aa r X i v : . [ m a t h . AG ] A p r ON ZHANG’S SEMIPOSITIVE METRICS
WALTER GUBLER AND FLORENT MARTIN
Abstract.
Zhang introduced semipositive metrics on a line bundle of a proper variety. Inthis paper, we generalize such metrics for a line bundle L of a paracompact strictly K -analyticspace X over any non-archimedean field K . We prove various properties in this setting suchas density of piecewise Q -linear metrics in the space of continuous metrics on L . If X is properscheme, then we show that algebraic, formal and piecewise linear metrics are the same. Ourmain result is that on a proper scheme X over an arbitrary non-archimedean field K , the setof semipositive model metrics is closed with respect to pointwise convergence generalizing aresult from Boucksom, Favre and Jonsson where K was assumed to be discretely valued withresidue characteristic 0.MSC: Primary 14G40; Secondary 14G22 Contents
1. Introduction 12. Formal and piecewise linear metrics 43. Semipositive metrics 124. Plurisubharmonic model functions 165. Semipositivity and pointwise convergence 21Appendix A. Divisorial points 27References 301.
Introduction
An arithmetic intersection theory on arithmetic surfaces was introduced by Arakelov and usedby Faltings to prove the Mordell conjecture. In higher dimensions, the theory was developed byGillet and Soul´e which proved to be a very useful tool in diophantine geometry. To producearithmetic intersection numbers from a given line bundle L on a proper variety X over a numberfield K , one has to endow the complexification of L with a smooth hermitian metric and one hasto choose an O K -model ( X , L ) for ( X, L ).Zhang [Zha95] realized that the contribution of a non-archimedean place v to this arithmeticintersection number is completely determined by a metric on L ( K v ) associated with L , where K v is the completion of K at v . This adelic point of view is very pleasant as it allows to dealwith archimedean and non-archimedean places in a similar way. Motivated by his studies ofthe Bogomolov conjecture [Zha93], Zhang [Zha95] introduced semipositive adelic metrics as auniform limit of metrics induced by nef models and he showed that every polarized dynamicalsystem has a canonical metric inducing the canonical height of Call and Silverman.In [Gub98], it became clear that Zhang’s metrics can be generalized over any non-archimedeanfield K working with formal models of the line bundle over the valuation ring. It turned out thatsuch metrics are continuous on the Berkovich analytification of the line bundle and so we callthem continuous semipositive metrics. Chambert-Loir introduced measures c ( L, k k ) ∧ n on the Berkovich space X an for a continuoussemipositive metric k k of a line bundle L over X ([Cha06], [Gub07a]). These measures arenon-archimedean equidistribution measures as in Yuan’s equidistribution theorem [Yua08] overnumber fields (see also [CT09]). The analogue over function fields was proven in [Fab09], [Gub08]and gave rise to progress for the geometric Bogomolov conjecture [Gub07a], [Yam13, Yam16].Continuous semipositive metrics played an important role in the study of the arithmeticgeometry of toric varieties due to Burgos-Gil, Philippon and Sombra, see [BPS14], [BPS15],[BPS16], [BMPS16] with Moriwaki and [BPRS15] with Rivera-Letelier. Katz–Rabinoff–Zureick-Brown [KRZ16] used semipositive model metrics to give explicit uniform bounds for the numberof rational points in situations suitable to the Chabauty–Coleman method.For the non-archimedean Monge–Amp`ere problem, continuous semipositive metrics are ofcentral importance. Uniqueness up to scaling was shown by Yuan and Zhang [YZ17]. In case ofresidue characteristic 0, a solution was given by Boucksom, Favre and Jonsson [BFJ16, BFJ15]using an algebraicity condition which was removed in [BGJKM].Semipositive model metrics also played a role in the thesis of Thuillier [Thu05] on potentialtheory on curves, in the work of Chambert-Loir and Ducros on forms and currents on Berkovichspaces [CD12] and in the study of delta-forms in [GK17, GK15].Looking at the above references, one observes that the authors work either under the hypoth-esis that the valuation is discrete or that K is algebraically closed. The reasoning behind theformer is that the valuation ring and hence the models are noetherian. If K is algebraicallyclosed, then the valuation ring is not noetherian (unless the valuation is trivial, but we excludethis case here). Working with formal models using Raynaud’s theory, this is not really a prob-lem. The assumption that K is algebraically closed is used to have plenty of formal modelswhich have locally the form Spf( A ), where A is the subring of power bounded elements in an K -affinoid algebra A . It has further the advantage that finite base changes are not necessaryin the semistable reduction theorem or in de Jong’s alteration theorems. This division has theannoying consequence that many results obtained under one of these hypotheses cannot be usedunder the other hypothesis. Moreover, there is a growing group of people who would like to useZhang’s metrics over any non-archimedean base field . The goal of this paper is to remedy thissituation and to study these metrics in the utmost generality which is available to us.From now on, we assume that K is a non-archimedean field which means in this paper that K is a field endowed with a non-trivial non-archimedean complete absolute value. We denotethe valuation ring by K ◦ .We first restrict our attention to the case of a line bundle L on a proper scheme X over K .We call a metric k k on L an algebraic (resp. formal ) if it is induced by a line bundle L on aflat proper scheme X (resp. a line bundle L on an admissible formal scheme X ) over K ◦ withgeneric fibre X and with L = L | X . We use the notation k k = k k L . Such a metric is called semipositive if L (resp. L ) restricts to a nef line bundle on the special fibre of X (resp. X ).More generally, we call k k a model metric if there is a non-zero k ∈ N such that k k ⊗ k is analgebraic metric. Then a model metric k k is called semipositive if k k ⊗ k is semipositive in theprevious sense. We say that k k is a continuous semipositive metric if it is the uniform limit ofa sequence of semipositive model metrics on L an .We note that the above definitions are global definitions. It is desirable to have local analyticdefinitions. Let V be a paracompact strictly K -analytic space and L a line bundle on V . First,we say that a metric k k on L is a piecewise linear metric if there is a G-covering ( V i ) i ∈ I of V (i.e.a covering with respect to the G-topology on V ) and frames s i of L over V i with k s i k ≡
1. Notethat such metrics are already considered in [Gub98], but they were called formal there which is The new paper of Boucksom and Eriksson [BE18] is an example for this point of view.
N ZHANG’S SEMIPOSITIVE METRICS 3 a bit confusing. We say that a metric k k is piecewise Q -linear if there is a G-covering ( V i ) i ∈ I of V and some integers ( k i ) i ∈ I such that for each i ∈ I , the restriction of k k ⊗ k i to V i is a piecewiselinear metric on V i . We refer to Section 2 for details and properties.Following a suggestion of Tony Yue Yu, we call a piecewise linear metric k k semipositive at x ∈ V if x has a strictly K -affinoid domain W of V as a neighborhood (in the Berkovichtopology) such that the restriction of k k to L | W is a semipositive formal metric. This notionwas studied in [GK15] for K algebraically closed. A semipositive piecewise linear metric on L isa piecewise linear metric which is semipositive at every x ∈ V . Semipositive metrics are studiedin Section 3. We highlight here the following result which is useful in comparing the variousdefinitions mentioned above. Theorem 1.1.
The following are equivalent for a metric k k on the line bundle L an over aproper scheme X : (a) k k is an algebraic metric; (b) k k is a formal metric; (c) k k is a piecewise linear metric.The equivalence remains true if we replace “metric” by “semipositive metric” in every item. As seen in Remark 2.6, the equivalence of (a) and (b) follows from [GK17, Proposition 8.13](as the argument does not use the assumption that K is algebraically closed). The equivalenceof (b) and (c) holds more generally over any paracompact strictly K -analytic space as shown inProposition 2.10. This equivalence was known before only in case of a compact reduced spaceover an algebraically closed field. Neither base change nor the old argument can be used andso we give an entirely new argument here. In the semipositive case, the equivalence of (a) and(b) follows immediately from Proposition 3.5. Finally, the equivalence of (b) and (c) is shown inProposition 3.11. It holds more generally for a separated paracompact strictly K -analytic space.We also prove the following result (Theorem 2.17) which generalizes [Gub98, Theorem 7.12]from the compact to the paracompact case. Theorem 1.2.
Let V be a paracompact strictly K -analytic space with a line bundle L . If k k isa continuous metric on L , then there is a sequence ( k k n ) n ∈ N of piecewise Q -linear metrics on L which converges uniformly to k k . Let us come back to semipositive metrics. For this, let us consider X a proper scheme over K .It is a natural question if the notion of semipositivity is closed in the space of model metrics of agiven line bundle L of X . First, we look at this question for uniform convergence of metrics. Weconsider a model metric k k on L an which is semipositive as a continuous metric, which meansby definition that it is uniform limit of semipositive model metrics on L an . Then the closednessproblem is equivalent to show that k k is semipositive as a model metric. By passing to a tensorpower, we may assume that k k = k k L for a line bundle L on a model X of X . By assumption, k k is the uniform limit of semipositive model metrics k k n on L an . For every n ∈ N , there is anon-zero k n ∈ N such that k k ⊗ k n n is an algebraic metric associated with a nef line bundle L n living on a proper flat scheme X n over K ◦ with generic fiber X . Since the models X n might becompletely unrelated to X , it is non-obvious to show that L is nef if all the line bundles L n are nef.An even more challenging problem is to show that the space of model metrics is closed withrespect to pointwise convergence. The solution of this problem is the main result of this paper: Theorem 1.3.
Let X be a proper scheme over K with a line bundle L . We assume that themodel metric k k on L an is a pointwise limit of semipositive model metrics on L an . Then k k isa semipositive model metric. WALTER GUBLER AND FLORENT MARTIN If K is discretely valued of residue characteristic zero and if X is a smooth projective variety,then this theorem was proven by Boucksom, Favre and Jonsson [BFJ16, Theorem 5.11] usingmultiplier ideals. They said in [BFJ16] Remark 5.13 that it would be interesting to have a proofalong the lines of Goodman’s paper [Goo69, p.178, Proposition 8]. This is what we providein Theorem 1.3 with a proof holding for any non-archimedean field and hence we obtain as animmediate consequence: Corollary 1.4.
A model metric is semipositive as a model metric if and only if it is semipositiveas a continuous metric.
For arbitrary non-archimedean fields, this result was first proven in [GK15, Proposition 8.13]using a lifting theorem for closed subvarieties of the special fibre. Amaury Thuillier told us thathe found a similar (unpublished) lifting argument to prove Corollary 1.4.Theorem 1.3 will follow from Theorem 5.5 which is a slightly more general version aboutpointwise convergence of θ -plurisubharmonic model functions for a closed (1 , θ . Thesenotions from [BFJ16] will be introduced in Section 4. In Theorem 1.3 and in Theorem 5.5, itis enough to require pointwise convergence over all divisorial points of X an . Such points will beintroduced and studied in Appendix A.1.1. Terminology.
For sets, in A ⊂ B equality is not excluded and A \ B denotes the comple-ment of B in A . N includes 0. All the rings and algebras are commutative with unity. For a ring A , the group of units is denoted by A × . If V is a topological space, for a set U ⊂ V we denoteby U ◦ the topological interior of U in V . A variety over a field k is an irreducible and reducedscheme which is separated and of finite type over k .For the rest of the paper we fix a non-archimedean field K . This means here that the field K isequipped with a non-archimedean absolute value | | : K → R + which is complete and non-trivial.Let v := − log | | be the corresponding valuation. We have a valuation ring K ◦ := { x ∈ K | v ( x ) ≥ } with maximal ideal K ◦◦ := { x ∈ K | v ( x ) > } and residue field ˜ K := K ◦ /K ◦◦ . Weset Γ := v ( K × ). It is a subgroup of ( R , +) called the value group of K . We denote by K analgebraic closure of K and we set C K for the completion of K . It is a minimal algebraicallyclosed non-archimedean field extension of K [BGR84, § Acknowledgements.
We thank Vladimir Berkovich, Antoine Ducros and Tony Yue Yufor helpful discussions. We thank Ofer Gabber for thoroughly answering a question posed byemail and we thank the referees for their helpful comments. This work was supported by thecollaborative research center SFB 1085 funded by the Deutsche Forschungsgemeinschaft.2.
Formal and piecewise linear metrics
For line bundles on paracompact strictly K -analytic spaces, we will introduce the global notionof formal metrics and the local notion of piecewise linear metrics. We will collect many propertiesand we will show that both notions agree. At the end, we will prove a density result for piecewise Q -linear metrics. Let X be a proper scheme over K . Then an algebraic K ◦ -model of X is a proper flat scheme X over K ◦ with a fixed isomorphism from the generic fiber X η to X . Usually, we will identify X η with X along this fixed isomorphism. It follows from Nagata’s compactification theorem[Con07, Theorem 4.1] that an algebraic K ◦ -model of X exists. The set of isomorphism classesof algebraic K ◦ -models of X is partially ordered by morphisms of K ◦ -models of X (where bydefinition such a map extends the identity on X ). A diagonal argument shows easily that theset of isomorphism classes is directed with respect to this partial order. N ZHANG’S SEMIPOSITIVE METRICS 5
Let L be a line bundle on X . An algebraic K ◦ -model ( X , L ) of ( X, L ) consists of an algebraic K ◦ -model X of X and of a line bundle L on X with a fixed isomorphism from L | X to L whichwe use again for identification.It follows from Vojta’s version of Nagata’s compactification theorem [Voj07, Theorem 5.7]and noetherian approximation that ( X, L ) has always an algebraic K ◦ -model. Alternatively,one can use the non-noetherian version of Nagata’s compactification theorem [Con07, Theorem4.1] to get an algebraic K ◦ -model X of X , then by [Sta16, Tag 01PI] one can extend L to an O X -module of finite presentation F , and finally by [RG71, Th´eor`eme 5.2.2], replacing X by adominating K ◦ -model, one can ensure that F is flat, hence a line bundle on X . Let V be a paracompact strictly K -analytic space. We use here the analytic spaces andthe terminology introduced by Berkovich in [Ber93, Section 1]. Then a formal K ◦ -model is anadmissible formal scheme V over K ◦ [Bos14, § V η ∼ = V on thegeneric fiber V η which we again use for identification. Note that we have a canonical reductionmap π : V → V s to the special fiber V s (see [GRW17, Section 2]). We say that a covering ( V i ) i ∈ I of V is of finite type if for each i ∈ I , the intersection V i ∩ V j is nonempty only for finitely many j ∈ I .The category of paracompact strictly K -analytic spaces is equivalent to the category of qua-siseparated rigid analytic varieties over K with a strictly K -affinoid G-covering of finite type(see [Ber93, § K ◦ -model of V exists and that the set of isomorphism classesof formal K ◦ -models is again directed. Some of the references in the following require that V iscompact, because the original formulation of Raynaud’s theorem in [BL93a, Theorem 4.1] usedthat the underlying rigid space is quasicompact and quasiseparated. This will be bypassed byusing the more general version in [Bos14, Theorem 8.4.3] for paracompact V (remember thatparacompact includes Hausdorff).We always consider the G-topology on V induced by the strictly K -affinoid domains in V (see[Ber93, § K -affinoiddomains may be seen as the basic open subsets of the G-topology while they are compact in theBerkovich topology of V . There is a canonical structure sheaf O X G on the G-topology of V suchthat for every strictly K -affinoid domain W of V the corresponding strictly K -affinoid algebrais O X G ( W ). A G-covering ( V i ) i ∈ I is called of finite type if for every i ∈ I there are only finitelymany j ∈ I with V i ∩ V j = ∅ .Let L be a line bundle on V which means that L is a locally free sheaf of rank 1 on theG-topology. A formal K ◦ -model ( V , L ) of ( V, L ) consists of a formal K ◦ -model V of V and aline bundle L on V with a fixed isomorphism from L | V to L which we use for identification. By[CD12, Proposition 6.2.13], a formal K ◦ -model of ( V, L ) always exists.
