µ v = µ ( v ) · c ( H ,v ) ∧ · · · ∧ c ( H b,v ). Furthermore, weclaim that Z B gen v ρ ( p ) d µ v ( p ) = µ ( v ) Z X an v log k s n k − n,v n − ^ j =0 c ( L j,v ) ∧ b ^ i =1 c ( π ∗ H i,v ) . (3.19)This two claims will be shown in a rather elaborate argument below.Assuming these two claims, we will first show that the theorem now followseasily. By Proposition 3.1.16, the integral in (3.19) is zero for all but finitely many v ∈ M F because the line bundles π ∗ H , . . . , π ∗ H b , L , . . . , L n are quasi-algebraic.We conclude that the function ρ = ρ + ρ is µ -integrable and obtain, by using theinduction hypothesis (3.15), (3.17), (3.19) and the global induction formula 3.1.16,h L ,...,L n ( X ) = Z M ρ ( w ) d µ ( w ) + Z B (1) ρ ( w ) d µ fin ( w ) + X v ∈ M F Z B gen v ρ ( p ) d µ v ( p )= h π ∗ H ,...,π ∗ H b , L ,..., L n − (cyc( s n ) hor /B )+ h π ∗ H ,...,π ∗ H b , L ,..., L n − (cyc( s n ) vert /B )+ X v ∈ M F µ ( v ) Z X an v log k s n k − n,v n − ^ j =0 c ( L j,v ) ∧ b ^ i =1 c ( π ∗ H i,v )= h π ∗ H ,...,π ∗ H b , L ,..., L n ( X ) , proving the theorem.We will prove more generally that for any non-trivial s n ∈ Γ( X v , L v ), the function ρ in (3.18) is µ v -integrable and that (3.19) holds. If v ∈ M F is an archimedeanplace, then the proof of [12, Theorem 2.4] shows that ρ is µ v -integrable on B gen v and that the equation (3.19) holds.From now on, we consider the case where v ∈ M F is non-archimedean. We firstassume that, for each j = 0 , . . . , n − i = 1 , · · · , b , the metrics on L j,v and H i,v are algebraic. Then the function ρ is µ v -integrable because µ v is a discretefinite measure.We choose, for each j , a proper model ( X j , L j , e j ) of ( X v , L j,v ) over Spec C ◦ v that induces the metric of L j,v . Note that we omit the place v in the notation ofthe models in order not to burden the notation. By linearity, we may assume that e j = 1 for all j . Furthermore, we can suppose that the models X j agree with acommon model X with reduced special fiber (cf. Remark 1.2.12). In the sameway, we have a proper C ◦ v -model B of B v with reduced special fiber and, for each i = 1 , . . . , b , a model H i of H i,v on B inducing the corresponding metric. As in theproof of Proposition 3.2.5, we can asume that the morphism π v : X v → B v extendsto a morphism τ : X → B over C ◦ v .To construct a suitable model of X v,p = X × K Spec C v,p over C ◦ v,p , we considerthe commutative diagram Spec C v,p / / (cid:15) (cid:15) B (cid:15) (cid:15) Spec C v / / Spec
F .
OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 47
By the universal property of the fiber product, we have a unique morphism Spec C v,p → B v . Because B is proper over C ◦ v and by the valuative criterion, this morphismextends to Spec C ◦ v,p → B . Let X p be the fiber product X × B Spec C ◦ v,p . This isa model of X v,p over C ◦ v,p . We denote the special fibers of B , X and X p by e B , f X and f X p respectively. By 1.1.7, there exists a formal admissible scheme X p over C ◦ v,p with generic fiber X an p = X an v,p and with reduced special fiber e X p such that thecanonical morphism ι p : e X p → f X p is finite and surjective. We obtain the followingcommutative diagram X an v,p = / / red (cid:15) (cid:15) X an v,p i p / / red (cid:15) (cid:15) X an v π v / / red (cid:15) (cid:15) B an v red (cid:15) (cid:15) e X p ι p / / f X p j p / / f X ˜ τ / / e B , where red is the reduction map from 1.1.5 and 1.1.7. We have f X p = f X × e B Spec e C v,p .By Definition 1.2.13, the left-hand side of equation (3.19) is equal to Z B gen v (cid:18) Z X an v,p log i ∗ p k s n k − n,v n − ^ j =0 c ( i ∗ p L j,v ) (cid:19) b ^ i =1 c ( H i,v )( p ) (3.20)= X Z ∈ e B (0) (cid:18) X V ∈ e X (0) ξZ log k s n ( i ξ Z ( ξ V )) k − n,v deg ι ∗ ξZ j ∗ ξZ e L ,...,ι ∗ ξZ j ∗ ξZ e L n − ( V ) (cid:19) deg H ,..., H b ( Z ) , where ξ Z (resp. ξ V ) denotes the unique point whose reduction is the generic pointof Z (resp. V ).First, we consider the inner sum. Let Z be an irreducible component of e B withgeneric point η Z = red( ξ Z ). For V ∈ X (0) ξ Z , we consider the irreducible component W := ι ξ Z ( V ) of f X ξ Z and the irreducible component Y := j ξ Z ( W ) of f X . It followsfrom the compatibility of the reduction with morphisms of models that i ξ Z ( ξ V ) isthe unique point ξ Y of X an v with reduction equal to the generic point of Y . Weconclude that log k s n ( i ξ Z ( ξ V )) k − n,v = log k s n ( ξ Y ) k − n,v . (3.21)Applying the projection formula in [25, Proposition 4.5], we deduce that( ι ξ Z ) ∗ (cyc( e X ξ Z )) = cyc( f X ξ Z ) (3.22)Now the geometric projection formula, (3.21) and (3.22) yield X V ∈ e X (0) ξZ log k s n ( i ξ Z ( ξ V )) k − n,v deg (cid:0) ι ∗ ξZ j ∗ ξZ e L k (cid:1) k =0 ,...,n − ( V )= X W ∈ e X (0) ξZ log k s n ( ξ Y ) k − n,v m (cid:0) W, f X ξ Z (cid:1) deg (cid:0) j ∗ ξZ e L k (cid:1) k =0 ,...,n − ( W ) , (3.23)where m (cid:0) W, f X ξ Z (cid:1) denotes the multiplicity of W in f X ξ Z and where Y = j ξ Z ( W ).By [17, Ch. 0, (2.1.8)], there is a bijective map n Y ∈ f X (0) | ˜ τ ( Y ) = Z o −→ f X (0) η Z , Y Y η Z . (3.24)The special fiber of B is reduced and hence, applying [1, 2.4.4(ii)] and using the com-patibility of reduction and algebraic closure, we get e C v,ξ Z = ^ H ( ξ Z ) = κ ( η Z ). Thus, f X ξ Z = f X × e B Spec e C v,ξ Z is the base change of the fiber f X η Z = f X × e B Spec κ ( η Z )by κ ( η Z ) → κ ( η Z ). Thus, by [43, Lemma 32.6.10], we obtain a surjective map f X (0) ξ Z −→ f X (0) η Z . (3.25)Composing the maps (3.24) and (3.25), we get a canonical surjective map f X (0) ξ Z −→ n Y ∈ f X (0) | ˜ τ ( Y ) = Z o with finite fibers. More precisely, for each irreducible component Y in f X with˜ τ ( Y ) = Z , the scheme Y ξ Z = Y × Z Spec e C v,ξ Z is a finite union of (non-necessarilyreduced) irreducible components of f X (0) ξ Z . Since i ξ Z ( ξ W ) = ξ Y for W ∈ Y (0) ξ Z , wededuce X W ∈ e X (0) ξZ log k s n ( ξ j ξZ ( W ) ) k − n,v m (cid:0) W, f X ξ Z (cid:1) deg (cid:0) j ∗ ξZ e L k (cid:1) k =0 ,...,n − ( W )= X Y ∈ e X (0) ˜ τ ( Y )= Z log k s n ( ξ Y ) k − n,v deg (cid:0) j ∗ ξZ e L k (cid:1) k =0 ,...,n − ( Y ξ Z ) . (3.26)Let Y be an irreducible component of f X with generic point η Y such that ˜ τ ( Y ) = Z . It follows from the definitions in algebraic intersection theory that we havedeg L ,..., L n − ,τ ∗ H ,...,τ ∗ H b ( Y ) = deg j ∗ ηZ e L ,...,j ∗ ηZ e L n − ( Y η Z ) deg H ,..., H b ( Z ) . Since the degree is stable unter base change, we deducedeg L ,..., L n − ,τ ∗ H ,...,τ ∗ H b ( Y ) = deg j ∗ ξZ e L ,...,j ∗ ξZ e L n − ( Y ξ Z ) deg H ,..., H b ( Z ) . (3.27)Combining the equations (3.20), (3.23), (3.26) and (3.27), we obtain Z B gen v (cid:18) Z X an v,p log i ∗ p k s n k − n,v n − ^ j =0 c ( i ∗ p L j,v ) (cid:19) b ^ i =1 c ( H i,v )( p )= X Z ∈ e B (0) X Y ∈ e X (0) ˜ τ ( Y )= Z log k s n ( ξ Y ) k − n,v deg L ,... L n − ,τ ∗ H ,...,τ ∗ H b ( Y )= X Y ∈ e X (0) log k s n ( ξ Y ) k − n,v deg L ,... L n − ,τ ∗ H ,...,τ ∗ H b ( Y )= Z X an v log k s n k − n,v n − ^ j =0 c ( L j,v ) ∧ b ^ i =1 c ( π ∗ H i,v ) , using in the next-to-last equality that, for an irreducible component Y of f X withdim(˜ τ ( Y )) ≤ b −