Remark 2.3. If X is a proper scheme over K with a line bundle L , then we denote the an-alytifications by X an and L an (in the category of Berkovich spaces). By formal completion,every algebraic K ◦ -model ( X , L ) of ( X, L ) induces a formal K ◦ -model ( ˆ X , ˆ L ) of ( X an , L an ).Note that the special fiber X s of X is canonically isomorphic to the special fiber of the formalcompletion ˆ X and hence the above yields a reduction map π : X an → X s . Lemma 2.4.
Let X be a proper scheme over K and let X be a formal K ◦ -model of X an . Thenthere exists an algebraic K ◦ -model X of X such that ˆ X dominates X .Proof. Let us fix an algebraic K ◦ -model X of X . As recalled in 2.2, the set of isomorphismclasses of formal K ◦ -models of X an is a directed set, hence there exists a formal K ◦ -model V which dominates both c X and X . Replacing V by a larger formal K ◦ -model of X an , it followsfrom [Bos14, Lemma 8.4.4 (d)] that the canonical map ϕ : V → c X may be assumed to be WALTER GUBLER AND FLORENT MARTIN an admissible formal blowing up in an open coherent ideal b of c X . Using the formal GAGA-principle proved by Fujiwara–Kato [FK18, Theorem I.10.1.2], b is actually the formal completionof a coherent vertical ideal a on X . Hence if X is the vertical blowing up of X in the ideal a ,by [Bos14, Proposition 8.2.6] we have V ∼ = ˆ X which dominates X . (cid:3) Definition 2.5.
Let ( V , L ) be a formal K ◦ -model of ( V, L ) as in 2.2. Then we get an associated formal metric k k L on L uniquely determined by requiring k s k L = 1 on the generic fibre W ofany frame s of L over any formal open subset W of V . This is well-defined because a change offrame involves an invertible function f on W and we have | f | = 1 on W . Remark 2.6.
If ( X , L ) is an algebraic K ◦ -model of ( X, L ) as in 2.1, then we get an associated algebraic metric k k L on L an by using the above construction for the formal K ◦ -model ( ˆ X , ˆ L )of ( X an , L an ) from Remark 2.3. By construction, every algebraic metric is a formal metric. Theconverse is also true as shown in [GK17, Proposition 8.13] (as the argument does not use theassumption that K is algebraically closed).We have the following extension result from [GK15, Proposition 5.11] Proposition 2.7.
Let L be line bundle on a paracompact strictly K -analytic space V and let W be a compact strictly K -analytic domain of V . Then every formal metric on the restriction of L to W extends to a formal metric on L .Proof. Since this is stated here under more general assumptions than in [GK15, Proposition5.11], we sketch the argument. Let ( W , L ) be the formal K ◦ -model for the given formal metricon L | W . We may assume that W is a formal open subset of a formal K ◦ -model V of V [Bos14,Lemma 8.4.5]. By the argument in [BL93a, Lemma 5.7], there is a coherent O V -module F on V which extends L . This works even for paracompact V as noted in the proof of [CD12,Proposition 6.2.13] and the argument there (or in the proof of [Gub98, Lemma 7.6]) shows thatafter replacing V by a suitable admissible blowing-up, we may assume that F is a line bundle.Then the associated formal metric satisfies the claim. (cid:3) Definition 2.8.
Let V be a paracompact strictly K -analytic space with a line bundle L . Ametric k k on L is called piecewise linear if there is a G-covering ( V i ) i ∈ I and frames s i of L over V i for every i ∈ I such that k s i k = 1 on V i . A function ϕ : V → R is called a piecewise linearfunction if it induces a piecewise linear metric on the trivial line bundle O V . Note that theseare G-local definitions (see [GK15, Proposition 5.10] for the argument). In particular, theseproperties are local with respect to the Berkovich topology. Lemma 2.9.
Any given G -covering of a connected paracompact strictly K -analytic space V canbe refined to an at most countable G -covering ( W i ) i ∈ I of finite type (see 2.2) made by strictly K -affinoid domains W i . Moreover, there is always a second G -covering ( U i ) i ∈ I of finite typemade by compact strictly K -analytic domains U i such that every W i is contained in the interior U ◦ i of U i with respect to the Berkovich topology.Proof. By [Bou71, chap. 1, §
9, Th´eor`eme 5], V is countable at infinity. For the proof of thelemma, we assume that V is not compact (the compact case is similar and even easier). Sincecompact strictly K -analytic domains of V form a basis of neighborhoods of V , we deduce thatthere is a sequence ( T j ) j ∈ N of compact strictly K -analytic domains T j of V such that T j ⊂ T ◦ j +1 for all j ∈ N and V = ∪ j ∈ N T j .For each j ∈ N , T ◦ j +3 \ T j is an open neighborhood of the compact set T j +2 \ T ◦ j +1 . Since compactstrictly K -analytic domains of T j +3 \ T j contain a basis of neighborhoods of T j +3 \ T j , for each j ∈ N , we can find m j ∈ N and finitely many compact strictly K -analytic domains ( T jk ) k =0 ,...,m j contained in T ◦ j +3 \ T j and covering T j +2 \ T ◦ j +1 . Since V = T ∪ S j ∈ N ( T j +2 \ T ◦ j +1 ), we deduce N ZHANG’S SEMIPOSITIVE METRICS 7 that the covering ( V h ) h ∈ H defined after re-indexing the covering { T } ∪ { T jk } j ∈ N , k =0 ,...,m j is acountable G-covering of finite type (to see that ( V h ) h ∈ H is a G-covering, one can use [Ber93,Lemma 1.6.2 (ii)]). Since any compact strictly K -analytic domain is a finite union of strictly K -affinoid domains, we can easily assume that every V h is actually a strictly K -affinoid domain.Let us first construct the covering ( W i ) i ∈ I refining a given G-covering ( Z l ) l ∈ L of V . Wecan replace the latter by a finer G-covering and hence we may assume that every Z l is astrictly K -affinoid domain. For each index h ∈ H , there are finitely many l h , . . . , l hq h ∈ L such that the family { Z l hq ∩ V h } q =1 ,...,q h is a G-covering of V h . Hence the countable family( Z l hq ∩ V h ) h ∈ H,q =1 ,...,q h is a G-covering of V of finite type refining ( Z l ) l ∈ L . By assumption, theunderlying topological space of V is Hausdorff and hence it follows from [Ber93, Theorem 1.6.1]that the intersection of two strictly K -affinoid domains is a finite union of strictly K -affinoiddomains. We conclude that every Z l hq ∩ V h is a finite union of strictly K -affinoid domains. Usingthem all, we get a G-covering ( W i ) i ∈ I of V of finite type by strictly K -affinoid domains W i refining ( Z l ) l ∈ L .Finally, for any G-covering ( W i ) i ∈ I of finite type by strictly K -affinoid domains W i , we con-struct a G-covering ( U i ) i ∈ I with the required properties. For i ∈ I and using the above notations,let j ∈ N be the largest number such that W i ∩ T j = ∅ . If T ∩ W i is non-empty, then we set j := − T j := ∅ . Since the strictly K -analytic domains form a basis of neighborhoods in V ,there is for every x ∈ W i a strictly K -analytic domain U x of V such that U x is a neighborhood of x contained in the complement of T j . Since W i is compact, it is covered by finitely many U ◦ x . Let U i be the union of these finitely many U x . Then U i is a compact strictly K -analytic domain of V contained in the complement of T j and with W i ⊂ U ◦ i . Since ( W i ) i ∈ I is a G-covering refining( U i ) i ∈ I , the latter is also a G-covering of V .It remains to prove that ( U i ) i ∈ I is a covering of finite type. We pick k ∈ I . Since U k iscompact, there is a j ∈ J such that U k ⊂ T ◦ j . Since ( W i ) i ∈ I is a G-covering, [Ber93, Lemma1.6.2 (ii)] again shows that the compact set T j is covered by finitely many of the W i . Since( W i ) i ∈ I is a covering of finite type, we conclude that T j is intersected by at most finitely many W i . Let us choose any i ∈ I with W i ∩ T j = ∅ . By construction of U i , we have U i ∩ T j = ∅ andhence U i is disjoint from U k ⊂ T j . We conclude that the covering ( U i ) i ∈ I is of finite type. (cid:3) Proposition 2.10.
Let k k be a metric on a line bundle L on a paracompact strictly K -analyticspace V . Then k k is formal if and only if it is piecewise linear.Proof. Clearly, every formal metric is piecewise linear. Let us prove the converse. For a piecewiselinear metric k k on L , there is a G-covering ( V i ) i ∈ I of V with frames s i of L | V i such that k s i k = 1on V i . By Lemma 2.9, we may assume that the G-covering is of finite type and that every V i isa strictly K -affinoid domain. By Raynaud’s theorem [Bos14, Theorem 8.4.3] and using [Bos14,Lemma 8.4.5], there is a formal K ◦ -model V and a covering ( V i ) i ∈ I of V of finite type by quasi-compact formal open subschemes V i with generic fiber V i . Note that any formal K ◦ -modelis quasi-separated [Bos14, bottom of p. 204]. For every i, j ∈ I , we conclude that the formalopen subscheme V i ∩ V j is a finite union of formal affine open subschemes V ijk = Spf( A ijk ).For f ij := s i /s j ∈ O ( V i ∩ V j ) × , the identity k s i k ≡ k s j k on V i ∩ V j yields that | f ij | ≡ V i ∩ V j . Then [Bos14, Lemma 8.4.6] shows that V ′ ijk = Spf( A ijk [ f ijk , f jik ]) is an admissibleformal scheme and that the canonical morphism V ′ ijk → V ijk is an admissible formal blowingup. By construction, we have f ij ∈ O ( V ′ ijk ) × .We apply now [Bos14, Proposition 8.2.14] to the covering { V ijk } of V of finite type. Thisgives the existence of an admissible formal blowing up ϕ : V ′ → V which factorizes through V ′ ijk → V ijk for every ijk . We note that ( ϕ − ( V i )) i ∈ I is a formal open covering of V ′ of finite WALTER GUBLER AND FLORENT MARTIN type and that ϕ − ( V i ) ∩ ϕ − ( V j ) = ϕ − ( V i ∩ V j ) = [ k ϕ − ( V ijk ) . Since ϕ − ( V ijk ) is the preimage of V ′ ijk with respect to V ′ ijk → V ijk , the above factorizationyields f ij ◦ ϕ ∈ O (( ϕ − ( V ijk )) × for every ijk and hence f ij ◦ ϕ ∈ O ( ϕ − ( V i ) ∩ ϕ − ( V j )) × . Let L be the model of L on V ′ given by the transition functions f ij ◦ ϕ with respect to the covering( ϕ − ( V i )) i ∈ I . Then the construction shows that k k = k k L . (cid:3) Definition 2.11.
Let V be a paracompact strictly K -analytic space with a line bundle L . Ametric k k on L is called piecewise Q -linear if for every x ∈ V there exists an open neighborhood W of x and a non-zero n ∈ N such that k k ⊗ n | W is a piecewise linear metric on L ⊗ n | W . A function ϕ : V → R is called a piecewise Q -linear function if it induces a piecewise Q -linear metric on thetrivial line bundle O V . Proposition 2.12.
Let V be a paracompact strictly K -analytic space with a line bundle L . Thenthe following properties hold: (a) A piecewise Q -linear metric on L is continuous. (b) The isometry classes of piecewise linear (resp. piecewise Q -linear) metrics on line bundlesof V form an abelian group with respect to ⊗ . (c) The pull-back f ∗ k k of a piecewise linear (resp. piecewise Q -linear) metric k k on L with respect to a morphism f : W → V of paracompact strictly K -analytic spaces is apiecewise linear (resp. piecewise Q -linear) metric on f ∗ L . (d) The minimum and the maximum of two piecewise linear (resp. piecewise Q -linear) met-rics on L are again piecewise linear (resp. piecewise Q -linear) metrics on L .Proof. These properties are proved in [Gub98, Section 7] under the assumption that K is al-gebraically closed and V is compact. The assumption K algebraically closed was not used inthe arguments. Since (a)–(d) are local statements, we can deduce them from the correspondingstatements in loc. cit. (cid:3) Let V be a paracompact strictly K -analytic space. Recall that for U ⊂ V , we denote thetopological interior of U in V by U ◦ . Lemma 2.13.
Let W ⊂ U ⊂ V where W, U are compact strictly K -analytic domains of V with W ⊂ U ◦ . Let f : W → R be a piecewise linear function. Then f extends to a piecewise linearfunction ϕ : V → R such that supp( ϕ ) ⊂ U .Proof. By compactness of U \ U ◦ , there exists a compact strictly K -analytic domain Z ⊂ V such that Z is a neighborhood of U \ U ◦ and W ∩ Z = ∅ . Hence W ` Z is a compact strictly K -analytic domain of V and we consider the piecewise linear function on W ` Z defined by f on W and by 0 on Z . Then we apply Proposition 2.7 to L = O V , in which case formal metricscorrespond to piecewise linear functions (see Proposition 2.10). We deduce that there exists apiecewise linear function g : V → R which agrees with f on W and which agrees with 0 on Z .But since Z is a neighborhood of U \ U ◦ , we deduce that the function ϕ : V → R defined by ϕ ( x ) = ( g ( x ) if x ∈ U x / ∈ U is still piecewise linear as this is a local property. Since ϕ extends f and supp( ϕ ) ⊂ U , we getthe claim. (cid:3) N ZHANG’S SEMIPOSITIVE METRICS 9
Lemma 2.14.
Let V be a paracompact strictly K -analytic space. Let W ⊂ V be a compactstrictly K -analytic domain of V and let f : W → R be a continuous function with f ≥ . Thenfor any ε > there exists a piecewise Q -linear function ϕ on V such that ϕ ≥ and for all x ∈ W we have f ( x ) − ε ≤ ϕ ( x ) ≤ f ( x ) .Proof. Since piecewise Q -linear functions are dense in the compact case [Gub98, Theorem 7.12],there exists a piecewise Q -linear function g : W → R such that f − ε ≤ g ≤ f on W . Since W is compact, there is a non-zero k ∈ N such that kg is piecewise linear. By Proposition 2.7 andProposition 2.10 applied to the formal metric on O V associated with kg , there exists a piecewise Q -linear function ψ : V → R which extends g . We then set ϕ := max( ψ, ϕ is piecewise Q -linear. By definition, we have ϕ ≥
0. We have ψ ≤ f on W and f is non-negative, hence we have ϕ ≤ f on W . Finally, since f − ε ≤ ψ on W we also have that f − ε ≤ max( ψ,
0) = ϕ on W . (cid:3) Proposition 2.15.
Let V be a paracompact strictly K -analytic space. Let f : V → R be a con-tinuous function on V . Then f can be uniformly approximated by piecewise Q -linear functions.In other words, for every ε > there exists a piecewise Q -linear function ϕ : V → R such that sup x ∈ V | f ( x ) − ϕ ( x ) | ≤ ε .Proof. We will use that the result holds when V is compact [Gub98, Theorem 7.12]. Note that in[Gub98, § K was assumed to be algebraically closed, but the argument for [Gub98, Theorem7.12] does not use this assumption and so we can use the result over any non-archimedean field.Let f + := max( f,
0) and f − := max( − f,
0) so that f = f + − f − . Hence replacing f by f + or f − we can assume that f ≥ V , hence we may assume that V isconnected. By Lemma 2.9, we can find ( W i ) i ∈ I and ( U i ) i ∈ I two G -coverings of V of finite typewith a finite or countable I and W i ⊂ U ◦ i for all i ∈ I . In the following, we assume I = N \ { } .The finite case is similar and easier. Let us now fix ε > Q -linear functions ( ϕ i ) i ∈ I with ϕ i : V → R such that(i) for all i ∈ I , supp( ϕ i ) ⊂ U i and ϕ i ≥ n ∈ I we have f ≥ P ni =1 ϕ i ≥ f − ε on ∪ ni =1 W i .(iii) f ≥ P ni =1 ϕ i on V .Observe that this will conclude the proof of the proposition since then ϕ := P i ∈ I ϕ i is a welldefined piecewise Q -linear function such that | f − ϕ | ≤ ε . The rest of the proof is dedicated toconstruct inductively a family ( ϕ i ) i ∈ I satisfying the conditions (i), (ii) and (iii).Let us consider n ≥ Q -linear functions ϕ , . . . , ϕ n satisfying the above conditions. We will now construct a piecewise Q -linear function ϕ n +1 such that ϕ , . . . , ϕ n +1 satisfies the conditions (i), (ii) and (iii).By the density result in the compact case [Gub98, Theorem 7.12], we know that there existsa piecewise Q -linear function g : W n +1 → R such that(2.15.1) f − n X i =1 ϕ i − ε ≤ g ≤ f − n X i =1 ϕ i on W n +1 Then by Lemma 2.13 applied to g and W n +1 ⊂ U n +1 ⊂ V , there exists a piecewise Q -linearfunction Ψ : V → R which extends g and with supp(Ψ) ⊂ U n +1 . Then (2.15.1) becomes(2.15.2) f − ε ≤ Ψ + n X i =1 ϕ i ≤ f on W n +1 . Then we set ψ := max(0 , Ψ) . From this definition, we get that supp( ψ ) ⊂ supp(Ψ) ⊂ U n +1 . It is a piecewise Q -linear functionby Proposition 2.12 (d) and it satisfies ψ ≥
0. Now, (2.15.2) combined with the condition (iii)for n yields(2.15.3) ψ + n X i =1 ϕ i ≤ f on W n +1 . Also, since Ψ ≤ ψ , we deduce from (2.15.2) that(2.15.4) f − ε ≤ ψ + n X i =1 ϕ i on W n +1 . On the other hand, since ψ ≥
0, the condition (ii) for n yields(2.15.5) f − ε ≤ ψ + n X i =1 ϕ i on n [ i =1 W i . From (2.15.4) and (2.15.5), we deduce that(2.15.6) f − ε ≤ ψ + n X i =1 ϕ i on n +1 [ i =1 W i . Lemma 2.14 applied to the non negative function f − P ni =1 ϕ i : V → R and to the compact K -analytic domain ∪ n +1 i =1 U i yields a piecewise Q -linear function χ : V → R such that χ ≥ f − n X i =1 ϕ i − ε ≤ χ ≤ f − n X i =1 ϕ i on n +1 [ i =1 U i . We then set ϕ n +1 := min( ψ, χ ) . By Proposition 2.12 (d), ϕ n +1 is a piecewise Q -linear function. Since ψ ≥ χ ≥ ϕ n +1 ≥ x ∈ V , ψ ( x ) = 0 ⇒ ϕ n +1 ( x ) = 0. This implies thatsupp( ϕ n +1 ) ⊂ supp( ψ ) ⊂ U n +1 . Hence (i) is satisfied for ϕ n +1 .Let us now prove that(2.15.8) n +1 X i =1 ϕ i ≤ f on V. Let x ∈ V . We first suppose that x ∈ U n +1 . Then by (2.15.7), we have χ ( x )+ P ni =1 ϕ i ( x ) ≤ f ( x ).By definition of ϕ n +1 , we have ϕ n +1 ≤ χ hence ϕ n +1 ( x ) + n X i =1 ϕ i ( x ) ≤ χ ( x ) + n X i =1 ϕ i ( x ) ≤ f ( x ) . If x / ∈ U n +1 , then we have ψ ( x ) = 0 since supp( ψ ) ⊂ U n +1 , hence ϕ n +1 ( x ) = 0. So by thecondition (iii) for n , we get n +1 X i =1 ϕ i ( x ) = n X i =1 ϕ i ( x ) ≤ f ( x ) . This proves (2.15.8), whence condition (iii) holds for n + 1.Let us finally prove that f − ε ≤ n +1 X i =1 ϕ i ≤ f on n +1 [ i =1 W i . N ZHANG’S SEMIPOSITIVE METRICS 11
The right inequality has been proven in (2.15.8) so it only remains to prove the left inequality.By (2.15.6), we have(2.15.9) f − ε ≤ ψ + n X i =1 ϕ i on n +1 [ i =1 W i and by construction (see (2.15.7) having in mind that W i ⊂ U i ), we have(2.15.10) f − ε ≤ χ + n X i =1 ϕ i on n +1 [ i =1 W i . Hence (2.15.9) and (2.15.10) yield that f − ε ≤ min( ψ, χ ) + n X i =1 ϕ i = n +1 X i =1 ϕ i on n +1 [ i =1 W i which proves condition (ii) for ϕ , . . . , ϕ n +1 . By induction, this proves the existence of a family( ϕ i ) i ∈ I satisfying conditions (i), (ii) and (iii). (cid:3) Remark 2.16.
The proof of Proposition 2.15 also gives that if ϕ : V → R is a piecewise Q -linearfunction on a paracompact strictly K -analytic space V , then there exists a family ( ϕ i ) i ∈ I ofpiecewise Q -linear functions on V such that the family supp( ϕ i ) i ∈ I is a locally finite family ofcompact sets subordinate to any given open covering of V and such that ϕ = P i ∈ I ϕ i . Indeed,in the above proof we may construct the covering U i finer than the given open covering and thenwe may use ε = 0 in the construction due to piecewise Q -linearity. Theorem 2.17.
Let V be a paracompact strictly K -analytic space with a line bundle L . If k k is a continuous metric on L , then there is a sequence ( k k n ) n ∈ N of piecewise Q -linear metrics on L which converges uniformly to k k .Proof. We have seen at the end of 2.2 that L admits a formal metric. Hence, tensoring by L − , wecan assume that L = O V and we are reduced to prove that for any continuous function f : V → R there exists a sequence of piecewise Q -linear functions ( ϕ n ) n ∈ N which converges uniformly to f which was done in Proposition 2.15. (cid:3) The next result deals with base change of piecewise linear metrics. We denote by ˆ ⊗ K F thebase change functor from the base field K to a non-archimedean extension field F applied to thecategory of strictly K -analytic spaces or to the line bundles on such spaces. The argument for(b) is due to Yuan (see [Yua08, Lemma 3.5]). Proposition 2.18.
Let L be a line bundle on a paracompact strictly K -analytic space V and let F/K be a non-archimedean field extension. (a)
The base change of a piecewise linear (resp. piecewise Q -linear) metric on L is a piecewiselinear (resp. piecewise Q -linear) metric on L ˆ ⊗ K F . (b) If F is a subfield of C K and if V is compact, then every piecewise linear (resp. piecewise Q -linear) metric on L ˆ ⊗ K F is the base change of a unique piecewise linear (resp. piecewise Q -linear) metric on L ˆ ⊗ K K ′ for a suitable finite subextension K ′ /K of F/K .Proof.
It follows from [Ber93, Theorem 1.6.1] that the base change of V to F is a paracompactstrictly F -analytic space. Property (a) is obvious.To prove (b), we assume that k k is a piecewise linear metric on L ˆ ⊗ K F . We have seen in 2.2that ( V, L ) has a formal K ◦ -model ( V , L ) and so we may assume that L = O V by passing to k k / k k L ˆ ⊗ K ◦ F ◦ . By Proposition 2.10, there is a formal F ◦ -model ( V ′′ , L ′′ ) of ( V ˆ ⊗ K F, L ˆ ⊗ K F )such that k k = k k L ′′ . By Raynaud’s theorem [BL93a, Theorem 4.1], we may assume that there is an admissible formal blowing up V ′′ → V ˆ ⊗ K ◦ F ◦ . Note that L = O V yields that L ′′ = O ( E )for a vertical Cartier divisor E on V ′′ . Replacing k k by a suitable multiple, we may assumethat E is an effective Cartier divisor.An approximation argument based on the density of the algebraic closure of K in F showsthat the coherent ideal of the admissible formal blowing up is defined over ( K ′ ) ◦ for a finitesubextension K ′ /K of F/K . We conclude that V ′′ → V ˆ ⊗ K ◦ F ◦ is the base change of an admis-sible formal blowing up V ′ → V ˆ ⊗ K ◦ ( K ′ ) ◦ for a formal ( K ′ ) ◦ -model V ′ of V ˆ ⊗ K K ′ . We choosea finite covering ( U ′ i ) i ∈ I of V ′ by formal affine open subsets U ′ i of V ′ . Then the coherent sheafof ideals O ( − E ) restricted to U ′ i ˆ ⊗ ( K ′ ) ◦ F ◦ is generated by finitely many regular functions. Asimilar approximation argument as above shows that all these generators can be replaced byregular functions on U ′ i if we replace K ′ by a larger finite subextension of F/K . We concludethat L ′′ = O ( E ) is defined on V ′ proving (b). Note that uniqueness is obvious. (cid:3) Semipositive metrics
We will first introduce semipositive formal metrics. We have seen in Proposition 2.10 thatformal metrics are the same as piecewise linear metrics and hence everything applies to piecewiselinear metrics as well.
Let X be a proper scheme over K with a line bundle L over X . We call an algebraic K ◦ -model ( X , L ) of ( X, L ) numerically effective (briefly nef ) if deg L ( C ) ≥ C in X which is proper over K ◦ . Of course, properness implies that C is contained inthe special fiber X s . An algebraic metric k k on L an is said to be semipositive if there is a nefalgebraic K ◦ -model ( X , L ) of ( X, L ) such that k k = k k L . We say that a line bundle L on X is numerically trivial if deg L ( C ) = 0 for every closed curve C in X which is proper over K ◦ . Equivalently, we can require that L and L − are both nef. We say that L is numericallyequivalent to a line bundle L ′ on X if L ′ ⊗ L − is numerically trivial. The above definition is easily generalized to the analytic setting: Let L be a line bundle ona paracompact strictly K -analytic space V . A formal K ◦ -model ( V , L ) of ( V, L ) is called nef ifdeg L ( C ) ≥ C in the special fiber V s which is proper over the residue field˜ K . A formal metric k k on L is called semipositive if there is a nef formal K ◦ -model ( V , L ) of( V, L ) such that k k = k k L . Obviously, the trivial metric on O V is a semipositive formal metric.It will follow from Proposition 3.5 below that we may use any model to test semipositivity ofthe associated metrics. Based on this result, it is easy to check that the tensor product of twosemipositive formal metrics is again a semipositive formal metric. Lemma 3.3.
Let V be a paracompact strictly K -analytic space, L a line bundle on V and ( V , L ) a formal K ◦ -model of ( V, L ) . Let F be a non-archimedean extension of K and ( V F , L F ) theformal F ◦ -model of ( V F , L F ) obtained by base change. Then L is nef if and only if L F is nef.Proof. We remark that V s ⊗ ˜ K ˜ L ∼ = ( V ˆ ⊗ K ◦ L ◦ ) s . Hence the result follows from the fact that aline bundle on a proper scheme over ˜ K is nef if and only if its pull-back to ˜ F is nef. This isproven in the projective case in [EFM, Remark 1.3.25] and the proper case follows from Chow’slemma and the projection formula. (cid:3) Lemma 3.4.
Let V be a paracompact strictly K -analytic space, L a line bundle on V and ( V , L ) a formal K ◦ -model of ( V, L ) . Let ( V red , L red ) be the formal K ◦ -model of ( V red , L red ) obtained byputting the induced reduced structure. Then L is nef if and only if L red is nef.Proof. Let V red be the induced reduced structure on V . Since V red → V is finite (in fact aclosed immersion), we deduce that the induced map ( V red ) s → V s between the special fibers isfinite. By the projection formula, we conclude that L is nef if and only if L red is nef. (cid:3) N ZHANG’S SEMIPOSITIVE METRICS 13
Proposition 3.5.