1, the degree is zero. This proves equation (3.19) in the algebraiccase.In a next step, we assume that, for each j = 0 , . . . , n , the metric k · k j,v on L j,v is algebraic, but that the metrics on H i,v , i = 1 , . . . , b , are not necessarily algebraic.For this case, we once again show that ρ is µ v -integrable and that the equality(3.19) holds.As in the previous case, we may assume that, for each j = 0 , . . . , n , there is aproper model ( L j , X ) of ( L j,v , X v ) over C ◦ v inducing the corresponding metric. Wechoose any projective model B over C ◦ v of the projective variety B v and suppose, asin the previous case, that π v : X v → B v extends to a proper morphism τ : X → B .Because X v is projective over C v and by [27, Proposition 10.5], we may assumethat X is projective over C ◦ v and thus, τ is projective. Using Serre’s theorem (see OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 49 [23, Theorem 13.62]), the line bundle L j is the difference of two very ample linebundles relative to τ . By multilinearity of the height, we reduce to the case where L j is very ample relative to τ . Because B is projective over C ◦ v , we deduce by[23, Summary 13.71 (3)] that there is a closed immersion f j : X ֒ → P N j B such that L j ≃ f ∗ j O P Nj B (1).For projective spaces P N j , j = 0 , . . . , n , let P := P N × · · · × P N n be the multi-projective space and let O P ( e j ) be the pullback of O P Nj (1) by the j -th projection.Since B is geometrically integral, we have the function field K v = C v ( B v ) and wedefine X v = X v × B v Spec K v and L j,v = L j,v ⊗ K v . The product of f , . . . , f n is aclosed immersion f : X ֒ → P B with L j ≃ f ∗ O P B ( e j ) for j = 0 , . . . , n . We obtainthe following commutative diagram X × B Spec C ◦ v,p (cid:31) (cid:127) f p / / j p (cid:15) (cid:15) P C ◦ v,p (cid:15) (cid:15) X v,p (cid:31) (cid:127) g p / / h p (cid:15) (cid:15) : : ✉✉✉✉✉✉✉ P C v,p (cid:15) (cid:15) : : ✉✉✉✉✉✉✉ X (cid:31) (cid:127) f / / P B .X v (cid:31) (cid:127) g / / : : ✉✉✉✉✉✉✉✉ P K v : : ✉✉✉✉✉✉✉✉ Note that each horizontal arrow is a closed immersion because f is a closed immer-sion and the other morphisms are obtained by base change.Let p ∈ B gen v . Then the metric k · k v,p = i ∗ p k · k v on L j,v,p = g ∗ p O P C v,p ( e j ) isinduced by j ∗ p L j = j ∗ p f ∗ O P B ( e j ) = f ∗ p O P C ◦ v,p ( e j ) . Hence, L j,v,p = g ∗ p O P C v,p ( e j ), where O P C v,p ( e j ) is endowed with the canonicalmetric. By Proposition 3.2.2, the field K v together with ( B gen v , µ v ) is a B gen v -field in the sense of Definition 3.1.1. Therefore, [26, Proposition 5.3.7(d)] saysthat every n -dimensional cycle on P K v is µ v -integrable on B gen v with respect to O P Kv ( e ) , . . . , O P Kv ( e n ). Since integrability is closed under tensor product andpullback (see 3.1.10), the local height ρ is µ v -integrable on B gen v . By the inductionhypothesis, we deduce that ρ = ρ − ρ is also µ v -integrable on B gen v .For proving the equality (3.19), we study ρ in more detail. We may always replace s n by λs n for a non-zero λ ∈ C v and hence we may assume that s n ∈ Γ( X , L n ).Using L j ≃ f ∗ O P B ( e j ) and by possibly changing the closed immersion f , we mayassume that s n = g ∗ t n for a global section of O P Kv ( e n ). We choose global sections t j of O P Kv ( e j ), j = 0 , . . . , n −
1, such that | div( t ) | ∩ · · · ∩ | div( t n ) | ∩ X v = ∅ . Note that the original s , . . . , s n − do not play a role anymore and so we may set s j := g ∗ t j for j = 0 , . . . , n −
1. By Proposition 1.3.5, we get ρ ( p ) = λ ( L ,s ) ,..., ( L n ,s n ) ( X v , p )= λ ( O PC v ( e ) ,t ) ,..., ( O PC v ( e n ) ,t n ) ( X v , p ) . (3.28)We can express ρ in terms of the Chow form of the n -dimensional subvari-ety X v of the multiprojective space P K v . This is a multihomogenous polynomial F X v ( ξ , . . . , ξ n ) with coefficients in K v and in the variables ξ j = ( ξ j , . . . , ξ jN j )viewed as dual coordinates on P N j K v (see [26, Remark 2.4.17] for details). By (3.28) and [26, Example 4.5.16], we obtain ρ ( p ) = log | F X v | v,p − log | F X v ( t , . . . , t n ) | v,p , (3.29)where in the first term we use the Gauss norm and in the second term t j denotesthe dual coordinates of t j .For each i = 1 , . . . , n , we choose a sequence of algebraic semipositive metrics( k ·k i,v,k ) k ∈ N on H i,v that converges to the semipositive metric k ·k i,v on H i . Denote H i,v,k = ( H i,v , k · k i,v,k ) and set µ v,k = µ ( v ) · c ( H ,v,k ) ∧ · · · ∧ c ( H b,v,k ) . By Corollary 1.4.6, we obtainlim k →∞ Z B gen v ρ ( p ) d µ v,k ( p ) = Z B gen v ρ ( p ) d µ v ( p ) . (3.30)Analogously we can show this for the local height ρ and hence we getlim k →∞ Z B gen v ρ ( p ) d µ v,k ( p ) = Z B gen v ρ ( p ) d µ v ( p ) . (3.31)On the other hand, Corollary 1.4.6 again showslim k →∞ Z X an v log k s n k n,v n − ^ j =0 c ( L j,v ) ∧ b ^ i =1 c ( π ∗ H i,v,k )= Z X an v log k s n k n,v n − ^ j =0 c ( L j,v ) ∧ b ^ i =1 c ( π ∗ H i,v ) . (3.32)Thus, the equality (3.19) for semipositive metrics on H i,v and algebraic metrics on L j,v follows from (3.31), (3.32) and the algebraic case.In the last step, we assume that the metrics on H i,v and L j,v are semipositiveand not necessarily algebraic. We choose, for each j = 0 , . . . , n −
1, a sequence ofalgebraic semipositive metrics ( k · k j,v,k ) k ∈ N on L j,v that converges to k · k j,v . For p ∈ B gen v , we set ρ ,k ( p ) := Z X an v,p log i ∗ p k s n k − n,v,k n − ^ j =0 c ( i ∗ p L j,v,k ) . By the induction formula 1.2.15 and Proposition 1.2.11 (iii), we obtain for each k, l ∈ N , | ρ ,k ( p ) − ρ ,l ( p ) | = (cid:12)(cid:12)(cid:12) λ ( L ,k ,s ) ,..., ( L n,k ,s n ) ( X v , p ) − λ ( L ,k ,s ) ,..., ( L n − ,k ,s n − ) (cyc( s n | X v ) , p ) − λ ( L ,l ,s ) ,..., ( L n,l ,s n ) ( X v , p ) + λ ( L ,l ,s ) ,..., ( L n − ,l ,s n − ) (cyc( s n | X v ) , p ) (cid:12)(cid:12)(cid:12) ≤ n X j =0 d( k · k j,v,k , k · k j,v,l ) deg L ,...,L j − ,L j +1 ,...,L n ( X v )+ n − X j =0 d( k · k j,v,k , k · k j,v,l ) deg L ,...,L j − ,L j +1 ,...,L n − (cyc( s n | X v )) . Hence, the sequence ( ρ ,k ) k ∈ N converges uniformly to ρ on B gen v . Because themeasure µ v has finite total mass and, by the previous case, the functions ρ ,k are µ v -integrable, we deduce that ρ is µ v -integrable and thatlim k →∞ Z B gen v ρ ,k ( p ) d µ v ( p ) = Z B gen v ρ ( p ) d µ v ( p ) . OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 51
Thus, using (3.19) for the functions ρ ,k and the induction formula 1.4.3, the equal-ity (3.19) also holds in the case when all the metrics are semipositive. (cid:3) Global heights of toric varieties over finitely generated fields.