Let ( V , L ) be a formal K ◦ -model of ( V, L ) . Then k k L is a semipositive formalmetric if and only if L is a nef formal K ◦ -model.Proof. By definition if L is nef, then k k L is semipositive, so we only have to prove the reverseimplication. Hence we assume that k k L is a semipositive formal metric and we have to showthat L is nef. Using Lemma 3.3, we can replace K by C K and hence we may assume that K isalgebraically closed.By definition of semipositivity, there is a nef formal K ◦ -model M of L on some formal K ◦ -model W of V with k k L = k k M . There exists a model X of V which dominates both V and W . Let π : X → V be the induced morphism. Since the induced morphism on the special fibers π s : X s → V s is proper and surjective, by the projection formula, L is nef if and only π ∗ L is nef.Hence replacing ( V , L ) by ( X , π ∗ L ), we can assume that V dominates W .Let V red be the induced reduced structure on V . Hence V red → V is finite. Locally, V red is given by Spf( A ) for some reduced admissible K ◦ -algebra A . Let A := A ⊗ K ◦ K . It is astrictly K -affinoid algebra, and by [BGR84, 6.4.3] A ′ := A ◦ is an admissible K ◦ -algebra as K isalgebraically closed, and by [BPR16, Proposition 3.8], A → A ′ is finite. By definition of A ′ wehave an isomorphism A ˆ ⊗ K ◦ K ∼ = A ′ ˆ ⊗ K ◦ K ∼ = A . By [BGR84, Proposition 7.2.6/3], we can gluethe morphisms Spf( A ′ ) → Spf( A ) to get a formal K ◦ -model V ′ of V red such that V ′ → V red isfinite. In particular, we deduce that the induced morphisms V ′ s → ( V red ) s → V s are proper andsurjective, and we conclude from the projection formula that L is nef if and only if its pull-back L ′ to V ′ is nef.By construction, V ′ is locally of the form Spf( A ◦ ), hence we deduce that V ′ s is locally givenby Spec( ˜ A ) which is reduced. Now we use the fact that on an admissible formal scheme withreduced special fibre and with K algebraically closed, the metric k k L ′ determines the model L ′ up to isomorphism (see [Gub98, Proposition 7.5]). Using that k k M ′ = k k L = k k L ′ for thepull-back M ′ of M to V ′ , we deduce that M ′ ∼ = L ′ . As above, the pull-back M ′ of M is nef andhence L ′ is nef. (cid:3) Proposition 3.6.
Let f : V ′ → V be a morphism of paracompact strictly K -analytic spaces andlet k k be a formal metric on a line bundle L of V . (a) If k k is a semipositive formal metric, then f ∗ k k is a semipositive formal metric. (b) If f is a surjective proper morphism and if f ∗ k k is a semipositive formal metric, then k k is a semipositive formal metric.Proof. There is a formal K ◦ -model ( V , L ) of ( V, L ) such that k k = k k L . We use Raynaud’stheorem to extend f to a morphism ϕ : V ′ → V of formal K ◦ -models. Then(3.6.1) f ∗ k k = k k ϕ ∗ L shows that f ∗ k k is a formal metric. To check semipositivity, Lemma 3.3 shows that K may beassumed to be algebraically closed which will allow us to use the results of [Kle66]. Then theclaims follow from [Kle66, Proposition I.4.1] applied to ϕ s . For (b), we use additionally that ϕ s is proper by [Tem00, Corollary 4.4] and surjective (as f and the reduction map V → V s aresurjective). (cid:3) Lemma 3.7.
Let X be a proper scheme over K , L a line bundle on X and ( X , L ) an algebraic K ◦ -model of ( X, L ) with k k := k k L . Let ( X i ) i ∈ I be the irreducible components of X equippedwith their induced reduced structures. Then k k is semipositive if and only if k k | X i is semipositivefor all i ∈ I .Proof. For each i ∈ I , let X i be the closed subscheme of X defined as the topological closure of X i in X equipped with the induced reduced structure. We then get for each i ∈ I a cartesian diagram X i / / (cid:15) (cid:15) X (cid:15) (cid:15) X i / / X Since the morphism ` i ∈ I X i → X is finite surjective, the projection formula shows that L isnef on X if and only if L | X i is nef on X i for all i . (cid:3) Following a suggestion of Tony Yue Yu, we can define semipositivity locally on V . Wesay that a piecewise linear metric on L is semipositive at x ∈ V if there is a compact strictly K -analytic domain W in V which is a neighborhood of x such that the restriction of k k to L | W is a semipositive formal metric in the sense of 3.2 (using the equivalence of Proposition 2.10).We say that k k is a semipositive piecewise linear metric if it is semipositive at all x ∈ V . Wewill see in Proposition 3.11 that this fits with the definition in 3.2 assuming that V is separated. Definition 3.9.
Let k k be a piecewise Q -linear metric on the line bundle L over V and let x ∈ V . Then k k is called semipositive at x ∈ V if and only if we may choose a compact strictly K -analytic domain W which is a neighborhood of x and some integer k ≥ k k ⊗ k | W isa semipositive formal metric.It follows easily from Proposition 3.5 that a piecewise linear metric on L is semipositive as apiecewise linear metric if and only if it is semipositive as a piecewise Q -linear metric. Proposition 3.10.
Let L be a line bundle on a paracompact strictly K -analytic space V . Let x ∈ V and let k k be a piecewise Q -linear metric on L . (a) The set of points in V where k k is semipositive is open in V . (b) The tensor product of two piecewise Q -linear metrics which are semipositive at x is againsemipositive at x . (c) Let f : V ′ → V be a morphism of paracompact strictly K -analytic spaces. If k k issemipositive at x , then f ∗ k k is semipositive at any point of f − ( x ) .Proof. Property (a) is obvious from the definitions. Property (b) follows easily from Proposition3.5 and the linearity of the degree of a proper curve with respect to the divisor. Finally (c)follows from Proposition 3.6. (cid:3)
Proposition 3.11.
Let L be a line bundle on the separated paracompact strictly K -analytic space V and let k k be a formal metric on L . Then k k is a semipositive formal metric as globallydefined in 3.2 if and only if k k is a semipositive piecewise linear metric in every x ∈ V as definedin 3.8.Proof. The proof follows mainly the arguments in [GK15, Proposition 6.4]. It is clear that asemipositive formal metric is a semipositive piecewise linear metric in every x ∈ X and we willprove now the converse. Let ( V , L ) be a formal K ◦ -model of ( V, L ) with k k = k k L . Since thegeneric fiber V is separated, it follows from [BL93a, Proposition 4.7] that the model V is alsoseparated and hence we may apply [CD12, Lemma 6.5.1]. This is a criterion which characterizesthe points v in the relative interior Int( V ) over the base K (defined in [Ber93, Definition 1.5.4])by the property that the closure of the reduction of v in the special fiber V s is proper over ˜ K .We assume that k k is semipositive at every x ∈ V . We choose a closed curve C in V s whichis proper over ˜ K . We have to show that deg L ( C ) ≥
0. By surjectivity of the reduction map π : V → V s , there is x ∈ V such that π ( x ) is the generic point of C . Using [CD12, Lemma 6.5.1],the properness of C yields that x ∈ Int( V ). Since k k is semipositive at x , there is a compactstrictly K -analytic neighborhood W of x , a nef formal K ◦ -model ( W , M ) of ( W, L | W ) and a N ZHANG’S SEMIPOSITIVE METRICS 15 non-zero k ∈ N such that k k ⊗ k = k k M over W . Using Proposition 3.5, we may always replacethe models W and V by dominating formal K ◦ -models and the line bundles M and L by theirpull-backs. By [BL93b, Corollary 5.4 (b)] , we may therefore assume that W is a formal opensubset of V . Then L | ⊗ k W is also a formal K ◦ -model of L | ⊗ kW and hence Proposition 3.5 impliesthat L | W is nef.Since W is a neighborhood of x and since Int( W ) is contained in the intersection of Int( V ) withthe topological interior of W in V by [Ber93, Proposition 1.5.5], we conclude that x ∈ Int( W ).Using [CD12, Lemma 6.5.1] again, the closure of the reduction of x in W s is proper over ˜ K andhence equal to C . Since L | W is nef, it follows that deg L ( C ) ≥ (cid:3) Proposition 3.12.
Let k k and k k be algebraic metrics of the line bundle L over the properscheme X over K . Then k k := min( k k , k k ) is an algebraic metric on L . If k k and k k are semipositive at x ∈ X an , then k k is semipositive at x .Proof. Since formal and algebraic metrics are the same as noted in Remark 2.6 and hence alsothe same as piecewise linear metrics, we deduce from Proposition 2.12 (d) that k k is an algebraicmetric. If the given metrics are semipositive at x , then it remains to prove that k k is semipositiveat x . By base change again, we may assume that K is algebraically closed. By Lemma 3.7, wemay assume that X is a proper variety over K . Let us pick models L , L and L of L definingthe algebraic metrics k k , k k and k k . There is an algebraic K ◦ -model X of X on which L , L and L are defined. There is an open neighborhood W of x in X an such that k k and k k are semipositive at all points of W . We will show that k k is semipositive at every pointof W . By [GK15, 6.5], it is equivalent to show that deg L ( C ) ≥ C of X s contained in the reduction of W . Moreover, the same result yields that L and L restrict tonef line bundles on C . By [GK15, Theorem 4.1], there is a closed curve Y in X such that C is an irreducible component of the special fibre of the closure Y in X . By restriction, we mayassume that X = Y is a curve and hence C is an irreducible component of X s . Let X be theformal completion of X and let L , L , L be the line bundles on X induced by the pull-backs of L , L , L .We have seen in the proof of Proposition 3.5 that we can associate to X a canonical formal K ◦ -model X ′ of X an with reduced special fibre and a canonical finite surjective morphism ι : X ′ → X .So there is a closed curve C ′ in X ′ s which maps onto C in X s = X s . Let L ′ , L ′ , L ′ be the linebundles on X ′ given by pull-back of L , L , L . Note that L ′ , L ′ , L ′ are formal K ◦ -models of themetrics k k , k k , k k on L an . By the projection formula, the line bundles L ′ , L ′ restrict to nefline bundles on C ′ and it remains to show that(3.12.1) deg L ′ ( C ′ ) ≥ . Let ζ be the generic point of C ′ . Then there is a unique point ξ in X an with reduction ζ . Thisfollows from [Ber90, Proposition 2.4.4] since ζ has a formal affine open neighborhood in X ′ of theform Spf( A ◦ ) for a strictly K -affinoid algebra A . Using k k = min( k k , k k ), we may assume k k ( ξ ) = k k ( ξ ). Since L an is algebraic, there is a non-trivial meromorphic section t of L ′ . Notethat the restriction of t to the generic fibre L an induces also a meromorphic section t of L ′ . Themeromorphic section t/t of M := L ′ ⊗ ( L ′ ) − restricts to the trivial section 1 of O X an and wehave k t/t k M = k t k / k t k = k t k / k t k ≤ . By [Gub98, Proposition 7.5], we deduce that t/t is a global section of M . The definition offormal metrics and k t/t k M ( ξ ) = k t k ( ξ ) / k t k ( ξ ) = 1 yield that { y ∈ X an | k t/t k M ( y ) ≥ } isthe generic fibre of a formal open neighborhood U of ζ . Hence [Gub98, Proposition 7.5] againshows that t/t is a nowhere vanishing regular section of M on U . We conclude that the restriction Note the misprint in [BL93b, Corollary 5.4 (b)]: immersion should be replaced by open immersion . of the global section t/t to C ′ is not identically zero inducing an effective Cartier divisor D on C ′ . This shows deg M ( C ′ ) = deg D ( C ′ ) ≥ . Using that L ′ is nef on C ′ and L ′ = M ⊗ L ′ , we getdeg L ′ ( C ′ ) ≥ deg L ′ ( C ′ ) ≥ (cid:3) Plurisubharmonic model functions
We will introduce closed (1 , θ on a proper scheme X over K and θ -psh model functionsfollowing the terminology in [BFJ16]. Let L be a line bundle on X . We say that a metric k k on L an is a model metric if thereis a non-zero d ∈ N such that k k ⊗ d is an algebraic metric on ( L an ) ⊗ d . By Proposition 2.10 andRemark 2.6, k k is a model metric if and only if it is a piecewise Q -linear metric. We say that a function ϕ : X an → R is a model function if there exists d ∈ N > and k k an algebraic metric on O X an such that ϕ = − d log k k . If we can take d = 1, we say that ϕ is a Z -model function . The set of model functions on X is denoted by D ( X ). Let X be an algebraic K ◦ -model of X . A vertical Cartier divisor on X is a Cartierdivisor D on X which is supported on the special fiber X s . A vertical Cartier divisor D on X determines a model O ( D ) of O X hence an associated model function ϕ D := − log k k O ( D ) : X an → R Note that every Z -model function has this form. Indeed, if L is an algebraic K ◦ -model of O X with ϕ = − log k k L , then the section 1 of O X extends to a meromorphic section s of L andthe vertical Cartier divisor D := div( s ) satisfies ϕ = ϕ D .For example, the constant functions on X an with values in the value group Γ = − log | K × | arethe Z -model functions of the form ϕ D with D = div( α ) for non-zero elements α ∈ K . Moreover,the constant functions on X an with values in Q Γ are Q -model functions. We set Pic( X ) R := Pic( X ) ⊗ Z R . We define the N´eron-Severi group as the R -vector spacePic( X ) R modulo the subspace generated by numerically trivial line bundles. We denote thisspace by N ( X /S ), where S := Spec( K ◦ ). The space of closed (1 , -forms on X is defined asthe direct limit Z , ( X ) := lim −→ N ( X /S )where the limit is taken over all algebraic K ◦ -models of X . We say that a closed (1 , θ is determined on some model X if it is in the image of the map N ( X /S ) → Z , ( X ). We denote by c Pic( X ) the group of isomorphism classes of line bundles on X equipped witha model metric. There is a map c : c Pic( X ) → Z , ( X ) which sends the class of ( L, k k L ) to theclass of L . We will show below that this map is well defined. We denote its image by c ( L, k k L )and call it the curvature form of ( L, k k L ). We get a natural linear map dd c : D ( X ) → Z , ( X )which maps a Z -model function ϕ to c ( O X an , k k ϕ ), where k k ϕ is the corresponding algebraicmetric on O X an .To show that the curvature c ( L, k k L ) is well defined in Z , ( X ), we consider algebraic K ◦ -models L , L ′ of L with k k L = k k L ′ . We have to show that L and L ′ are numericallyequivalent on a suitable algebraic K ◦ -model of X . Since the isomorphism classes of algebraic For a dd c -lemma, see [BFJ16, Theorem 4.3] in the discretely valued case and [Jel16, Theorem 4.2.7] for ageneralization to non-discrete valuations. N ZHANG’S SEMIPOSITIVE METRICS 17 K ◦ -models of X form a directed set, we may assume that L and L ′ are line bundles on thesame algebraic K ◦ -model X . Using base change to C K , Lemma 3.3 shows that we may assume K algebraically closed. By Lemma 3.4, we may assume that X is reduced. Then the formalcompletion X of X is also reduced. We consider the canonical formal K ◦ -model X ′ with reducedspecial fibre and with a finite surjective morphism X ′ → X extending id X an as in the proof ofProposition 3.5. We may apply [Gub98, Proposition 7.5] to the line bundles L , L ′ on X ′ inducedby L , L ′ . Since k k L = k k L = k k L ′ = k k L ′ , we deduce that L ∼ = L ′ . Using that X s = X s ,the projection formula applied to the finite surjective morphism X ′ s → X s shows that L isnumerically equivalent to L ′ . We say that an element of N ( X /S ) is ample if it is the class of a non-empty sum P i a i c ( L i ) for some real numbers a i > L i . For an algebraic K ◦ -model X of X , a closed (1 , θ is called X -positive if it is determined on X by some θ X ∈ N ( X /S ) which is ample. We say that a model metric k k of a line bundle L is X -positive if the same holds for the curvature form c ( L, k k ). We say that an element θ ∈ N ( X /S ) is nefif θ · C ≥ C ⊂ X s . A closed (1 , θ is said to be semipositive if it isdetermined by a nef class θ X ∈ N ( X /S ) on a model X .If θ is a closed (1 , ϕ is θ -plurisubharmonic (briefly θ -psh ) if θ + dd c ϕ is semipositive. If θ is the closed (1 , L on X and if D is a vertical Cartier divisor on X , then by definition ϕ D is a θ -psh functionif and only if L ⊗ O ( D ) is nef if and only if k k L ⊗O ( D ) is a semipositive metric. Let L be a line bundle on X . Let k k be a model metric on L an and θ := c ( L, k k ). Let k k ′ be another metric on L an and let ϕ := − log( k k ′ / k k ). Then k k ′ is a model metric if andonly if ϕ is a model function. Moreover k k ′ is a semipositive model metric if and only if ϕ is a θ -psh model function. The N´eron–Severi group N ( X ) of X is the group Pic( X ) ⊗ Z R modulo the subspacegenerated by the numerically trivial line bundles. For a closed (1 , θ , let { θ } be theassociated de Rham class, given by { θ } = θ X | X ∈ N ( X ) for any algebraic K ◦ -model X onwhich θ is determined by θ X ∈ N ( X /S ). If θ is semipositive, then { θ } is nef.To see this, we choose any closed curve C in X and non-zero ρ in the maximal ideal of thevaluation ring K ◦ . Then using the divisorial intersection theory in [Gub98], we have v ( ρ ) deg { θ } ( C ) = deg(div( ρ ) .θ X .C ) = deg( θ X . div( ρ ) .C ) = v ( ρ ) deg θ X ( C s ) . Since θ X is nef, the degree of the special fibre C s of the closure C in X is non-negative provingthe claim. Lemma 4.9.