Follow-ing [12, § 3] closely, we use our preceding results to get some combinatorial formulasfor the height of a fibration with generic toric fiber. Indeed, our non-discrete non-archimedean toric geometry is necessary since the measure space M from (3.11)contains arbitrary non-archimedean absolute values, in contrast to the measurespace considered in [12, § 1].As usual, we fix a lattice M ≃ Z n with dual M ∨ = N and use the respectivenotations from the sections on toric geometry. At first, we consider an arbitrary M -field K with associated positive measure µ . Let Σ be a complete fan in N R andlet X Σ be the associated proper toric variety over K with torus T = Spec K [ M ]. Let L be a toric line bundle on X Σ . An M -metric k · k = ( k · k v ) v ∈ M on L is toric if, for each v ∈ M , the metric k · k v on L v is toric (see Definition 2.4.1).The canonical M -metric on L , denoted k · k can , is given, for each v ∈ M , by thecanonical metric on L v (see Definition 2.4.11). We will write L can = ( L, k · k can ).Let s be a toric section on L and Ψ the associated virtual support function. Thena toric M -metric ( k·k v ) v on L induces a family (cid:0) ψ L,s,v (cid:1) v ∈ M of real-valued functionson N R as in Definition 2.4.6. If k · k is semipositive, then each ψ L,s,v is concaveand we obtain a family (cid:0) ϑ L,s,v (cid:1) v ∈ M of concave functions on ∆ Ψ called v -adic rooffunctions (cf. Definition 2.6.4 which works the same way in the archimedean case).When L and s are clear from the context, we also denote ψ L,s,v by ψ v and ϑ L,s,v by ϑ v . Proposition 3.3.2.
For each i = 0 , . . . , t , let L i be a toric line bundle on X Σ equipped with a DSP toric M -metric and denote by L can i the same toric line bundleendowed with the canonical M -metric. Let Y be either the closure of an orbit orthe image of a proper toric morphism of dimension t . Then Y is integrable withrespect to L can0 , . . . , L can t and h L can0 ,...,L can t ( Y ) = 0 . (3.33) Furthermore, if Y is integrable with respect to L , . . . , L t , then the global height isgiven by h L ,...,L t ( Y ) = Z M λ tor L ,...,L t ( Y, v ) d µ ( v ) , (3.34) where λ tor L ,...,L t ( Y, v ) = λ tor L ,v ,...,L t,v ( Y v ) is the toric local height from Definition2.6.1.Proof. The first statement and equation (3.33) can be shown using the same argu-ments as in [11, Proposition 5.2.4]. The proof is based on an inductive argumentusing the local induction formula from Theorem 1.2.15. The second equation followseasily from the first one. (cid:3)
Corollary 3.3.3.
Let L = ( L, ( k · k v ) v ) be a toric line bundle on X Σ equipped witha semipositive toric M -metric. Choose any toric section s of L and denote by Ψ the corresponding support function on Σ . If X Σ is integrable with respect to L , then h L ( X Σ ) = ( n + 1)! Z M Z ∆ Ψ ϑ L,s,v dvol M d µ ( v ) . Proof.
This is a direct consequence of Proposition 3.3.2 and Theorem 2.6.6 whichholds also in the archimedean case by [11, Theorem 5.1.6]. (cid:3)
Now we consider the particular case of an M -field which is induced by a varietyover a global field as in § 3.2. Let B be a b -dimensional normal proper variety overa global field F and let H , . . . , H b be nef quasi-algebraic metrized line bundleson B . This provides the function field K = F ( B ) with the structure ( M , µ ) of an M -field as in (3.11). Let X be an n -dimensional normal proper toric variety over K with torus T = Spec K [ M ], corresponding to a complete fan Σ in N R . We choose abase-point-free toric line bundle L on X together with a toric section s and denoteby Ψ the associated support function on Σ.Let π : X → B be a dominant morphism of proper varieties over F such that X isthe generic fiber of π . We equip L with a toric M -metric k · k such that L = ( L, k · k )is induced by a semipositive quasi-algebraic M F -metrized line bundle L on X as in(3.13). Then it follows easily that L is also semipositive and so, for each v ∈ M ,the function ψ v is concave.The following result generalizes Corollary 3.1 in [12], where the global field is Q and the metrized line bundles are induced by models over Z . It is based on ourmain theorems 2.6.6 and 3.2.6. Corollary 3.3.4.
Let notation be as above. Then the function M −→ R , w Z ∆ Ψ ϑ L,s,w ( m ) dvol M ( m ) (3.35) is integrable with respect to µ and h π ∗ H ,...,π ∗ H b , L ,..., L ( X ) = h L ( X ) = ( n + 1)! Z M Z ∆ Ψ ϑ w ( m ) dvol M ( m ) d µ ( w ) . (3.36) Proof.
By Theorem 2.6.6 (non-archimedean case) and [11, Theorem 5.1.6] (archi-medean case), we have( n + 1)! Z ∆ Ψ ϑ w dvol M = λ tor L ,w ,...,L n,w ( X w ) . Hence, Theorem 3.2.6 implies the µ -integrability of the function (3.35). The firstequality of (3.36) is Theorem 3.2.6. The second follows readily from (3.34) and(3.35). (cid:3) Proposition 3.3.5.
We use the same notation as above.(i) For each m ∈ ∆ Ψ , the function M −→ R , w ϑ w ( m ) is µ -integrable.(ii) The function ϑ L,s : ∆ Ψ −→ R , m Z M ϑ L,s,w ( m ) d µ ( w ) is continuous and concave.(iii) The function M × ∆ Ψ −→ R , ( w, m ) ϑ w ( m ) is integrable with respectto the measure µ × vol M .(iv) We have h π ∗ H ,...,π ∗ H b , L ,..., L ( X ) = h L ( X ) = ( n + 1)! Z ∆ Ψ ϑ L,s ( m ) dvol M ( m ) , where ϑ L,s is the function in (ii).Proof.
The proof of (i)–(iii) respectively (iv) is analogous to [12, Theorem 3.2 re-spectively Corollary 3.4] using Corollary 3.3.4 in place of [12, Corollary 3.1]. Itutilizes in an essential way that ϑ w is concave (see Theorem 2.5.8). (cid:3) OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 53
Heights of projectively embedded toric varieties over the functionfield of an elliptic curve.
Similarly as in [12, § 4], we consider the formulas in§ 3.3 in the case where X is the normalization of a translated subtorus in a projectivespace using canonical metrics. Then we illustrate the resulting formulas in the caseof the function field K of an elliptic curve which is a natural example where thecanonical polarizations at a place of bad reduction lead to non-discrete valuationson K .Let B be a b -dimensional normal proper variety over a global field F and let H , . . . , H b be nef quasi-algebraic M F -metrized line bundles on B . We equip K = F ( B ) with the structure ( M , µ ) of an M -field as in (3.11).For r ≥
1, let us consider the projective space P rB = P rF × F B over B andSerre’s twisting sheaf O P rB (1). We equip O P rB (1) with the metric obtained by pullingback the canonical M F -metric of O P rF (1) and denote this metrized line bundle by O (1) = O P rB (1).For m j ∈ Z n and f j ∈ K × , j = 0 , . . . , r , we regard the morphism G n m ,K −→ P rK , t ( f t m : · · · : f r t m r ) , where f j t m j = f j t m j, · · · t m j,n n . For simplicity, we suppose that m = , f = 1and that m , . . . , m r generate Z n as an abelian group. Let Y be the closure ofthe image of this morphism. Then Y is a toric variety over K , but not necessarilynormal.Let Y be the closure of Y in P rB and let π : Y → B be the restriction of P rB → B .Our goal is to compute the height h π ∗ H ,...,π ∗ H b , O (1) ,..., O (1) ( Y ) using formula (3.36).Since Y is not necessarily normal, we consider the normalization X of Y and theinduced dominant morphism X → B which we also denote by π . Then the genericfiber X = X × B Spec K is a normal G n m ,K -toric variety over K . Let L be thepullback of O (1) via X → P rB and let L be the induced M -metrized line bundle on X as in (3.13). Then L is a toric semipositive M -metrized line bundle on X .Analogously to [12, Proposition 4.1], we can explicitly describe the associated w -adic roof functions as follows: Proposition 3.4.1.
We keep the above notations and let s be the toric section of L induced by the global section x of O (1) . Then the polytope ∆ associated to ( L, s ) is determined by ∆ = conv( m , . . . , m r ) . For w ∈ M , the graph of the w -adic roof function ϑ w : ∆ → R is the upper envelopeof the polytope ∆ w ⊆ R n × R which is given by ∆ w = ( conv (cid:0) ( m j , − h H ,...,H b ( V ) ord V ( f j )) j =0 ,...,r (cid:1) , if w = V ∈ B (1) , conv (cid:0) ( m j , log | f j ( p ) | v ) j =0 ,...,r (cid:1) , if w = p ∈ B gen v , v ∈ M F . Now we differ from the setting in [12, § 4] and consider the special case of thefunction field of an elliptic curve equipped with a canonical metrized line bundle.Note that in this case non-discrete non-archimedean absolute values naturally occur.
From now on, we assume that B is an elliptic curve E over the global field F and let H be an ample symmetric line bundle on E . We choose any rigidification ρ of H , i. e. ρ ∈ H ( F ) \ { } . By the theorem of the cube, we have, for each m ∈ Z , acanonical identification [ m ] ∗ H = H ⊗ m of rigidified line bundles. Then there existsa unique M F -metric k · k ρ = ( k · k ρ,v ) v on H such that, for all v ∈ M F , m ∈ Z ,[ m ] ∗ k · k ρ,v = k · k ⊗ m ρ,v . For details, see [5, Theorem 9.5.7]. We call such an M F -metric canonical becauseit is canonically determined by H up to ( | a | v ) v ∈ M F for some a ∈ F × . By [28, 3.5], the canonical metric k · k ρ is quasi-algebraic and, since H is ample and symmetric,it is semipositive.The global height associated to H = ( H, k · k ρ ) is equal to the Néron-Tate heightˆh H (see [5, Corollary 9.5.14]). In particular, it does not depend on the choice ofthe rigidification ρ . Since H is ample, we have h H = ˆh H ≥ v ∈ M F , the canonically metrized line bundle H induces the canonicalmeasure c ( H v ) = c ( H v , k · k ρ,v ) which does not depend on the choice of therigidification (cf. [29, 3.15]) and which is positive. It has the propertiesc ( H v )( E an v ) = deg H ( E ) and [ m ] ∗ c ( H v ) = m c ( H v ) for all m ∈ Z . For a detailed description of these measures, we have to consider three kinds ofplaces v ∈ M F .(i) The set of archimedean places is denoted by M ∞ F . For v archimedean, E an v = E ( C ) is a complex analytic space which is biholomorphic to a complex torus C / ( Z + Z τ ), ℑ τ >
0. We have c ( H v ) = deg H ( E ) µ Haar for the Haar probability measure µ Haar on this torus.(ii) The set of non-archimedean places v with E of good reduction at v is denotedby M g F . For such a v , the canonical measure c ( H v ) is a Dirac measure at a singlepoint of E an v . Since E has good reduction at v , there is a smooth proper scheme E v over C ◦ v with generic fiber E v . The special fiber e E v is an elliptic curve over e C v . Let ξ v be the unique point of E an v such that red( ξ v ) is the generic point of e E v . Thenc ( H v ) = deg H ( E ) δ ξ v .(iii) The set of non-archimedean places v with E of bad reduction at v is denotedby M b F . Then E an v is a Tate elliptic curve over C v , i. e. E an v is isomorphic as ananalytic group to G anm ,v /q Z , where G m ,v is the multiplicative group over C v withfixed coordinate x and q is an element of G m ,v ( C v ) = C × v with | q | v < G anm ,v → R , p
7→ − log p ( x ), the tropicalizationmap and set Λ v := − log | q | v Z . Then we obtain a commutative diagram G anm ,v trop / / (cid:15) (cid:15) R (cid:15) (cid:15) E an v trop / / R / Λ v . Consider the continuous section ρ : R → G anm ,v of trop, where ρ ( u ) is given by X m ∈ Z α m x m max m ∈ Z | α m | exp( − m · u ) (3.37)as in 2.4.7. Using E an v = G anm ,v /q Z , this section descends to a continuous section¯ ρ : R / Λ v → E an v of trop. The image of ¯ ρ is a canonical subset S ( E an v ) of E an v which is called the skeleton of E an v . By [1, Ex. 5.2.12 and Thm. 6.5.1], this isa closed subset of E an v and trop restricts to a homeomorphism from S ( E an v ) onto R / Λ v . By [29, Corollary 9.9], the canonical measure c ( H v ) on E an v is supported onthe skeleton S ( E an v ) and we have c ( H v ) = deg H ( E ) µ Haar for the Haar probabilitymeasure µ Haar on R / Λ v .Let m = ∈ Z n and m . . . , m r ∈ Z n generating Z n as a group and let f , . . . f r ∈ K × = F ( E ) × with f = 1. Recall that we consider the morphism G n m ,K −→ P rK , t ( f t m : · · · : f r t m r ) . The closure of the image of this morphism in P rE is denoted by Y . OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 55
Corollary 3.4.3.
With the assumptions and notations from 3.4.2, we have n + 1)! · deg H ( E ) h π ∗ H, O (1) ,..., O (1) ( Y )= 1deg H ( E ) X P ∈ C Z ∆ ϑ P ( x ) dvol( x ) + X v ∈ M ∞ F Z E ( C v ) Z ∆ ϑ p ( x ) dvol( x ) d µ Haar ( p )+ X v ∈ M g F Z ∆ ϑ ξ v ( x ) dvol( x ) + X v ∈ M b F Z R / Λ v Z ∆ ϑ ¯ ρ ( u ) ( x ) dvol( x ) d µ Haar ( u ) , where C ⊂ E (1) is the set of irreducible components of the divisors cyc( f ) , . . . , cyc( f r ) and vol is the standard measure on R n .Proof. We have h π ∗ H, O (1) ,..., O (1) ( Y ) = h π ∗ H, L ,..., L ( X ) because the global height isinvariant under normalization. We get the result by Theorem 3.2.6, Corollary 3.3.3,Proposition 3.4.1 and the description in 3.4.2. (cid:3) Appendix A. Convex geometry
In this appendix, we collect the notions of convex geometry that we need for thestudy of toric geometry. We follow the notation of [11, § 2] and [32] which is basedon the classical book [42].Let M be a free abelian group of rank n and N := M ∨ := Hom( M, Z ) its dualgroup. The natural pairing between m ∈ M and u ∈ N is denoted by h m, u i := u ( m ).If G is an abelian group, we set N G := N ⊗ Z G = Hom( M, G ). In particular, N R = N ⊗ Z R is an n -dimensional real vector space with dual space M R = Hom( N, R ).We denote by Γ a subgroup of R . A.1. A polyhedron Λ in N R is a non-empty set defined as the intersection of finitelymany closed half-spaces, i. e.Λ = r \ i =1 { u ∈ N R | h m i , u i ≥ l i } where m i ∈ M R , l i ∈ R . (A.1)A polytope is a bounded polyhedron. A face Λ ′ of a polyhedron Λ, denoted byΛ ′ (cid:22) Λ, is either Λ itself or of the form Λ ∩ H where H is the boundary of a closedhalf-space containing Λ. A face of Λ of codimension 1 is called a facet , a face ofdimension 0 is a vertex . The relative interior of Λ, denoted by ri Λ, is the interiorof Λ in its affine hull. A.2.
Let Λ be a polyhedron in N R . We call Λ strongly convex if it does not containany affine line. We say that Λ is Γ -rational if there is a representation as (A.1) with m i ∈ M and l i ∈ Γ. If Γ = Q , we just say Λ is rational . We say that a polytope in M R is lattice if its vertices lie in M . A.3. A polyhedral cone in N R is a polyhedron σ such that λσ = σ for all λ ≥
0. Its dual is defined as σ ∨ := { m ∈ M R | h m, u i ≥ ∀ u ∈ σ } . A polyhedral cone is strongly convex if and only if dim( σ ∨ ) = 0. We denote by σ ⊥ the set of m ∈ M R with h m, u i = 0 for all u ∈ σ . The recession cone of apolyhedron Λ is defined asrec(Λ) := { u ∈ N R | u + Λ ⊆ Λ } . If Λ has a representation as (A.1), the recession cone can be written asrec(Λ) = r \ i =1 { u ∈ N R | h m i , u i ≥ } . A.4. A polyhedral complex Π in N R is a non-empty finite set of polyhedra such that(i) every face of Λ ∈ Π lies also in Π;(ii) if Λ , Λ ′ ∈ Π, then Λ ∩ Λ ′ is empty or a face of Λ and Λ ′ .A polyhedral complex Π is called Γ -rational (resp. rational , resp. strongly convex )if each Λ ∈ Π is Γ-rational (resp. rational, resp. strongly convex). The support ofΠ is defined as the set | Π | := S Λ ∈ Π Λ. We say that Π is complete if | Π | = N R . Wewill denote by Π k the subset of k -dimensional polyhedra of Π.A fan in N R is a polyhedral complex in N R consisting of strongly convex rationalpolyhedral cones. A.5.
Let Π be a polyhedral complex in N R . The recession rec(Π) of Π is definedas rec(Π) = { rec(Λ) | Λ ∈ Π } . If Π is a complete Γ-rational strongly convex polyhedral complex, then rec(Π) is acomplete fan in N R . A.6.
Let C be a convex set in a real vector space. A function f : C → R is concave if f ( tu + (1 − t ) u ) ≥ tf ( u ) + (1 − t ) f ( u ) (A.2)for all u , u ∈ C and 0 < t < A.7.
Let f be a concave function on N R . We define the stability set of f as∆ f := { m ∈ M R | h m, ·i − f is bounded below } . This is a convex set in M R . The (Legendre-Fenchel) dual of f is the function f ∨ : ∆ f −→ R , m inf u ∈ N R ( h m, u i − f ( u )) . It is a continuous concave function.
A.8.
Let f : N R → R be a concave function. The recession function rec( f ) of f isdefined as rec( f ) : N R −→ R , u lim λ →∞ f ( λu ) λ . By [42, Theorem 13.1], rec( f ) is the support function of the stability set ∆ f , i. e. itis given by rec( f )( u ) = inf m ∈ ∆ f h m, u i for u ∈ N R . Proposition A.9.
Let Σ be a complete fan in N R and let Ψ : N R → R be a supportfunction on Σ (Definition 2.1.9). Then the assignment ψ ψ ∨ gives a bijectionbetween the sets of(i) concave functions ψ on N R such that | ψ − Ψ | is bounded,(ii) continuous concave functions on ∆ Ψ .Proof. This follows from the propositions 2.5.20 (2) and 2.5.23 in [11]. (cid:3)
A.10.