Let us assume that K is algebraically closed. Let ϕ be a model function determinedby a vertical Cartier divisor D on the algebraic K ◦ -model X of X . We assume that the specialfibre X s is reduced. Then ϕ ≥ if and only if the Cartier divisor D is effective.Proof. By definition of a Cartier divisor, O X ( − D ) is a coherent subsheaf of the sheaf of mero-morphic functions on X , and D is effective if and only if O X ( − D ) happens to be a subsheafof O X . Since D is a vertical Cartier divisor, there exists a ∈ K ◦ \ { } such that D + div( a )is effective, i.e. a O X ( − D ) is a subsheaf of O X . Now D is effective if and only if a O X ( − D )is a subsheaf of a O X when they are both considered as coherent subsheaves of O X . Since thecompletion functor is fully faithful on the category of coherent sheaves by [FK18, PropositionI.9.4.2], it is equivalent to check the associated inclusion on the formal completion of ˆ X .We now get the claim from the corresponding statement for admissible formal schemes whichis proven in [GRW16, Proposition A.7] and hence applies to the formal completion ˆ X of X andits Cartier divisor ˆ D given by pull-back of D . (cid:3) Remark 4.10.
If we assume that X is normal instead of assuming that X s is reduced, thenLemma 4.9 holds for any non-archimedean field K (see [GS15, Corollary 2.12]).We recall the following result from [BFJ16, Corollary 1.5]. For convenience of the reader andto check that no noetherian hypotheses are used, we give here a proof. Proposition 4.11.
Let L be an ample line bundle on the projective scheme X over K and let X be any algebraic K ◦ -model of X . Then there is an algebraic K ◦ -model X of X dominating X and an ample line bundle L on X which is a K ◦ -model of L ⊗ m for a suitable m ∈ N .Proof. Every algebraic K ◦ -model of a projective scheme X is dominated by a projective algebraic K ◦ -model [Gub03, Proposition 10.5]. Hence we may assume that X is projective. There is m ∈ N and a closed immersion of X into P NK such that L ⊗ m = O P NK (1) | X . Then the schematicclosure of X in P NK ◦ is an algebraic K ◦ -model X of X . Moreover, X has an ample line bundle L such that L | X = L ⊗ m . Then the schematic closure of the diagonal in X × K ◦ X isa projective K ◦ -model X of X . Now the claim follows from the following lemma applied to f := p . (cid:3) Lemma 4.12.
Let f : X → X be a morphism of projective algebraic K ◦ -models of X extending id X and assume that L ⊗ m extends to an ample line bundle L on X for some m ∈ N . Thenthere is a positive multiple m of m such that L ⊗ m extends to an ample line bundle on X .Proof. Note that f is a projective morphism, hence there is a closed immersion of X into aprojective space P k X over X . Let E be the restriction of O P k X (1) to X . Since E is relativelyample with respect to f and since L is an ample line bundle on X , there is m ∈ N such that L := f ∗ ( L ) ⊗ m ⊗ E is ample on X . Then L is an ample K ◦ -model of L ⊗ m for m := m m . (cid:3) Proposition 4.13.
Let ω be X -positive and let θ be any closed (1 , -form determined on X .Then ω + εθ is X -positive for ε ∈ R sufficiently close to .Proof. Since Spec( K ◦ ) is affine, ample is the same as relatively ample. It remains to check thatthe restriction of ω + εθ to the special fibre is ample (see [Gro66, 9.6.4 and 9.6.5]). The amplecone on the special fiber is the interior of the nef cone. This proves immediately the claim. (cid:3) The following result generalizes [BFJ16, Proposition 5.2]. Again, we follow their arguments.
Proposition 4.14.
Let θ be a closed (1 , -form with ample de Rham class { θ } ∈ N ( X ) and let X be any algebraic K ◦ -model of X . Then there is an algebraic K ◦ -model X of X dominating X such that θ is determined on X and a model function ϕ such that θ + dd c ϕ is X -positive.If θ is semipositive and ε > , then we may find such a model function with − ε ≤ ϕ ≤ .Proof. For any algebraic K ◦ -model X , the canonical homomorphism N ( X /S ) → N ( X s ) isinjective by definition of numerical equivalence. The ample cone of N ( X /S ) is the preimageof the ample cone of N ( X s ) (see the proof of Proposition 4.13).By assumption, θ can be represented by a finite sum P i λ i c ( L i ) with line bundles L i onalgebraic K ◦ -models X i of X and λ i ∈ R . We set L i := L i | X . Hence P i λ i c ( L i ) represents { θ } . Since { θ } is ample, there are finitely many ample line bundles H j on X and µ j > P j µ j c ( H j ) also represents { θ } . Clearly, we may assume that every H j is very ample andhence H j has a very ample K ◦ -model H j on a projective algebraic K ◦ -model Y j of X . Let ϑ ∈ Z , ( X ) be the class of P j µ j c ( H j ). We recall that the isomorphism classes of K ◦ -modelsof X form a directed set and that any K ◦ -model of the projective variety X is dominated by aprojective K ◦ -model. We conclude that there is a projective K ◦ -model X of X such that id X extends to morphisms X → X i and X → Y j for every i and j . Replacing L i by its pull-back to X , we may assume that X i = X meaning that L i lives on X for every i . Moreover, it follows N ZHANG’S SEMIPOSITIVE METRICS 19 from Lemma 4.12 that H j extends to an ample line bundle on X and hence we may assume thatevery H j lives on X as well. This means that ϑ is X -positive.We now consider the linear equation in the variables ( λ ′ i ), ( µ ′ j )(4.14.1) X i λ ′ i c ( L i ) − X j µ ′ j c ( H j ) = 0which we consider as an equation in Pic( X ) ⊗ Z Q ⊂ Pic( X ) ⊗ Z R . By assumption, ( λ i ), ( µ j ) isa real solution of (4.14.1). Hence, by a linear algebra argument, for any δ >
0, we can find asolution ( λ ′ i ), ( µ ′ j ) of (4.14.1) with λ ′ i , µ ′ j ∈ Q such that | λ ′ i − λ i | < δ and | µ ′ j − µ j | < δ . We cantake δ small enough to get µ ′ j >
0. We may assume that the Q -line bundles L ′ := O i L ⊗ λ ′ i i and H ′ := O j H ⊗ µ ′ j j agree on the generic fibre X . Let ϕ be the model function corresponding to H ′ ⊗ ( L ′ ) − . As all µ ′ j are positive and all H j are ample on X , H ′ is an ample Q -line bundle on X . By definitionwe have θ + dd c ϕ = θ + c ( H ′ ) − c ( L ′ ) = ( θ − c ( L ′ )) + ( c ( H ′ ) − ϑ ) + ϑ. Taking δ small enough, we can make the first two terms ( θ − c ( L ′ )) and ( c ( H ′ ) − ϑ ) as smallas we want in N ( X /S ). In addition, we know that ϑ is ample on X . It follows from our remarkat the beginning that the ample cone is open in N ( X /S ). For δ small enough, we deduce that θ + dd c ϕ is ample on X .Now let us assume that θ is semipositive. Since a function in D ( X ) is continuous on X an , itis bounded. We may replace ϕ by ϕ − c for any sufficiently large c in the value group Γ withoutchanging dd c ϕ and hence we may assume ϕ ≤
0. Since the sum of a nef and an ample classin N ( X /S ) remains ample (as we can check that on the special fibre, see the remark at thebeginning of the proof), we know that θ + dd c ( εϕ ) = ε ( θ + dd c ϕ ) + (1 − ε ) θ is also X -positive for all 0 < ε ≤
1. Using a rational ε > εϕ . (cid:3) We want to recall a result due to Kiehl [Kie72, Theorem 2.9] that we will use later. Let Y be a K ◦ -scheme. Let f : X → Y be a proper morphism of finite presentation and let M bea coherent O X -module. Then f ∗ ( M ) is a coherent O Y -module. For some explanation on whyKiehl’s result implies this, we refer to Example 3.3 and 3.5 in [Ull95].We will apply this result in the case of proper flat schemes X , Y over K ◦ . By [Raynaud-Gruson, Corollaire 3.4.7], they are finitely presented over K ◦ and hence any f : X → Y is properand of finite presentation. We also recall a non-noetherian version of the Stein factorization theorem that will beused later. We quote the following result from [Sta16, Tag 03GY].Let f : X → S be a universally closed and quasi-separated morphism of schemes. Then thereexists a factorization X f ′ / / f (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ S ′ g (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦ S with the following properties:(1) the morphism f ′ is universally closed, quasi-compact, quasi-separated and surjective;(2) the morphism g : S ′ → S is integral; (3) f ′∗ O X = O S ′ ;(4) The relative spectrum of f ∗ O X over S is equal to S ′ ;In the following, we consider an admissible formal scheme X over K ◦ . A vertical coherentfractional ideal a on X is an O X -submodule of O X ⊗ K ◦ K with generic fibre a η = O X η such thatfor every formal open affine subset U , there is a non-zero α ∈ K ◦ with α a | U a coherent idealsheaf on U . Definition 4.17.
Let a be a vertical coherent fractional ideal on X . We define the function(4.17.1) log | a | : X η → R x max { log | f ( x ) | (cid:12)(cid:12) f ∈ a π ( x ) } where π : X η → X s is the reduction map. This is a continuous map because a is a coherent sheaf.We will now use divisorial points as introduced and studied in Appendix A. Lemma 4.18.
Let X be an admissible formal scheme over K ◦ and let f ∈ O ( X η ) . Let I be theset of divisorial points associated with X . Then sup x ∈ X η | f ( x ) | = sup x ∈ I | f ( x ) | . Proof.
Since we can work locally on X , we can assume that X = Spf( A ) is formal affine, in whichcase the supremum is not + ∞ . It follows from Proposition A.3 that the set of divisorial points I is the Shilov boundary of X η proving precisely our claim. (cid:3) Corollary 4.19.
Let X be an admissible formal scheme over K ◦ and let ϕ := log | a | for a verticalcoherent fractional ideal a on X . Let I be the set of divisorial points associated with X . Then sup x ∈ X η ϕ ( x ) = sup x ∈ I ϕ ( x ) . Proof.
We can easily replace X by one of its open formal subschemes because divisorial pointsare compatible with formal open subsets. So we can assume that X is an admissible for-mal affine scheme. Hence a is generated by finitely many functions f , . . . , f n . Then ϕ ( x ) =max j =1 ...n log | f j ( x ) | . The result follows from Lemma 4.18. (cid:3) Let us return to our algebraic setting with an algebraic K ◦ -model X of the proper scheme X . We will apply the above to the formal completion X ′ of the algebraic K ◦ -model X ′ from thefollowing result. Lemma 4.20.
Let θ be a closed (1 , -form determined on X and let ϕ be a θ -psh model functionon X . Then there is an algebraic K ◦ -model X ′ of X with a finite morphism X ′ → X extending id X and a sequence a m of vertical coherent fractional ideals on X ′ such that m log | a m | convergesuniformly to ϕ .Proof. The proof of [BFJ16, 5.7] can be adapted in our non-noetherian context. For the conve-nience of the reader we detail this.
Step 1.
We have seen in 4.3 that there is an algebraic K ◦ -model Y and a vertical Cartierdivisor D on Y with ϕ = ϕ D . By 2.1, we may assume that the identity id X extends to a morphism π : Y → X . It follows from Raynaud’s theorem that there is an admissible formal blowing up ψ : Y → ˆ X of the formal completion ˆ X in an open coherent ideal b ′ such that Y dominatesthe formal completion of Y . By the formal GAGA-principle for proper schemes over K ◦ provedby Fujiwara–Kato [FK18, Theorem I.10.1.2], the coherent ideal b ′ is the formal completion of acoherent vertical ideal b on X and hence ψ is the formal completion of the blowing up of X in b . Hence we may assume that π is precisely this algebraic blow up morphism. N ZHANG’S SEMIPOSITIVE METRICS 21
Step 2.
Note that π is a proper morphism and hence π ∗ ( O Y ) is a coherent sheaf by 4.15. Let π = g ◦ π ′ be the Stein factorization of π as in 4.16. It follows from coherence of π ∗ ( O Y ) andfrom 4.16(4) that the morphism g : X ′ → X is finite. By construction, X ′ is a model of X and π ′ , g restrict to the identity on X . Step 3.
Let C ⊂ Y s be a curve which is contracted by π i.e. such that π ( C ) = { x } for someclosed point x ∈ X s . Since ϕ is θ -psh, by definition we get that ( D + π ∗ ( θ )) · C ≥
0. On theother hand, by the projection formula, π ∗ ( π ∗ ( θ ) .C ) = θ.π ∗ ( C ) = 0. Hence π ∗ ( θ ) · C = 0 and so D · C ≥
0. By definition, this means that D is π -nef. Step 4.
By the construction in Step 1, there is a vertical ideal sheaf b on X such that π isthe blow up of X along b . By the universal property of the blow up, b O Y = O Y ( H ) for aneffective π -ample vertical Cartier divisor H on Y . Step 5.