A function f : N R → R is piecewise affine if there is a finite cover { Λ i } i ∈ I of N R by closed subsets such that f | Λ i is an affine function.Let Π be a complete polyhedral complex in N R . We say that f is a piecewiseaffine function on Π if f is affine on each polyhedron of Π. OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 57
A.11.
Let f : N R → R be a piecewise affine function on N R . Then there is acomplete polyhedral complex Π in N R such that, for each Λ ∈ Π, f | Λ ( u ) = h m Λ , u i + l Λ with ( m Λ , l Λ ) ∈ M R × R . (A.3)The set { ( m Λ , l Λ ) } Λ ∈ Π is called defining vectors of f . We call f a Γ -lattice functionif it has a representation as (A.3) with ( m Λ , l Λ ) ∈ M × Γ for each Λ ∈ Π. We saythat f is a Γ -rational piecewise affine function if there is an integer e > ef is a Γ-lattice function. A.12.
Let f be a concave piecewise affine function on N R . Then there are m i ∈ M R , l i ∈ R , i = 1 , . . . , r , such that f is given by f ( u ) = min i =1 ,...,r h m i , u i + l i for u ∈ N R . The stability set ∆ f is a polytope in M R which is the convex hull of m , . . . , m r .The recession function rec( f ) has integral slopes if and only if the stability set∆ f is a lattice polytope. A.13.
Let f be a piecewise affine function on N R . Then we can write f = g − h ,where g and h are concave piecewise affine functions on N R . The recesssion function of f is defined as rec( f ) = rec( g ) − rec( h ).In Theorem 2.5.8 we need the following assertion. Proposition A.14.
Let Γ be a non-trivial subgroup of R . Let Ψ be a supportfunction on a complete fan in N R (Definition 2.1.9) and ψ a concave function on N R such that | ψ − Ψ | is bounded. Then there is a sequence of Γ -rational piecewiseaffine concave functions ( ψ k ) k ∈ N with rec( ψ k ) = Ψ , that uniformly converges to ψ .Proof. Since Ψ is a support function with bounded | ψ − Ψ | , the stability set ∆ Ψ is a lattice polytope in M R with ∆ Ψ = ∆ ψ . Thus, by Proposition [11, Proposition2.5.23 (2)], there is a sequence of piecewise affine concave functions ( ψ k ) k ∈ N with∆ ψ k = ∆ Ψ , that converges uniformly to ψ . Because the divisible hull of Γ lies densein R , we may assume that the ψ k ’s are Γ-rational. Finally, Proposition 2.3.10 in[11] says that ∆ ψ k = ∆ Ψ implies rec( ψ k ) = Ψ. (cid:3) A.15.
Let f be a concave function on N R . The sup-differential of f at u ∈ N R isdefined as ∂f ( u ) := { m ∈ M R | h m, v − u i ≥ f ( v ) − f ( u ) for all v ∈ N R } . For each u ∈ N R , the sup-differential ∂f ( u ) is a non-empty compact convex set.For a subset E of N R , we set ∂f ( E ) := [ u ∈ E ∂f ( u ) . Remark A.16.
Let f be a concave function on N R which is piecewise affine on acomplete fan. Then the stability set ∆ f is equal to the sup-differential ∂f (0). Thisfollows easily from the definitions. A.17.
We denote by vol M the Haar measure on M R such that M has covolumeone. Let f be a concave function on N R . The Monge-Ampère measure of f withrespect to M is defined, for any Borel subset E of N R , as M M ( f )( E ) := vol M ( ∂f ( E )) . We have for the total mass M M ( f )( N R ) = vol M (∆ f ). Proposition A.18.
Let ( f k ) k ∈ N be a sequence of concave functions on N R thatconverges uniformly to a function f . Then the Monge-Ampère measures M M ( f k ) converge weakly to M M ( f ) . Proof.
This follows from [11, Proposition 2.7.2]. (cid:3)
Proposition A.19.
Let f be a piecewise affine concave function on a completepolyhedral complex Π in N R . Then M M ( f ) = X v ∈ Π vol M ( ∂f ( v )) δ v , where δ v is the Dirac measure supported on v .Proof. This is [11, Proposition 2.7.4]. (cid:3)
A.20.
Let ∆ be an n -dimensional lattice polytope in M R and let F be a face of ∆.Then we set σ F := { u ∈ N R | h m − m ′ , u i ≥ m ∈ ∆ , m ′ ∈ F } . This is a strongly convex rational polyhedral cone which is normal to F . By settingΣ ∆ := { σ F | F (cid:22) ∆ } , we obtain a complete fan in N R . We call Σ ∆ the normal fan of ∆. The assignment F σ F defines a bijective order reversing correspondencebetween faces of ∆ and cones of Σ ∆ . The inverse map sends a cone σ to the face F σ := { m ∈ ∆ | h m ′ − m, u i ≥ m ′ ∈ ∆ , u ∈ σ } . (A.4)For details, we refer to [16, § 2.3].We also use the notation F σ in the following situation. Let Σ be a fan in N R and Ψ a support function on Σ with associated lattice polytope ∆ Ψ . For σ ∈ Σ, wedenote by F σ the face of ∆ Ψ given as in (A.4). A.21.
Let ∆ be a lattice polytope in M R . We denote by aff(∆) the affine hull of ∆and by L ∆ the linear subspace of M R associated to aff(∆). Then M (∆) := M ∩ L ∆ defines a lattice in L ∆ . The measure vol M (∆) on L ∆ = M (∆) R (see A.17) inducesa measure on aff(∆) which we also denote by vol M (∆) .If ∆ is n -dimensional and F is a facet of ∆, we denote by v F ∈ N the uniqueminimal generator of the ray σ F ∈ Σ ∆ (see A.20). We call v F the primitive innernormal vector to F . Proposition A.22.
Let f be a concave function on N R such that the stability set ∆ f is a lattice polytope of dimension n . With the notations in A.21 we have − Z N R f d M M ( f ) = ( n + 1) Z ∆ f f ∨ dvol M + X F h F, v F i Z F f ∨ dvol M ( F ) , where the sum is over the facets F of ∆ f .Proof. This is [11, Corollary 2.7.10]. (cid:3)
Appendix B. Strictly semistable models
In this appendix, we will see that the theory of strictly semistable models hasstrong similarities to the theory of toric schemes and we will use that to prove asemipositivity statement on a toric scheme of relative dimension 1 which will beuseful in § 2.5. On the way, we will prove some new results for formal models relatedto regular subdivisons of the skeleton of a given strictly semistable formal scheme.In this appendix, K is an algebraically closed field endowed with a completenon-trivial non-archimedean absolute value | | , valuation val := − log | | and corre-sponding valuation ring K ◦ . B.1. A strictly semistable formal scheme X is a connected quasi-compact admissibleformal scheme X over K ◦ which is covered by formal open subsets U admitting anétale morphism ψ : U −→ Spf (cid:0) K ◦ h x , . . . , x d i / h x · · · x r − π i (cid:1) (B.1) OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 59 for r ≤ d and π ∈ K × with | π | < strictly semistable formal model over K ◦ of a proper algebraic variety X is astrictly semistable formal scheme which is a formal K ◦ -model of X . B.2.
Let V , . . . , V R be the irreducible components of the special fiber of a strictlysemistable formal scheme X over K ◦ . Then the special fiber e X has a stratification,where a stratum S is given as an irreducible component of T i ∈ I V i \ S i I V i for any I ⊂ { , . . . , R } . We get a partial order on the set of strata by using S ≤ T if andonly if S ⊂ T .A formal open subset U as in B.1 is called a building block if the special fiber e U has a smallest stratum. This stratum is called the distinguished stratum of thebuilding block U . It is given on e U by ψ − ( x = · · · = x r = 0) in terms of the étalemorphism ψ in B.1. We note from [31, Proposition 5.2] that the building blocksform a basis of topology for the strictly semistable formal scheme X . B.3.
Berkovich showed in [3] and in [4] that for a formal strictly semistable scheme X over K ◦ there is a canonical deformation retraction τ : X η → S ( X ) to a canonicalpiecewise linear subspace S ( X ) of X η called the skeleton of X . We sketch theconstruction referring to [31, §5] for details.In the construction of S ( U ) for a building block U , one uses the skeleton S ( D ) ofthe closed affine torus D in G d +1 m given by the equation x · · · x r = π . Note that theskeleton S ( D ) can be canonically identified with { u ∈ R r +1 | u + · · · + u r = val( π ) } and hence it contains ∆( r, π ) × { } ⊂ R r +1 × R d − r for the Γ-rational standardsimplex ∆( r, π ) := (cid:8) u ∈ R r +1 ≥ | u + · · · + u r = val( π ) (cid:9) (B.2)associated to U . Then we define S ( U ) := ψ − (∆( r, π ) × { } ). One further noticesthat S ( U ) depends only on the stratum S of e X which contains the distinguishedstratum of the building block U . We call S ( U ) the canonical simplex associatedto S and we denote it by ∆ S . We note that any stratum of e X contains the distin-guished stratum of a suitable building block as a dense subset. We use the canonicalhomeomorphismVal : ∆ S −→ ∆( r, π ) , x (val( x ) , . . . , val( x r ))to see ∆ S as a Γ-rational simplex. Now the skeleton S ( X ) is defined as the unionof all S ( U ) and hence it is a compact subset of X η . The piecewise linear structureis given by the closed faces ∆ S and we have the order-reversing correspondence∆ S ⊂ ∆ T if and only if T ≤ S for strata S, T of e X . It is clear that the integralstructure is preserved on overlappings. The retraction map τ : X η → S ( X ) is givenon U η by Val and the natural identification ∆ S ∼ = ∆( r, π ). It is shown in [3,Theorem 5.2] that τ is a proper strong deformation retraction. The stratum–facecorrespondence is an order-reversing bijective correspondence between the strata S of e X and the canonical simplices ∆ S of S ( X ) given byri(∆ S ) = τ (red − ( S )) , red( τ − (ri(∆ S ))) = S. Moreover, we have dim( S ) = d − dim(∆ S ). In particular, we get a bijective corre-spondence between the irreducible components of e X and the vertices of S ( X ). Avertex u means a canonical simplex of dimension 0 and we will denote the associateirreducible component by Y u . B.4.
For every smooth projective curve X over K and every admissible formal K ◦ -model X of X , there is a strictly semistable formal model X over K ◦ such thatthe canonical isomorphism on the generic fibers extends to a morphism X → X .This is proved in [9, § 7]. In the case of a curve, the skeleton is also called the dualgraph of X . B.5.
Let X and X ′ be strictly formal schemes over K ◦ . A morphism ϕ : X ′ → X of formal schemes over K ◦ induces the map τ ◦ ϕ : S ( X ′ ) → S ( X ) which is integralΓ-affine on each canonical simplex (see [4, Corollary 6.1.3]). Here, integral Γ -affine means that the map is a translate of a linear homomorphism of the underlyingintegral structure by a Γ-rational vector.Let L be a line bundle on the proper algebraic variety X over K . We consider astrictly semistable formal K ◦ -model X of X with a line bundle L which is a formalmodel of L . This model induces a formal metric k · k L on L (see § 1.2). Proposition B.6.
Let s be an invertible meromorphic section of L and let ψ bethe restriction of − log k s k L to the skeleton S ( X ) . (i) Then ψ is a well-defined function on S ( X ) . (ii) Any canonical simplex ∆ S is covered by finitely Γ -rational polyhedra ∆ j such that ψ | ∆ j is a Γ -lattice function. (iii) If s is a nowhere vanishing global section, then the restriction of ψ to ∆ S is even a Γ -lattice function and we have ψ ◦ τ = − log k s k L on X an = X η .Proof. Since the building blocks form a basis, we may assume that X is a buildingblock U and that L = O X . Then s is a meromorphic function on X and ψ isthe restriction of − log | f | to S ( X ). Then the claim follows from the proofs ofpropositions 5.2 and 5.6 in [34]. (cid:3) Proposition B.7.
For a strictly semistable formal K ◦ -model X of the proper alge-braic variety X , there are canonical isomorphisms between the following groups: (a) the group {k · k L | L formal model of O X on X } endowed with ⊗ ; (b) the additive group of functions ψ : S ( X ) → R which are Γ -lattice functionson every canonical simplex ∆ S ; (c) the group of Cartier divisors on X which are trivial on the generic fiber X η .Proof. Since the special fiber e X of a strictly semistable formal scheme is reduced,it follows from [25, Proposition 7.5] that the map D
7→ k · k O ( D ) is an isomorphismfrom the group in (c) onto the group in (a).By Proposition B.6 and using the canonical invertible global section s := 1 of O X , we get a homomorphism k · k L ψ L := − log k s k L from the group in (a) tothe group in (b).Next, we will define a map ψ D ψ from the group in (b) to the group in (c). Let U be a building block given as in (B.1). Let S be the unique stratum of e X containingthe distinguished stratum of U in e X . Then there is m ∆ ∈ Z r +1 and α ∆ ∈ K × suchthat ψ ( u ) = h m, u i + val( α ∆ ) for all u ∈ ∆ := ∆ S = ∆( r, π ). Using the coordinates x , . . . , x r from B.1, we define the equation α ∆ ψ ∗ ( x ) m · · · ψ ∗ ( x r ) m r on U η . Weclaim that this defines a Cartier divisor D ψ on X . By construction, the absolutevalues of the equations agree on overlappings and hence the equations are equalup to units as claimed. It is clear from the construction that D ψ is trivial on thegeneric fiber.Our goal is to show that all these homomorphisms are isomorphisms. Let usconsider the formal metric k · k L for a line bundle L as in (a) and let ψ := ψ L . Weclaim that L = O ( D ψ ) as formal models for O X . We cover X by building blocks U . Then the Cartier divisor div(1) associated to the meromorphic section 1 of L is given on U by some γ ∈ O ( U η ) × . Let S be the unique stratum of e X containingthe distinguished stratum of U and let ∆ := ∆ S . We use the same equations for theCartier divisor D ψ as above. It follows from [29, Proposition 2.11] that γ agreeswith α ∆ ψ ∗ ( x ) m · · · ψ ∗ ( x r ) m r up to multiplication by a unit in K ◦ . This provesdiv(1) = D ψ and hence L = O ( D ψ ). OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 61
Conversely, if D is a Cartier divisor on X which is trivial on X η , then we alwaysfind a covering of X by building blocks U such that D is given on U by γ ∈ O ( U η ) × .Let S be the unique stratum of e X containing the distinguished stratum of U andlet ∆ := ∆ S . Then as above, we may assume that γ = α ∆ ψ ∗ ( x ) m · · · ψ ∗ ( x r ) m r for suitable m ∆ ∈ Z r +1 and α ∆ ∈ K × . Let ψ : S ( X ) → R be the restriction of − log k s D k O ( D ) to S ( X ). By construction, we have D ψ = D on U and hence allmaps between the groups in (a)–(c) are isomorphisms. (cid:3) B.8.
Let X be a strictly semistable formal model of X over K ◦ . We consider a regular subdivision D of the skeleton S ( X ) which means the following:(a) Every ∆ ∈ D is a subset of a canonical face of S ( X ) and is integral Γ-affineisomorphic to a standard simplex of the form (B.2).(b) For every canonical face ∆ S of S ( X ), the set { ∆ ∈ D | ∆ ⊂ ∆ S } is apolyhedral complex with support equal to ∆ S .We will show that the regular subdivision D induces a canonical strictly semistableformal model X ′′ of X over X with skeleton S ( X ′′ ) = S ( X ) as a subset of X an , with τ ′′ = τ for the canonical retractions and with canonical simplices of S ( X ′′ ) agreeingwith the subdivision D .We recall the construction of X ′′ from [31, 5.5, 5.6]. Let U be a building blockof X as in B.1 and let S be the unique stratum of e X containing the distinguishedstratum of e U . Then ∆ S denotes the associated canonical simplex of S ( X ). Let∆ ∈ D with ∆ ⊂ ∆ S . We identify ∆ S = ∆( r, π ) as in (B.3). Using the relation u = val( π ) − u − · · · − u r , the simplex ∆( r, π ) leads to the simplex∆ ( r, π ) = { u ∈ R r ≥ | ≤ u + · · · + u r ≤ val( π ) } and ∆ induces a Γ-rational simplex ∆ ⊂ ∆ ( r, π ). This allows us to work with the T rS -toric schemes U ∆ and U ∆( r,π ) , where T rS is the split affine torus over S of rank r with coordinates x , . . . , x r . Let U ∆ and U ∆( r,π ) be the associated formal schemesobtained by ρ -adic completion for some non-zero ρ ∈ K ◦◦ . Since ∆ ⊂ ∆ ( r, π ), wehave a canonical morphism ι : U ∆ → U ∆ ( r,π ) . Let B d − r := Spf( K ◦ h x r +1 , . . . , x d i )be the formal ball of dimension d − r . We may skip the coordinate x in (B.1) by therelation x · · · x r = π and then ψ may be seen as a morphism U → U ∆ ( r,π ) × B d − r .We form the cartesian square U ′′ ψ ′ −−−−→ U ∆ × B d − r p ′ −−−−→ U ∆ y ι ′ y ι y ι U ψ −−−−→ U ∆ ( r,π ) × B d − r p −−−−→ U ∆ ( r,π ) (B.3)where p is the first projection. This defines the building block U ′′ of X ′′ . We gluethe building blocks U ′′ along common faces of the subdivision D and then alongoverlappings of the building blocks U which leads to a formal model X ′′ of X over K ◦ . By construction, we have a canonical morphism ϕ : X ′′ → X extending theidentity on X . Since ∆ is isomorphic to a standard simplex of the form (B.2) andsince ψ ′ is étale, we conclude that X ′′ is strictly semistable. It follows from theabove construction and the definitions in B.3 that S ( X ′′ ) = S ( X ) as a set, that thecanonical faces of S ( X ′′ ) agree with the regular subdivision D and that τ ′′ = τ . B.9.