We choose a non-zero k ∈ N . Let x ∈ X s be a closed point. We denote by Y x the fiberover x with respect to the morphism Y s → X s . Note that Y x is a proper scheme over the residuefield of x . Since H is π -ample, O ( H ) | Y x is ample. Similarly, since D is π -nef, O ( D ) | Y x is nef.It follows from Kleiman’s criterion that O ( kD + H ) | Y x is ample. Hence by [Gro66, Corollaire9.6.5], kD + H is π -ample and hence kD + H is also π ′ -ample. Step 6.
By 4.16(3), we have π ′∗ O Y = O X ′ . It follows that π ′∗ maps vertical coherent fractionalideals on Y to vertical coherent fractional ideals on X ′ . It follows that a := π ′∗ O Y ( n ( kD + H ) + lD ) is a vertical coherent fractional ideal on X ′ for every n ∈ N and l = 0 , . . . , k − Step 7.
Hence by the characterization given in [Gro61, Proposition 4.6.8] of π ′ -ampleness,for all sufficiently large n ∈ N , the map π ′∗ π ′∗ O Y ( n ( kD + H ) + lD ) → O Y ( n ( kD + H ) + lD )is surjective which means that π ′∗ a → O Y ( n ( kD + H ) + lD ) is surjective. This implies thatlog | a | = ϕ n ( kD + H )+ lD and hence 1 m log | a | = ϕ D + nm ϕ H for m := nk + l . We have 0 ≤ nm ϕ H ≤ k ϕ H and this is arbitrarily small for sufficiently large k independently of the choice of n and l . This leads easily to the construction of an approximatingsequence as in the claim. (cid:3) Remark 4.21. If X is normal, then we have X = X ′ in Lemma 4.20. Proposition 4.22.
Let X be an algebraic K ◦ -model of the proper scheme X over K . Let I be the set of divisorial points of X an associated with X . Let θ be a closed (1 , -form which isdetermined on X and let ϕ be a θ -psh model function on X. Then sup x ∈ X an ϕ ( x ) = sup x ∈ I ϕ ( x ) . Proof.
Let X ′ be the model of X from Lemma 4.20. Since the constructed morphism X ′ → X is finite, the set of divisorial points of X an associated with X ′ agrees with I . Then the claimfollows from Lemma 4.20 and from Corollary 4.19 applied to the formal completion X ′ of X ′ . (cid:3) Semipositivity and pointwise convergence
Our goal is to generalize [BFJ16, Theorem 5.11] to a line bundle L on a proper scheme X over any non-archimedean field K . This is a generalization in various aspects as in [BFJ16], X was assumed to be a smooth projective variety and the valuation was discrete with residuecharacteristic zero (due to a use of the theory of multiplier ideals).In terms of metrics, the main result means that pointwise convergence of semipositive modelmetrics on L an to a model metric implies that the limit is a semipositive model metric. ByChow’s lemma, we will reduce to the case of projective varieties. We will first prove an analogue of [BFJ16, Lemma 5.12]. Recall that we denote by X div theset of divisorial points of the analytification X an (see Appendix A). Proposition 5.1.
Let X be a projective scheme over K with an ample line bundle L . Weconsider an algebraic K ◦ -model X of X and a line bundle L on X extending L . Let k k = k k L be the corresponding model metric on L an which is assumed to be the pointwise limit over X div of semipositive model metrics on L an . Then k k is a semipositive model metric. By Lemma 3.3 and Lemma 3.7, it is enough to check the claim for a projective variety overan algebraically closed field K . Here we have used that ( X ⊗ C K ) div is the preimage of X div with respect to base change morphism ( X ⊗ C K ) an → X an (see Proposition A.7), and that X div = S ( X i ) div where X i ranges over the irreducible components of X (see Proposition A.12)Then Proposition 5.1 follows immediately from Lemma 5.3 and Lemma 5.4 below.Recall that the base-ideal a m of L ⊗ m is defined as the image of the canonical map H ( X , L ⊗ m ) ⊗ L ⊗ ( − m ) → O X . Since L is ample on X , a m is a vertical coherent ideal sheaf for m sufficiently large.We give now the analogue of Definition 4.17 in the algebraic setting: Definition 5.2.
Let a be a coherent fractional ideal on the algebraic K ◦ -model X of the properscheme X over K . Then we setlog | a | ( x ) := max { log | f ( x ) | (cid:12)(cid:12) f ∈ a π ( x ) } ∈ [ −∞ , ∞ [where π : X an → X s is the reduction map. Lemma 5.3.
We keep the same hypotheses as in Proposition 5.1. We assume additionally that K is algebraically closed and that X is a variety. We fix a finite subset S of X div . Then there isa sequence of algebraic K ◦ -models Z m such that id X extends to finite morphisms g m : Z m → X with the following property: Let b m be the base-ideal of g ∗ m ( L ) ⊗ m . For m sufficiently large, b m is a vertical coherent ideal on Z m and m log | b m | converges pointwise to on S . This lemma is similar to the first step in the proof of [BFJ16, Lemma 5.12]. Note that we donot assume here that X and the model X are normal. This leads to additional complications.In case of a normal model X (as in loc. cit. ), the finite morphisms g m are the identity and b m is just the base-ideal a m of L ⊗ m on Z m = X . Then pointwise convergence holds on X div . Ifthe normalization Z of X would be finite over X , then we could use Z m = Z for all m .After we have submitted this paper, Boucksom and Eriksson [BE18, Theorem 4.20] showedthat the normalization Z is indeed finite over X and so we may use Z m = Z in the lemma.Moreover, the pointwise convergence holds on X div . We were informed by Ofer Gabber that thefiniteness of the normalization Z over X was also shown by Anantharaman in [Ana73, Th´eor`eme1’ in Appendice II]. We thank Ofer Gabber very much for providing us with this reference. Proof.
We have a m · a l ⊂ a m + l by definition of the base-ideal a m of L ⊗ m . It follows that thesequence (log | a m | ) is super-additive, i.e.log | a m + l | ≥ log | a m | + log | a l | for all m, l ∈ N . For m sufficiently large, a m is a coherent vertical ideal sheaf and hence log | a m | > −∞ . By Fekete’s super-additivity lemma, the limit of the sequence m log | a m | exists pointwisein ] − ∞ , ∞ ]. Since a m is an ideal sheaf, we have m log | a m | ≤ − ∞ < lim m →∞ m log | a m | = sup m m log | a m | ≤ X an . N ZHANG’S SEMIPOSITIVE METRICS 23
We choose ε > θ := c ( L, k k ). Now we use that k k is thepointwise limit of semipositive model metrics on L over X div . This is equivalent to the propertythat 0 is the pointwise limit of θ -psh model-functions over X div (see 4.7). It follows from 4.3 thatthere is a vertical Q -Cartier divisor D on a model X ′ of X such that ϕ D is θ -psh and such that(5.3.2) ϕ D ( x ) ≥ − ε, ϕ D ( y ) ≤ ε for all x ∈ S and all divisorial points y associated with X . We may assume D lives on a K ◦ -model X ′ with a morphism π : X ′ → X extending the identity on X . By Proposition 4.22, weget ϕ D ≤ ε .Let us consider the model function ϕ D ′ := ϕ D − ε on X with associated vertical Q -Cartierdivisor D ′ on X ′ . We conclude that(5.3.3) ϕ D ′ ( x ) ≥ − ε, ϕ D ′ ≤ x ∈ S . We note that O ( D ′ ) ∼ = O ( D ) as Q -line bundles (which means that D and D ′ are Q -linearly equivalent) and hence π ∗ ( L ) ⊗ O ( D ′ ) is nef using that ϕ D is θ -psh. Let θ ′ be thecorresponding semipositive closed (1 , X . Since { θ ′ } = { θ } is ample, we may applyProposition 4.14 to deduce that there is a sufficiently large K ◦ -model X ′′ dominating X ′ anda model function ϕ ′′ with(5.3.4) − ε ≤ ϕ ′′ ≤ θ ′ + dd c ϕ ′′ is X ′′ -positive. Let D ′′ be the vertical Q -Cartier divisor on X ′′ such that ϕ ′′ = ϕ D ′′ .To ease notation, we may assume that X ′ = X ′′ . Then we deduce that π ∗ ( L ) ⊗ O ( D ′ + D ′′ )is an ample Q -line bundle. Using that K is algebraically closed, the reduced fibre theorem[BLR95, Theorem 2.1’] shows that there is a proper model Y of X dominating X ′ with reducedspecial fibre. By Lemma 4.9, (5.3.3) and (5.3.4), the pull-backs − E ′ , − E ′′ of − D ′ , − D ′′ to Y are both effective vertical Q -Cartier divisors. Let ρ : Y → X be the morphism extending theidentity on the generic fibre. We note that ρ ∗ ( L ) ⊗ O ( E ′ + E ′′ ) is semiample which means that ρ ∗ ( L ) ⊗ m ⊗ O ( m ( E ′ + E ′′ )) is a honest line bundle on Y generated by global sections for asuitable m ∈ N \ { } .Since models are proper over K ◦ , any K ◦ -morphism between models is proper and hencewe may consider the Stein factorization ρ = g ◦ ρ ′ as in 4.16 for morphisms g : Z → X and ρ ′ : Y → Z of schemes over K ◦ . Similarly as in Step 2 of the proof of Lemma 4.20, we deducethat g : Z → X is a finite morphism of K ◦ -models of X extending id X . By (3) in 4.16, we have(5.3.5) ρ ′∗ ( O Y ) = O Z . By the projection formula and (5.3.5), we get ρ ′∗ ( ρ ∗ ( L ⊗ m )) ∼ = g ∗ ( L ⊗ m ) and hence(5.3.6) H ( Z , g ∗ ( L ⊗ m )) = H ( Y , ρ ∗ ( L ⊗ m )) . For all m ∈ N divisible by m , we have seen that − mE ′ − mE ′′ is an effective vertical Cartierdivisor and hence O ( mE ′ + mE ′′ ) is a vertical ideal sheaf in O Y . We get a canonical inclusion(5.3.7) ρ ∗ ( L ) ⊗ m ⊗ O ( m ( E ′ + E ′′ )) ⊂ ρ ∗ ( L ) ⊗ m for all m ∈ N divisible by m . The left hand side is globally generated. Let b m be the base idealof g ∗ ( L ⊗ m ) on Z . We claim that(5.3.8) O ( m ( E ′ + E ′′ )) ⊂ O Y b m ⊂ O Y . Note that the inclusion O ( m ( E ′ + E ′′ )) ⊂ O Y is given by multiplication with the canonical globalsection s − m ( E ′ + E ′′ ) of O ( − mE ′ − mE ′′ ). We check (5.3.8) at y ∈ Y . Using semiampleness,there is a global section s of ρ ∗ ( L ) ⊗ m ⊗ O ( m ( E ′ + E ′′ )) which does not vanish at y , i.e. s − is a local section at y . Let t be any section of O ( m ( E ′ + E ′′ )) around y . We have to show that t ⊗ s − m ( E ′ + E ′′ ) is a section of O Y b m around y . To see this, we write t ⊗ s − m ( E ′ + E ′′ ) = ( s ⊗ s − m ( E ′ + E ′′ ) ) ⊗ ( t ⊗ s − ) . Since − mE ′ − mE ′′ is an effective vertical Cartier divisor, s ⊗ s − m ( E ′ + E ′′ ) is a global sectionof ρ ∗ ( L ⊗ m ) and hence it is the pull-back of a global section of g ∗ ( L ⊗ m ) by (5.3.6). Moreover, t ⊗ s − is a local section of ρ ∗ ( L − m ) around y and hence it is an O Y -multiple of the pull-backof a local section of g ∗ ( L − m ) at ρ ′ ( y ). It follows from the definition of the base-ideal b m that t ⊗ s − m ( E ′ + E ′′ ) is a local section of O Y b m around y proving (5.3.8).It follows from (5.3.3), (5.3.4) and (5.3.8) that − ε ≤ ϕ E ′ + E ′′ ( x ) ≤ m log | b m | ( x )for all m ∈ N divisible by m and all x ∈ S . By (5.3.1) applied to the base ideals b m on Z instead of a m , we deduce that(5.3.9) − ε ≤ lim m →∞ m log | b m | ( x ) ≤ x ∈ S . Using a sequence ε →
0, we construct easily from (5.3.9) a sequence of finitemorphisms g m : Z m → X with the required property. (cid:3) The following result is similar to step 2 in the proof of [BFJ16, Lemma 5.12]. Note thatwe need here another argument as the multiplier ideals used in [BFJ16] do not work in residuecharacteristic p >
0. Let us recall that a line bundle L on a scheme is called semiample if L ⊗ m is globally generated for some m ∈ N > . Lemma 5.4.
Let L be a semiample line bundle on the projective variety X over the algebraicallyclosed non-archimedean field K with a K ◦ -model L on the algebraic K ◦ -model X of X . Supposethat for any finite S ⊂ X div , there is a sequence of algebraic K ◦ -models Z m of X with id X extending to finite morphisms g m : Z m → X such that m log | b m | converges pointwise to on S ,where b m is the base-ideal of g ∗ m ( L ) ⊗ m as before. Then k k L is a semipositive model metric.Proof. In this proof, we will need intersection theory on K ◦ -models. Since the base K ◦ is notnoetherian, we will use the intersection theory with Cartier divisors from [Gub98] (see also[GS15, Section 2] and [GRW16, Appendix] for algebraic versions). The main ingredient is thatevery vertical Cartier divisor D has an associated Weil divisor cyc( D ) with multiplicities in thevalue group Γ. To define the multiplicities, we pass to a dominating model with reduced specialfibre and use the projection formula (see [Gub98, 3.8, 3.10]). In the algebraic setting, such adominating model exists by the reduced fibre theorem [BLR95, Theorem 2.1’].Let n := dim( X ). Hence X is irreducible of dimension n + 1. We choose a closed curve Y in the special fibre X s . Then we have to show that deg L ( Y ) ≥
0. We follow the strategy of[Goo69] to use the blow-up π : X ′ → X along Y (as suggested in [BFJ16, Remark 5.13]). Then E := π − ( Y ) is an effective Cartier divisor on X ′ which is vertical. Note that any K ◦ -model of X is dominated by a projective K ◦ -model of X [Gub03, Proposition 10.5] and so we may replace X ′ by a projective dominating model. Then we have a very ample invertible sheaf H ′ on X ′ .We may view the vertical closed subscheme E of X ′ as a projective scheme of pure dimension n over the residue field ˜ K and we consider the surjective morphism E → Y induced by π . Then thesupport of the Weil divisor cyc( E ) is contained in E . Using a dominating model with reducedspecial fibre, using [GRW16, Proposition A.7] and the projection formula [Gub98, Proposition4.5], it is clear that cyc( E ) has a component mapping onto Y . It follows from using generic N ZHANG’S SEMIPOSITIVE METRICS 25 hyperplane sections and the fibre theorem [Har77, Exercise II.3.22] that π ∗ ( c ( H ′ ) n − . cyc( E ))is a positive multiple of Y . By the projection formula, it is enough to show(5.4.1) deg L ′ ( c ( H ′ ) n − . cyc( E )) ≥ L ′ := π ∗ ( L ).Let S be the set of divisorial points of X an associated with the K ◦ -model X ′ . By A.10 the set S is finite. By our standing assumptions in Lemma 5.4, there is a sequence of finite morphisms g m : Z m → X with base-ideal b m of g ∗ m ( L ) ⊗ m such that m log | b m | converges pointwise to 0 on S . The crucial new idea is to consider a sequence of morphisms ψ m : X m → X ′ related to thebase-ideals b m . In the following, m is a sufficiently divisible integer such that the base-ideal b m is vertical. Let π ′ : Z ′ m → Z m be the base change of π to Z m and let ψ ′ m : X m → Z ′ m be theblow up of Z ′ m in the closed subscheme ( π ′ ) − ( V ( b m )). Then we have a commutative diagram X m ψ m " " ❉❉❉❉❉❉❉❉ ψ ′ m / / Z ′ m π ′ / / g ′ m (cid:15) (cid:15) Z mg m (cid:15) (cid:15) X ′ π / / X of morphisms of K ◦ -models of X extending id X . Note that the base change g ′ m of g m is afinite morphism. Setting π ′ m := π ′ ◦ ψ ′ m , we have an effective vertical Cartier divisor D m :=( π ′ m ) − ( V ( b m )) on X m and we denote by s − D m the canonical meromorphic section of O ( − D m ).We define π m := π ◦ ψ m . Note that E m := π − m ( Y ) = ψ − m ( E ) is an effective Cartier divisor on X m and that H m := ψ ∗ m ( H ′ ) is a line bundle on X m which is generated by global sections. Weconclude from refined intersection theory that(5.4.2) cl( C ) = c ( H m ) n − . cyc( E m ) ∈ CH ( E m )for an effective 1-dimensional cycle C of X m with support over Y . We consider the invertiblesheaf L m := π ∗ m ( L ⊗ m ) ∼ = ψ ∗ m ( L ′⊗ m ) ∼ = ( π ′ m ) ∗ ( g ∗ m ( L ⊗ m )) of X m . We claim that(5.4.3) deg L m ( C ) ≥ deg O ( D m ) ( C ) . To prove this, let C m be any irreducible component of C . We choose ζ m ∈ C m and let ζ := π ′ m ( ζ m ). We note first that the stalk of L m ( − D m ) at ζ m is generated by global sections. Indeed,it follows from the definitions that there is a global section s m of g ∗ m ( L ⊗ m ) and an invertiblesection ℓ m of g ∗ m ( L ⊗ m ) at ζ such that ( π ′ m ) ∗ ( s m /ℓ m ) is an equation of the Cartier divisor D m at ζ m . Using that D m = π ′− m ( V ( b m )), a similar local consideration in any point of X m shows that t m := ( π ′ m ) ∗ ( s m ) ⊗ s − D m is a global section of L m ( − D m ) and the choice of s m yields that t m generates the stalk at ζ m . We deduce that the restriction of t m to C m is a global section whichis not identically zero and hencedeg L m ( C m ) = deg O ( D m ) ( C m ) + deg(div( t m | C m )) ≥ deg O ( D m ) ( C m )proving (5.4.3). By the projection formula [Gub98, Proposition 4.5] and (5.4.2), we have m deg L ′ ( c ( H ′ ) n − .E ) = deg L m ( C )and hence (5.4.3) leads to m deg L ′ ( c ( H ′ ) n − .E ) ≥ deg O ( D m ) ( C ) = deg O ( D m ) ( c ( H m ) n − . cyc( E m )) . Commutativity of intersection product [Gub98, Theorem 5.9] shows(5.4.4) m deg L ′ ( c ( H ′ ) n − . cyc( E )) ≥ deg( c ( H m ) n − .E m . cyc( D m )) . As we may replace X m in the above considerations by any dominating K ◦ -model of X , thereduced fibre theorem [BLR95, Theorem 2.1’] shows that we may assume that X m has reducedspecial fibre. We have cyc( D m ) = X W µ W W, where W ranges over all irreducible components of the special fibre of X m . Since the specialfibre of X m is reduced, [Gub98, Lemma 3.21] shows that there is a unique point ξ W of theanalytification X an of the generic fibre of X m with reduction equal to the generic point of W and the multiplicities µ W are given by µ W = − log k s D m ( ξ W ) k O ( D m ) . We insert this in (5.4.4) and use again projection formula to get(5.4.5) m deg L ′ ( c ( H ′ ) n − . cyc( E )) ≥ X V X W : ψ m ( W )= V µ W [ W : V ] deg( c ( H ′ ) n − .E.V ) , where V ranges over all irreducible components of ( X ′ ) s and W ranges over the irreduciblecomponents of ( X m ) s with ψ m ( W ) = V . Here, [ W : V ] is the degree of the induced map W → V . Note that ξ W = ψ m ( ξ W ) is a divisorial point of X an which reduces to the genericpoint of V in the model X ′ and hence ξ W is an element of the set S of divisorial points of X an associated with the K ◦ -model X ′ .We choose ε > b m and using that S is finite, there is a sufficiently divisible m such that0 ≤ − m log | b m | ( ξ W ) ≤ ε for all W as above. We conclude that(5.4.6) 0 ≤ µ W = − log k s D m ( ξ W ) k O ( D m ) = − log | b m | ( ξ W ) ≤ mε for all V and W as above with ψ m ( W ) = V . Let − R be the minimum of the finitely manyintersection numbers deg( c ( H ′ ) n − .E.V ) and 0. Then (5.4.5) and (5.4.6) lead todeg L ′ ( c ( H ′ ) n − . cyc( E )) ≥ − Rε X V X W : ψ m ( W )= V [ W : V ] . By the projection formula for ψ m applied to the Cartier divisor div( ρ ) on X ′ for any non-zero ρ in the maximal ideal of K ◦ and using that the special fibre of X m is reduced, we deduce easilythat X W : ψ m ( W )= V [ W : V ] = m V for the multiplicity m V of ( X ′ ) s along V . We conclude thatdeg L ′ ( c ( H ′ ) n − . cyc( E )) ≥ − Rε X V m V . The numbers R and m V are independent of ε . This proves (5.4.1) and hence the claim. (cid:3) In the following, we use the notation introduced in §
4. Recall that D ( X ) denotes the space ofmodel functions on X . Theorem 5.5.
Let X be a proper scheme over K and let θ be a closed (1 , -form on X . Thenthe set of θ -psh model functions is closed in D ( X ) with respect to pointwise convergence on X div . This is a generalization of Theorem 5.11 in [BFJ16] as we allow K to be an arbitrary non-archimedean field and also because we allow any proper scheme X . N ZHANG’S SEMIPOSITIVE METRICS 27
Proof.
We may check semipositivity for the pull-back with respect to a proper surjective mor-phism X ′ → X by Proposition 3.6 (b). Using Chow’s lemma and Proposition A.11, we concludethat we may assume X projective.Let ϕ be a model function on X which is the pointwise limit over X div of θ -psh model functionson X an . Replacing θ by θ + dd c ϕ , we may assume that ϕ = 0. Then the existence of a θ -pshmodel function ψ yields that θ + dd c ψ is semipositive and hence { θ } is nef (see 4.8). Let X be analgebraic K ◦ -model of X such that θ is determined on X . Then the restriction of θ X to X isnef. Since any K ◦ -model of X is dominated by a projective K ◦ -model of X [Gub03, Proposition10.5], we may assume that X is projective.The proof of Proposition 4.14 shows that N ( X /S ) is a finite dimensional R -vector space aswe can see it as a subspace of N ( X s ). We have also seen that the ample cone in N ( X /S ) isthe intersection of N ( X /S ) with the ample cone in N ( X s ) and hence it is open in N ( X /S ).We conclude that there are H , . . . , H n ample line bundles on X such that their numericalclasses α j form a basis of N ( X /S ). Then there are λ j ∈ R such that P j λ j c ( H j ) ∈ Pic( X ) R represents θ . Let ε j be small positive numbers such that the numbers λ j + ε j are rational. Weconsider the Q -line bundle L ε := O j H ⊗ ( λ j + ε j ) j on X and let L ε := L ε | X . Since { θ } is nef and ε j >
0, it follows that L ε is ample. For anymodel function ψ on X , we have c ( L ε , e − ψ k k L ε ) = dd c ψ + θ + X j ε j α j . We conclude that a θ -psh model function ψ yields a semipositive model metric e − ψ k k L ε . Since ϕ = 0 is the pointwise limit over X div of θ -psh model functions ψ on X , we deduce that k k L ε is the pointwise limit over X div of semipositive model metrics on L ε . It follows from Proposition5.1 that k k L ε is semipositive. This means that L ε is nef.By definition of nef and using N ( X /S ) ⊂ N ( X s ), we see that the cone in N ( X /S ) of nefclasses is the intersection of N ( X /S ) with the nef cone in N ( X s ). In particular, the cone ofnef classes is closed in N ( X /S ). Using ε = ( ε , . . . , ε n ) →
0, we deduce that P j λ j c ( H j ) isnef. Since the latter represents θ , we conclude that ϕ = 0 is θ -psh. (cid:3) Remark 5.6.
Note that Theorem 1.3 is a special case of Theorem 5.5 by using 4.7.
Appendix A. Divisorial points
Definition A.1.
Let V be a paracompact strictly K -analytic space. A divisorial point x of V is a point x ∈ V such that there is a formal K ◦ -model V with reduction map π : V → V s suchthat π ( x ) is the generic point of an irreducible component of V s . We call x also a divisorial point associated with the model V . A.2.
Let V be a strictly K -affinoid space. We recall the following facts from [Ber90, Proposition2.4.4]: The Shilov boundary of V is the unique minimal closed subset Γ of V with the propertythat max x ∈ Γ | f ( x ) | = max x ∈ V | f ( x ) | for every f ∈ A := O ( V ). Note that A is a strictly K -affinoid algebra and let ˜ A := { a ∈ A | | a | sup ≤ } / { a ∈ A | | a | sup < } be its canonicalreduction. There is a canonical reduction map V → Spec( ˜ A ) which is surjective. The genericpoint of an irreducible component E of Spec( ˜ A ) has a unique preimage in V denoted by ξ E .The Shilov boundary Γ is equal to the finite set of points ξ E with E ranging over all irreduciblecomponents of the canonical reduction Spec( ˜ A ). Proposition A.3.
Let V = Spf( A ) be a formal affine K ◦ -model of the strictly K -affinoid space V . Then the set of divisorial points of V associated with V is equal to the Shilov boundary of V .In particular, this set is finite.Proof. Let A := A ⊗ K ◦ K be the associated strictly K -affinoid algebra. By [GRW17, Proposition2.12], the canonical morphism ι : Spec( ˜ A ) → V s is finite and surjective. Let π : V → V s be thereduction map of V and let π ′ : V → Spec( ˜ A ) be the canonical reduction map of V . Using that π = π ′ ◦ ι , we conclude for x ∈ V that π ( x ) is the generic point of an irreducible component of V s if and only if π ′ ( x ) is the generic point of an irreducible point of Spec( ˜ A ). It follows thatthe set of divisorial points associated with V is given by the points ξ E with E ranging over theirreducible components of Spec( ˜ A ). We have seen in A.2 that this set is the Shilov boundary of V = V η proving precisely our claim. (cid:3) A.4.
For a point x of a paracompact strictly K -analytic space V , recall that O V,x is endowedwith a canonical seminorm p x which induces a canonical absolute value on the fraction field of O V,x / { p x = 0 } . The completion of this fraction field is a non-archimedean field extension of K denoted by H ( x ). As in [Ber90, 9.1], we define s ( x ) as the transcendence degree of the residuefield of H ( x ) over ˜ K .We define dim x ( V ) as the minimum of the dimensions of the strictly K -affinoid domains in V containing x . Let us pick any strictly K -affinoid domain W of V containing x . Then(A.4.1) dim x ( V ) = max i dim( O ( W i ))where W i ranges over the irreducible components of W containing x and where we use the Krulldimension of the strictly K -affinoid algebra O ( W i ) on the right. We refer to [Duc07, Section1] for more details and additional properties on the dimension of K -analytic spaces. It followseasily from [Ber93, Lemma 2.5.2] that(A.4.2) s ( x ) ≤ dim x ( V ) . Proposition A.5.
Let x be a point of a paracompact strictly K -analytic space V . Then x is adivisorial point of V if and only if s ( x ) = dim x ( V ) . Let us remark that the result and its proof are similar to [Poi13, Corollaire 4.18].
Proof.
Let us prove that if s ( x ) = dim x ( V ) then x is a divisorial point (the other implicationfollows easily from the definition of divisorial points, see for instance [Poi13, Lemme 4.4]). Let g , . . . , g n be elements in the residue field of H ( x ) which are algebraically independent over ˜ K where n = dim x ( V ). Let U be an n -dimensional strictly K -affinoid domain in V containing x and let A = O ( U ) be the corresponding strictly K -affinoid algebra. The residue field of H ( x ) can be identified with the residue field of the fraction field of A / p x for the prime ideal p x := { a ∈ A | | a ( x ) | = 0 } . For each i = 1 . . . n we can then find some functions α i , β i ∈ A such that | β i ( x ) | 6 = 0, | α i ( x ) /β i ( x ) | = 1 and such that the residue classes of α i ( x ) /β i ( x ) areequal to g i in the residue field of H ( x ). Shrinking U if necessary, we can then assume thatthe β i ’s are invertible on U (it suffices to consider strictly K -affinoid Laurent domains of theform {| β i | ≥ r } ). Replacing U by the Weierstrass domain {| α i /β i | ≤ , i = 1 . . . n } , we can evenassume that f i := α i /β i ∈ A ◦ for i = 1 , . . . , n . These functions have residue classes ˜ f i = g i inthe residue field of H ( x ). Let us now denote by ˜ x ∈ Spec( ˜ A ) the canonical reduction of x . Let κ (˜ x ) be the residue field of ˜ x . In the following diagram A ◦ → f A → κ (˜ x ) ֒ → ^ H ( x )the last map is injective. Since the g i ’s are algebraically independent, it follows that the ˜ f i ’s arealgebraically independent in κ (˜ x ). Since dim( ˜ A ) = dim( A ) = n (see the remark at the end of N ZHANG’S SEMIPOSITIVE METRICS 29 § x is a generic point of Spec( ˜ A ) and hence it is a divisorialpoint of U . According to [Bos14, Lemma 8.4.5], there exists an admissible formal scheme V withgeneric fibre V such that U is the generic fibre of a formal affine open subset U of V . It followsthat x is a divisorial point associated with V . (cid:3) Corollary A.6.