Let ϕ : X ′ → X be a morphism of strictly semistable models of the properalgebraic variety X over K extending the identity on X . Berkovich gave in [4,Theorem 4.3.1] an intrinsic characterization of the points of S ( X ) as the maximalpoints of a certain partial order on X an = X an depending canonically on the strictlysemistable formal model X . He mentioned after the definition at the beginning of [4, § 4.3.] that this partial order is compatible with morphisms of formal schemeswhich implies immediately that S ( X ) ⊂ S ( X ′ ) in our situation above and that thecontractions agree on S ( X ).It follows from the projection formula in [25, Proposition 4.5] that every irre-ducible component Y of e X is dominated by exactly one irreducible component Y ′ of e X ′ . Moreover, in this case the induced morphism Y ′ → Y is a proper birationalmorphism. We conclude from the stratum–face correspondence in B.3 that the ver-tices of S ( X ) are vertices of S ( X ′ ). It follows from B.5 that every canonical simplexof S ( X ′ ) with relative interior intersecting S ( X ) is in fact contained in a canonicalsimplex of S ( X ). Putting these two facts together, we get a regular subdivision D := { ∆ ′ ⊂ S ( X ) | ∆ ′ canonical simplex of S ( X ′ ) } of S ( X ). B.10.
Let ϕ : X ′ → X be a morphism of strictly semistable formal schemes over K ◦ extending the identity on X and let D be the regular subdivision of S ( X )constructed in B.9. We apply B.8 to this subdivision D and we get an associatedstrictly semistable formal scheme X ′′ and a canonical morphism ϕ : X ′′ → X extending the identity on X . Now let us consider a line bundle L on X which hasa formal model L on X . In this situation and for L ′ a formal model of L on X ′ , wedefine the following formal metric k · k L ′ ,S ( X ) on L : By Proposition B.7, the metric k · k L ′ / k · k L corresponds to a piecewise linear function ψ ′ on S ( X ′ ). Let ψ be therestriction of ψ ′ to S ( X ). Then ψ is also piecewise linear function satisfying therequirements of Proposition B.7(b) for the strictly semistable model X ′′ from B.8.By Proposition B.7, we get an associated vertical Cartier divisor D ψ on X ′′ and wedefine k · k L ′ ,S ( X ) := k · k O ( D ψ ) ⊗ k · k L . To get a geometric idea of this definition, we note that the value of ψ in a vertex u of S ( X ′′ ) is the multiplicity of the Weil divisor associated to D ψ in the correspondingirreducible component Y u of f X ′′ . We have a similar description for the Weil divisorof D ψ ′ in the vertices of the skeleton S ( X ′ ) and hence we may view D ψ as somesort of push-forward of D ψ ′ . This could be made more precise if the identity on X extends to a morphism X ′ → X , but we will not need it. Proposition B.11.
Under the hypotheses from B.10, we have the following prop-erties: (i)
The definition of k · k L ′ ,S ( X ) is independent of the choice of L . (ii) There is a formal model L ′′ of L on X ′′ with k · k L ′′ = k · k L ′ ,S ( X ) . (iii) The line bundle L ′′ in (ii) is unique up to isomorphism. (iv) The construction of the metric k · k L ′ ,S ( X ) is additive in L ′ . (v) If there is a morphism ϕ : X ′ → X ′′ extending the identity on X andfactorizing through ϕ with L ′ ∼ = ϕ ∗ ( L ) for a line bundle L on X ′′ , then L ′′ ∼ = L .Proof. Let L and L be two formal models of L on X . For i = 1 ,
2, we denote by ψ i the corresponding piecewise linear function on S ( X ) constructed in B.10. Thenthe canonical vertical Cartier divisor D on X with O ( D ) = L ⊗ L − correspondsin Proposition B.7 to the piecewise linear function ψ − ψ on S ( X ). By linearity,this proves (i). By construction, we get (ii). Property (iii) follows from [25, Propo-sition 7.5] using that a strictly semistable formal scheme has reduced special fiber.Property (iv) and (v) are obvious from the construction. (cid:3) Remark B.12.
Let ϕ : X ′ → X and ψ : X → Y be morphisms of strictly semistableformal models of X extending the identity on X . We assume that the line bundle L on X has a formal model on Y . For the line bundle L ′′ on X ′′ with k · k L ′′ = OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 63 k · k L ′ ,S ( X ) , we have the transitivity property k · k L ′ ,S ( Y ) = k · k L ′′ ,S ( Y ) . This follows easily from S ( X ′′ ) = S ( X ) and the construction in B.10. B.13.
Now we return to the case of a proper algebraic curve X over K with strictlysemistable formal K ◦ -model X . Then the skeleton S ( X ) is the dual graph which iscanonically a metrized graph using lattice length on each segment. For a function ψ : S ( X ) → R which is piecewise affine on each segment, we define the formal sumdiv( ψ ) := X v ∈ S ( X ) X e ∋ v d e ψ ( v )[ v ]on S ( X ), where e ranges over all edges containing v and where d e ( ψ )( v ) is the slopeof ψ at v along the edge e . We view div( ψ ) as a divisor on the dual graph. Forany P ∈ X ( K ), we define τ ∗ ([ P ]) :=[ τ ( P )] and we extend the map τ ∗ linearly to allcycles of X .The following result is due to Katz, Rabinoff and Zureick-Brown. Note that theyuse another sign in the definition of div( ψ ). Theorem B.14.
Let X be a smooth proper curve over K and let X be a strictlysemistable formal model of X over K ◦ . Let s be an invertible meromorphic sectionof the line bundle L on X with formal model L on X . Then − log k s k L restricts toa piecewise linear function ψ on S ( X ) and we have the slope formula τ ∗ (div( s )) + div( ψ ) = X v deg( L | Y v )[ v ] , where v ranges over the vertices of the dual graph S ( X ) .Proof. See [37, Proposition 2.6]. (cid:3)
Proposition B.15.
Let ϕ : X ′ → X be a morphism of strictly semistable formal K ◦ -models of the proper curve X extending the identity. We assume that the linebundle L has a formal model on X and that L ′ is a formal model of L on X ′ . Forthe canonical retraction τ : X an → S ( X ) , we have τ ∗ ( c ( L, k · k L ′ ) = c ( L, k · k L ′ ,S ( X ) ) . Proof.
We first assume that L ′ ∼ = ϕ ∗ ( L ) for a formal model L of L on X . Then theprojection formula (Proposition 1.3.10 (ii)) shows that c ( L, k · k L ′ ) = c ( L, k · k L ).By Proposition B.11(v), we have k · k L ′ ,S ( X ) = k · k L and since the Chambert-Loirmeasure c ( L, k · k L ) is supported in the vertices of S ( X ), we get the claim in thisspecial case.Now we handle the general case. Using the above and the linearity of the con-structions, we may assume that L = O X . Moreover, the above special case andProposition B.11(v) show that we may replace X and X ′ by models associated tocompatible Γ-rational subdivisions of the dual graphs S ( X ) and S ( X ′ ). Hence wemay assume that the piecewise linear map τ : S ( X ′ ) → S ( X ) from B.5 maps verticesto vertices.Now we consider a vertex u ′ of S ( X ′ ) which is not contained in S ( X ). Weclaim that any edge e ′ of S ( X ′ ) with vertex u ′ is contracted by τ . To prove that,we note that the projection formula shows that the irreducible component Y u ′ of e X ′ corresponding to u ′ is mapped to a closed point by ϕ . By the stratum–facecorrespondence in B.3, this point is contained in the dense open stratum of theirreducible component Y u of e X ′ corresponding to the vertex u = τ ( u ′ ). In particular,we deduce that the double point corresponding to the edge e ′ is mapped to thisopen stratum and hence the whole edge is mapped to u . Since the dual graph S ( X ′ ) is connected, we conclude that u ′ is connected by anedge-path to the vertex u = τ ( u ′ ) in S ( X ). Since τ is a retraction, we conclude that τ − ( u ) ∩ S ( X ′ ) is a tree with finitely many edges e i for i = 0 , . . . , r . We denote thevertices of e i by v i and w i using some orientation. Let µ = c ( L, k · k L ′ ,S ( X ) ) andlet µ ′ = c ( L, k · k L ′ ).Let ψ ′ be the restriction of − log k k L ′ to S ( X ′ ). We have seen in Theorem B.14that ψ ′ is affine on each edge of S ( X ′ ). Moreover, for a vertex w = u in the tree,we have µ ′ ( { w } ) = X v i = w d e i ψ ′ ( w ) + X w i = w d e i ψ ′ ( w )and µ ′ ( { u } ) = X v i = u d e i ψ ′ ( u ) + X w i = u d e i ψ ′ ( u ) + X e ∋ u d e ψ ′ ( u ) , where e ranges over all edges of S ( X ) with vertex u . On the other hand, we have µ ( { u } ) = X e ∋ u d e ψ ′ ( u ) . Since ψ ′ is affine on each edge e i , we have d e i ψ ′ ( v i ) = − d e i ψ ′ ( w i ). Using that wedeal with a tree, we deduce for the multiplicity ( τ ∗ ( µ ′ ))( { u } ) of the discrete measure τ ∗ ( µ ′ ) in u that( τ ∗ ( µ ′ ))( { u } ) = X w µ ′ ( { w } ) = X e ∋ u d e ψ ′ ( u ) = µ ( { u } )where w ranges over all vertices of the tree contracted to u . (cid:3) Corollary B.16.
Let L be a line bundle on the proper smooth curve X over K which has a formal model L on X and let ϕ : X ′ → X be a morphism of strictlysemistable models of X extending the identity. If L ′ is a formal model of L on X ′ such that the formal metric k · k L ′ is semipositive, then k · k L ′ ,S ( X ) is a semipositiveformal metric.Proof. It follows from Proposition B.15 that c ( L, k · k L ′ ,S ( X ) ) is a positive measureand hence k · k L ′ ,S ( X ) is a semipositive formal metric. (cid:3) Proposition B.17.
Let X Π be the toric scheme of relative dimension n over K ◦ associated to a complete Γ -rational polyhedral complex Π (see 2.2.9). Then X Π is strictly semistable if every maximal polyhedron of Π has an integral Γ -affineisomorphism onto a polyhedron of the form ∆( r, π ) × R n − r ≥ for suitable r ∈ N ≤ n and non-zero π in K ◦◦ .Proof. This follows easily from the definitions. (cid:3)
In 2.4.3, we have introduced the torification k · k S of the metric on a toric linebundle over a toric variety over K . In the next result, we apply the above theoryto the toric variety P K . Corollary B.18.
The torification of a semipositive formal metric on P K is semi-positive.Proof. Note that algebraic metrics and formal metrics are the same (see Proposition1.2.7). We choose the toric model P K ◦ . Let L be the underlying line bundle of thesemipositive formal metric k · k in question. Then L is isomorphic to O P K ( k ) forsome k ≥ O P K ◦ ( k ) on P K ◦ . Let s be anon-trivial global section and let ψ be the restriction of − log( k s k / k s k O ( k ) ) to theskeleton S ( T ) = N R of the dense torus T = P K \ { , } (see 2.4.7). Then ψ is apiecewise affine function on the skeleton S ( T ) and there is a Γ-rational polyhedral OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 65 subdivision of the fan of P K such that ψ is a Γ-lattice function on the segmentsand halflines (see Proposition 2.5.5). Let X be the associated toric model of P K .By Proposition B.17, we note that X is a strictly semistable formal scheme and itis obvious that S ( X ) is the bounded part of the polyhedral subdivision. Note alsothat ψ is constant on the two halflines as the recession function is associated to thetrivial line bundle (see Proposition 2.4.10).The formal metric k · k is given by a formal model L ′ on a formal model X ′ of P K . By B.4, we may assume that X ′ is strictly semistable and that we havea canonical morphism ϕ : X ′ → X extending the identity. By construction, themetric k · k L ′ ,S ( X ) from B.10 is the torification of k · k . Then the claim follows fromCorollary B.16. (cid:3) References [1] V.G. Berkovich.
Spectral theory and analytic geometry over non-Archimedeanfields , volume 33. American Mathematical Society, 1990.[2] V.G. Berkovich. Étale cohomology for non-Archimedean analytic spaces.
Publ.Math. IHÉS , 78(78):5–161, 1993.[3] V.G. Berkovich. Smooth p -adic analytic spaces are locally contractible. Invent.Math. , 137(1):1–84, 1999.[4] Vladimir G. Berkovich. Smooth p -adic analytic spaces are locally contractible.II. In Geometric aspects of Dwork theory. Vol. I, II , pages 293–370. Walter deGruyter GmbH & Co. KG, 2004.[5] E. Bombieri and W. Gubler.
Heights in Diophantine Geometry . CambridgeUniversity Press, 2006.[6] S. Bosch, U. Güntzer, and R. Remmert.
Non-Archimedean analysis . Springer,1984.[7] S. Bosch and W. Lütkebohmert. Formal and rigid geometry. II. Flatteningtechniques.
Math. Ann. , 296(3):403–429, 1993.[8] S. Bosch, W. Lütkebohmert, and M. Raynaud. Formal and rigid geometry. IV.The reduced fibre theorem.
Invent. Math. , 119(2):361–398, 1995.[9] Siegfried Bosch and Werner Lütkebohmert. Stable reduction and uniformiza-tion of abelian varieties. I.
Math. Ann. , 270(3):349–379, 1985.[10] J.B. Bost, H. Gillet, and C. Soulé. Heights of projective varieties and positiveGreen forms.
Journal of the American Mathematical Society , 7(4):903–1027,1994.[11] J.I. Burgos Gil, P. Philippon, and M. Sombra. Arithmetic geometry of toricvarieties. metrics, measures and heights.
Astérisque , 360, 2014.[12] J.I. Burgos Gil, P. Philippon, and M. Sombra. Height of varieties over finitelygenerated fields.
Kyoto J. Math. , 56(1):13–32, 2016.[13] J.I. Burgos Gil and M. Sombra. When do the recession cones of a polyhedralcomplex form a fan?
Discrete & Computational Geometry , 46(4):789–798,2011.[14] A. Chambert-Loir. Mesures et équidistribution sur les espaces de berkovich.
Journal für die reine und angewandte Mathematik (Crelles Journal) , 595:215–235, 2006.[15] A. Chambert-Loir and A. Thuillier. Mesures de mahler et équidistributionlogarithmique.
Annales de l’institut Fourier , 59(3):977–1014, 2009.[16] D.A. Cox, J.B. Little, and H.K. Schenck.
Toric Varieties . Graduate studies inmathematics. American Mathematical Society, 2011.[17] J.A. Dieudonné and A. Grothendieck. Éléments de géométrie algébrique: I. Lelangage des schémas.
Publ. Math. IHÉS , 4:5–228, 1960. [18] A. Ducros. Les espaces de Berkovich sont excellents.
Ann. Inst. Fourier (Greno-ble) , 59(4):1443–1552, 2009.[19] G. Faltings. Diophantine approximation on abelian varieties.
Ann. of Math.(2) , 133(3):549–576, 1991.[20] W. Fulton.
Introduction to Toric Varieties . Annals of mathematics studies.Princeton University Press, 1993.[21] W. Fulton.
Intersection Theory . Springer, 1998.[22] H. Gillet and C. Soulé. Arithmetic intersection theory.
Inst. Hautes ÉtudesSci. Publ. Math. , 72(72):93–174 (1991), 1990.[23] U. Görtz and T. Wedhorn.
Algebraic Geometry I: Schemes With Examples andExercises . Vieweg+Teubner Verlag, 2010.[24] W. Gubler. Heights of subvarieties over M -fields. In Arithmetic geometry(Cortona, 1994) , Sympos. Math., XXXVII, pages 190–227. Cambridge Univ.Press, Cambridge, 1997.[25] W. Gubler. Local heights of subvarieties over non-archimedean fields.
Journalfür die reine und angewandte Mathematik , 498:61–113, 1998.[26] W. Gubler.
Basic Properties of Heights of Subvarieties . ETH Zürich (Mathe-matikdepartement), 2002. Habilitation thesis.[27] W. Gubler. Local and canonical heights of subvarieties.
Annali della ScuolaNormale Superiore di Pisa-Classe di Scienze-Serie V , 2(4):711–760, 2003.[28] W. Gubler. The Bogomolov conjecture for totally degenerate abelian varieties.
Inventiones Mathematicae , 169(2):377–400, 2007.[29] W. Gubler. Tropical varieties for non-archimedean analytic spaces.
Inventionesmathematicae , 169(2):321–376, 2007.[30] W. Gubler. Equidistribution over function fields.
Manuscripta Mathematica ,127(4):485–510, 2008.[31] W. Gubler. Non-archimedean canonical measures on abelian varieties.
Com-positio Mathematica , 146:683–730, 4 2010.[32] W. Gubler. A guide to tropicalizations. In
Algebraic and combinatorial aspectsof tropical geometry , volume 589 of
Contemp. Math. , pages 125–189. Amer.Math. Soc., Providence, RI, 2013.[33] W. Gubler and K. Künnemann. Positivity properties of metrics and delta-forms. arXiv preprint arXiv:1509.09079 , 2015.[34] W. Gubler, J. Rabinoff, and A. Werner. Skeletons and tropicalizations.
Adv.Math. , 294:150–215, 2016.[35] W. Gubler and A. Soto. Classification of normal toric varieties over a valuationring of rank one.
Doc. Math. , 20:171–198, 2015.[36] T. Kajiwara. Tropical toric geometry. In
Toric topology , volume 460 of
Con-temp. Math. , pages 197–207. American Mathematical Society, Providence, RI,2008.[37] E. Katz, J. Rabinoff, and D. Zureick-Brown. Uniform bounds for the num-ber of rational points on curves of small mordell–weil rank.
Duke Math. J. ,165(16):3189–3240, 2016.[38] G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat.
Toroidal Embeddings1 . Lecture Notes in Mathematics. Springer, 1973.[39] S.L. Kleiman. Toward a numerical theory of ampleness.
Ann. of Math. (2) ,84:293–344, 1966.[40] A. Moriwaki. Arithmetic height functions over finitely generated fields.
Invent.Math. , 140(1):101–142, 2000.[41] S. Payne. Analytification is the limit of all tropicalizations.
MathematicalResearch Letters , 16(3):543–556, 2009.
OCAL HEIGHTS OF TORIC VARIETIES OVER NON-ARCHIMEDEAN FIELDS 67 [42] R. T. Rockafellar.
Convex analysis . Princeton Mathematical Series, No. 28.Princeton University Press, 1970.[43] Stacks Project Authors.
Stacks project. http://stacks.math.columbia.edu ,2015.[44] S.-W. Zhang. Small points and adelic metrics.
J. Algebraic Geom. , 4(2):281–300, 1995.[45] S.-W. Zhang. Equidistribution of small points on abelian varieties.
Ann. ofMath. (2) , 147(1):159–165, 1998., 147(1):159–165, 1998.