Let x be a point of a strictly K -affinoid domain W of the paracompact strictly K -analytic space V . Then x is a divisorial point of W if and only if x is a divisorial point of V .Proof. Since the invariants s ( x ) and dim x ( V ) do not change if we pass from V to W , the claimfollows from Proposition A.5. (cid:3) Proposition A.7.
Let F be a non-archimedean field extension of K which is a subfield of C K and let x be a point of the base change V ˆ ⊗ K F of the paracompact strictly K -analytic space V .Let ϕ : V ˆ ⊗ K F → V be the natural map. Then ϕ ( x ) is a divisorial point of V if and only if x isa divisorial point of V ˆ ⊗ K F .Proof. By [Duc07, Proposition 1.22] we have(A.7.1) dim x ( V ˆ ⊗ K F ) = dim ϕ ( x ) ( V ) . The equality s ( x ) = s ( ϕ ( x )) follows easily from [Ber90, Lemma 9.1.1]. Then the claim followsfrom (A.7.1) and Proposition A.5. (cid:3) Remark A.8.
In general, if F is an arbitrary non-archimedean extension of K , with the abovenotations, it is not true that divisorial points of V ˆ ⊗ K F are mapped to divisorial points of V .For instance, let r ∈ | F ∗ | with 0 < r < r n
6∈ | K ∗ | for all non-zero n ∈ N . If D denotes the closed unit disc over K , then the point η r ∈ D ˆ ⊗ K F given by the supremum over theclosed disc of radius r is a divisorial point of D ˆ ⊗ K F (it is a point of type 2 in D ˆ ⊗ K F ), but it ismapped to a point of type 3 in D , namely the point corresponding to the closed disc of radius r in D , which is not a divisorial point of D .Now we restrict to the algebraic setting. For a proper scheme X over K , let X div denote theset of divisorial points of X an . The next result shows that it is enough to consider algebraicmodels. Proposition A.9.
Let X be a proper scheme over K . Then x ∈ X div if and only if thereis an algebraic K ◦ -model X of X such that x is a divisorial point associated with the formalcompletion ˆ X .Proof. It is enough to show that a divisorial point x of X an associated with a formal K ◦ -model V of V is also associated with ˆ X for a suitable algebraic K ◦ -model X of X . This followseasily from the fact that V is dominated by the formal completion of an algebraic K ◦ -model (seeLemma 2.4). (cid:3) A.10.
We say that x ∈ X an is a divisorial point associated with the algebraic K ◦ -model X if x is a divisorial point associated with the formal completion ˆ X in the sense of Definition A.1.Equivalently, this means that the reduction of x is a generic point of an irreducible componentof the special fibre X s .Note that ˆ X has a finite covering by formal affine open subsets. It follows from PropositionA.3 and Corollary A.6 that the set of divisorial points of X an associated with X is finite. Proposition A.11.
Let f : X → Y be a generically finite surjective morphism of proper varietiesover K . Then we have X div = f − ( Y div ) . Proof.
There is an open dense subset U of Y such that f induces a finite surjective morphism f − ( U ) → U . For x ∈ ( f − ( U )) an and y := f ( x ), we note that H ( x ) / H ( y ) is a finite extension.Since dim x ( X ) = dim( X ) = dim( Y ) = dim y ( Y ), it follows from Proposition A.5 that x is adivisorial point of X an if and only if y is a divisorial point of Y an . The same criterion shows thatevery divisorial point of X (resp. Y ) is contained in the analytification of f − ( U ) (resp. U ). (cid:3) Proposition A.12.
Let ( X i ) i ∈ I be the irreducible components of a proper scheme X over K .Then we have X div = [ i ∈ I ( X i ) div . Proof.
It follows from (A.4.2) and Proposition A.5 that the divisorial points of X an or of any( X i ) an are contained in the Zariski open subset of X an consisting of those divisorial points whichare contained in the analytification of only one irreducible component of X . Now the claimfollows from the fact shown in Corollary A.6 that divisorial points can be checked G -locallyand hence Zariski-locally. Note also that divisorial points depend only on the induced reducedstructure. (cid:3) References [Ana73] Sivaramakrishna Anantharaman. Sch´emas en groupes, espaces homog`enes et espaces alg´ebriques surune base de dimension 1. In
Sur les groupes alg´ebriques , M´em. 33 of
Bull. Soc. Math. France , pages5–79. Soc. Math. France, Paris, 1973.[BPR16] Matthew Baker, Sam Payne, and Joseph Rabinoff. Nonarchimedean geometry, tropicalization, andmetrics on curves.
Algebr. Geom.
Spectral theory and analytic geometry over non-Archimedean fields , vol-ume 33 of
Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI,1990.[Ber93] Vladimir G. Berkovich. ´Etale cohomology for non-Archimedean analytic spaces.
Inst. Hautes ´EtudesSci. Publ. Math. , (78):5–161 (1994), 1993.[Bos14] Siegfried Bosch.
Lectures on formal and rigid geometry , volume 2105 of
Lecture Notes in Mathemat-ics . Springer, Cham, 2014.[BGR84] Siegfried Bosch, Ulrich G¨untzer, and Reinhold Remmert.
Non-Archimedean analysis , volume 261of
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sci-ences] . Springer-Verlag, Berlin, 1984. A systematic approach to rigid analytic geometry.[BL93a] Siegfried Bosch and Werner L¨utkebohmert. Formal and rigid geometry. I. Rigid spaces.
Math. Ann. ,295(2):291–317, 1993.[BL93b] Siegfried Bosch and Werner L¨utkebohmert. Formal and rigid geometry. II. Flattening techniques.
Math. Ann. , 296(3):403–429, 1993.[BLR95] Siegfried Bosch, Werner L¨utkebohmert, and Michel Raynaud. Formal and rigid geometry. IV. Thereduced fibre theorem.
Invent. Math. , 119(2):361–398, 1995.[BE18] S´ebastien Boucksom and Dennis Eriksson. Spaces of norms, determinant of cohomology and Feketepoints in non-Archimedean geometry. arXiv:1805.01016v1 , 2018.[BFJ15] S´ebastien Boucksom, Charles Favre, and Mattias Jonsson. Solution to a non-Archimedean Monge-Amp`ere equation.
J. Amer. Math. Soc. , 28(3):617–667, 2015.[BFJ16] S´ebastien Boucksom, Charles Favre, and Mattias Jonsson. Singular semipositive metrics in non-Archimedean geometry.
J. Algebraic Geom. , 25(1):77–139, 2016.[Bou71] Nicolas Bourbaki. ´El´ements de math´ematique. Topologie g´en´erale. Chapitres 1 `a 4 . Hermann, Paris,1971.[BGJKM] Jos´e Ignacio Burgos Gil, Walter Gubler, Philipp Jell, Klaus K¨unnemann, and Florent Martin. Dif-ferentiability of non-archimedean volumes and non-archimedean Monge-Amp`ere equations (with anappendix by Robert Lazarsfeld). arXiv:1608.01919 , 2016.[BMPS16] Jos´e Ignacio Burgos Gil, Atsushi Moriwaki, Patrice Philippon, and Mart´ın Sombra. Arithmeticpositivity on toric varieties.
J. Algebraic Geom. , 25(2):201–272, 2016.[BPRS15] Jos´e Ignacio Burgos Gil, Patrice Philippon, Juan Rivera-Letelier, and Mart´ın Sombra. The distribu-tion of Galois orbits of points of small height in toric varieties. arXiv:1509.01011 , 2015.
N ZHANG’S SEMIPOSITIVE METRICS 31 [BPS14] Jos´e Ignacio Burgos Gil, Patrice Philippon, and Mart´ın Sombra. Arithmetic geometry of toric vari-eties. Metrics, measures and heights.
Ast´erisque , (360):vi+222, 2014.[BPS15] Jos´e Ignacio Burgos Gil, Patrice Philippon, and Mart´ın Sombra. Successive minima of toric heightfunctions.
Ann. Inst. Fourier (Grenoble) , 65(5):2145–2197, 2015.[BPS16] Jos´e Ignacio Burgos Gil, Patrice Philippon, and Mart´ın Sombra. Height of varieties over finitelygenerated fields.
Kyoto J. Math. , 56(1):13–32, 2016.[Cha06] Antoine Chambert-Loir. Mesures et ´equidistribution sur les espaces de Berkovich.
J. Reine Angew.Math. , 595:215–235, 2006.[CD12] Antoine Chambert-Loir and Antoine Ducros. Formes diff´erentielles r´eelles et courants sur les espacesde Berkovich. arXiv:1204.6277 , 2012.[CT09] Antoine Chambert-Loir and Amaury Thuillier. Mesures de Mahler et ´equidistribution logarithmique.
Ann. Inst. Fourier (Grenoble) , 59(3):977–1014, 2009.[Con07] Brian Conrad. Deligne’s notes on Nagata compactifications.
J. Ramanujan Math. Soc. , 22:205–257,2007.[Duc07] Antoine Ducros. Variation de la dimension d’un morphisme analytique p-adique.
Compositio Math. http://homepages.math.uic.edu/~ein/DFEM.pdf .[Fab09] Xander Faber. Equidistribution of dynamically small subvarieties over the function field of a curve.
Acta Arith. , 137(4):345–389, 2009.[FK18] Kazuhiro Fujiwara and Fumiharu Kato.
Foundations of rigid geometry I . EMS Monographs in Math-ematics. European Mathematical Society (EMS), Z¨urich, 2018.[Goo69] Jacob Eli Goodman. Affine open subsets of algebraic varieties and ample divisors.
Ann. of Math.(2) , 89:160–183, 1969.[Gro61] Alexander Grothendieck. ´El´ements de g´eom´etrie alg´ebrique. II. ´Etude globale ´el´ementaire de quelquesclasses de morphismes.
Inst. Hautes ´Etudes Sci. Publ. Math. , (8):222, 1961.[Gro66] Alexander Grothendieck. ´El´ements de g´eom´etrie alg´ebrique. IV. ´Etude locale des sch´emas et desmorphismes de sch´emas. III.
Inst. Hautes ´Etudes Sci. Publ. Math. , (28):255, 1966.[Gub98] Walter Gubler. Local heights of subvarieties over non-Archimedean fields.
J. Reine Angew. Math. ,498:61–113, 1998.[Gub03] Walter Gubler. Local and canonical heights of subvarieties.
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) ,2(4):711–760, 2003.[Gub07a] Walter Gubler. The Bogomolov conjecture for totally degenerate abelian varieties.
Invent. Math. ,169(2):377–400, 2007.[Gub07b] Walter Gubler. Tropical varieties for non-Archimedean analytic spaces.
Invent. Math. , 169(2):321–376, 2007.[Gub08] Walter Gubler. Equidistribution over function fields.
Manuscripta Math. , 127(4):485–510, 2008.[Gub13] Walter Gubler. A guide to tropicalizations. In
Algebraic and combinatorial aspects of tropical geom-etry , volume 589 of
Contemp. Math. , pages 125–189. Amer. Math. Soc., Providence, RI, 2013.[GK17] Walter Gubler and Klaus K¨unnemann. A tropical approach to nonarchimedean Arakelov geometry.
Algebra Number Theory , 11(1):77–180, 2017.[GK15] Walter Gubler and Klaus K¨unnemann. Positivity properties of metrics and delta-forms.
J. ReineAngew. Math. doi.org/10.1515/crelle-2016-0060 .[GRW16] Walter Gubler, Joseph Rabinoff, and Annette Werner. Skeletons and tropicalizations.
Adv. Math. ,294:150–215, 2016.[GRW17] Walter Gubler, Joseph Rabinoff, and Annette Werner. Tropical Skeletons.
Ann. Inst. Fourier (Greno-ble) , 67(5):1905–1961, 2017.[GS15] Walter Gubler and Alejandro Soto. Classification of normal toric varieties over a valuation ring ofrank one.
Doc. Math. , 20:171–198, 2015.[Har77] Robin Hartshorne.
Algebraic geometry . Graduate Texts in Mathematics, No. 52. Springer-Verlag,New York-Heidelberg, 1977.[Jel16] Philipp Jell.
Differential forms on Berkovich analytic spaces and their cohomology . PhD thesis,Universit¨at Regensburg, 2016. urn:nbn:de:bvb:355-epub-347884 .[dJ96] Aise Johan de Jong. Smoothness, semi-stability and alterations.
Inst. Hautes ´Etudes Sci. Publ.Math. , (83):51–93, 1996.[KRZ16] Eric Katz, Joseph Rabinoff, and David Zureick-Brown. Uniform bounds for the number of rationalpoints on curves of small Mordell–Weil rank.
Duke Math. J. , 165(16):3189–3240, 2016. [Kie72] Reinhardt Kiehl. Ein ”Descente”-Lemma und Grothendiecks Projektionssatz f¨ur nichtnoetherscheSchemata.
Math. Ann. , 198:287-316, 1972.[Kle66] Steven L. Kleiman. Toward a Numerical Theory of Ampleness.
Annals of Mathematics , SecondSeries, Vol. 84, No. 3 (Nov., 1966), pp. 293-344.[Poi13] J´erˆome Poineau. Les espaces de Berkovich sont ang´eliques.
Bulletin de la Soci´et´e Math´ematique deFrance
Invent. Math. , 13:1–89, 1971.[Sta16] The Stacks Project Authors.
Stacks Project . http://stacks.math.columbia.edu , 2016.[Tem00] Michael Temkin. On local properties of non-archimedean analytic spaces. Math. Ann. , 318:585-607,2000.[Thu05] Amaury Thuillier.
Th´eorie du potentiel sur les courbes en g´eom´etrie analytique non-archim´edienne.Applications `a la th´eorie d’Arakelov . PhD thesis, Universit´e de Rennes I, 2005.[Ull95] Peter Ullrich. The direct image theorem in formal and rigid geometry.
Math. Ann. , 301:69-104,1995.[Voj07] Paul Vojta. Nagata’s embedding theorem. arXiv:0706.1907
Manuscripta Math. , 142(3-4):273–306, 2013.[Yam16] Kazuhiko Yamaki. Strict supports of canonical measures and applications to the geometric Bogo-molov conjecture.
Compos. Math. , 152(5):997–1040, 2016.[Yua08] Xinyi Yuan. Big line bundles over arithmetic varieties.
Invent. Math. , 173(3):603–649, 2008.[YZ17] Xinyi Yuan and Shou-Wu Zhang. The arithmetic Hodge index theorem for adelic line bundles.
Math.Ann. , 367(3-4):1123–1171, 2017.,[Zha93] Shouwu Zhang. Admissible pairing on a curve.
Invent. Math. , 112(1):171–193, 1993.[Zha95] Shouwu Zhang. Small points and adelic metrics.
J. Algebraic Geom. , 4(2):281–300, 1995., 4(2):281–300, 1995.