A comparison of the real and non-archimedean Monge–Ampère operator

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https://doi.org/10.1007/s00209-020-02527-3

Mathematische Zeitschrift

A comparison of the real and non-archimedean Monge–Ampère operator

Christian Vilsmeier¹

Received: 19 December 2019 / Accepted: 20 February 2020 / Published online: 2 April 2020

Abstract

LetXbe a proper algebraic variety over a non-archimedean, non-trivially valued field. We show that the non-archimedean Monge–Ampère measure of a metric arising from a convex function on an open face of some skeleton ofX^anis equal to the real Monge–Ampère measure of that function up to multiplication by a constant. As a consequence we obtain a regularity result for solutions of the non-archimedean Monge–Ampère problem on curves.

Keywords Monge–Ampère operator·Berkovich spaces·Metrics·Tropical geometry Mathematics Subject Classification Primary 32P05; Secondary 32W20·14G22·14T05

Contents

1 Introduction . . . 633

2 Skeletons, formal models and divisors. . . 635

3 Metrics . . . 641

4 Measures . . . 643

5 Comparison of the real and non-archimedean Monge–Ampère operator . . . 654

6 Applications to regularity . . . 660

Appendix A: Reduction of germs. . . 662

Appendix B: Convexity of psh-functions . . . 664

References. . . 667

1 Introduction

The non-archimedean analogue of the Calabi conjecture is still an open problem in non-archimedean geometry. In the complex case it states that for a complex compactn-

This work was partially supported by the collaborative research center ’SFB 1085: Higher Invariants’ funded by the Deutsche Forschungsgemeinschaft.

B

Christian Vilsmeier

Christian.Vilsmeier@mathematik.uni-regensburg.de

1 Mathematik, Universität Regensburg, 93053 Regensburg, Germany

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dimensional manifold M with a Kähler form ω and f ∈ C^∞(M), f > 0 such that

M fωⁿ=

Mωⁿthere exists a unique up to constantϕ∈C^∞(M)such thatω+dd^cϕ >0 and(ω+dd^cϕ)ⁿ = fωⁿ. This was solved by Calabi (uniqueness, [14]) and Yau (existence, [45]). A strategy to attack the non-archimedean Monge–Ampère equation was proposed by Kontsevich and Tschinkel in unpublished though influential notes dated around 2001. In the non-archimedean setting, we fix a non-archimedean, non-trivially valued fieldK and a smooth projective varietyXoverK of dimensionnwith a line bundleLonXand consider the correspondingK-analytic spaceX^anwith the line bundleL^anin the sense of Berkovich.

To any continuous semipositive metric · onLânone can associate a positive Radon mea- surec1(L, · )ⁿ on Xân, called the Monge–Ampère measure, which was introduced by Chambert-Loir in [16]. In a non-archimedean analogue of the Calabi conjecture one asks for a solution ofc₁(L, · )ⁿ = μfor a positive Radon measureμonXânof massLⁿ when Lis ample. The uniqueness up to addition of a constant of such a solution was proved by Yuan and Zhang in [46]. The existence was proved by Liu in [36] for the case of a totally degenerate abelian varietyXunder some regularity assumptions on the measure by reducing to the complex case. The best known existence result is due to Boucksom, Favre and Jonsson [5, Theorem A]. They prove existence of a solution to the non-archimedean Monge–Ampère equation ifK is discretely valued of residue characteristic zero andμis supported on the dual complex of some SNC model ofX. Note that they assumed also an algebraicity condition which was later removed by Burgos et al. [7, Theorem D]. As such a dual complex consists of faces which look like simplices inRⁿit would be tempting to observe a connection of the non-archimedean Monge–Ampère operator with the real one. This is the aim of the paper at hand. In particular we will prove the following result (a precise definition of the occurring measures is given in Sect.4.):

Theorem 1.1 Let X be an n-dimensional proper algebraic variety over K , L =(L, · ) a formally metrized line bundle on Xân andτ an open face of dimension n of a skeleton corresponding to a strictly semistable formal modelXof Xânon which L has a formal model L. Letϕbe a continuous function on Xânsuch that · e^−ϕis a semipositive metric. Suppose thatϕfactorizes through the retraction p_Xonto the skeleton. Then

c₁(L, · e^−ϕ)ⁿ = [ ˜K(S): ˜K] ·n! ·MA ϕ

τ

on p⁻¹_X (τ)whereMAdenotes the real Monge–Ampère operator onτ which is considered to be a measure on p_X⁻¹(τ)by pushforward via the inclusion and S denotes the point in the special fibre ofXwhich is the image ofτunder the reduction map.

The paper is organized as follows: In Sect.2we give an overview over basic concepts in formal geometry. We recall the definition of a strongly nondegenerate strictly polystable formal scheme and its associated skeleton introduced in [3] and explain the stratum face correspondence developed in [32]. At the end of the section we construct a Cartier divisor from a piecewise affine linear function on the skeleton and prove an important lemma dealing with the degree with respect to this divisor in the case of an affine linear function.

In Section3we collect basic definitions and facts on metrized line bundles. Following [25]

we introduce piecewise linear, algebraic and formal metrics and the notion of semipositivity for them. We also recall some useful properties and the situations in which the definitions coincide.

In Sect. 4we recall the definitions of the real and non-archimedean Monge–Ampère measure but we define the latter on strictlyK-analytic Hausdorff spaces which locally admit open immersions into analytifications of schemes of finite type over the fieldK. In order

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to do so, we prove a convergence result for these spaces. This will allow us to formulate Theorem1.1in a more general setting where everything is defined locally, see Corollary5.7.

Section5is subject to the proof of Theorem1.1. In fact in Corollary5.7, we prove a local generalization of this result. It will follow from Lemma4.8and CorollaryB.4that Corollary5.7implies Theorem1.1. We will also generalize the local result in Corollary5.10 to strongly nondegenerate polystable formal models ofX^ani.e. we will prove:

Theorem 1.2 Let X be an n-dimensional proper algebraic variety over K andXa strongly nondegenerate polystable formal model of X^anover K^◦with associated skeleton. Letτbe an n-dimensional open face ofwith associated point S in the special fibre ofX. Let h be a convex function onτand denote byO^h◦^p^Xthe trivial line bundle on the strictly K -analytic space p_X⁻¹(τ)endowed with the metric given by1 =e^−h◦p^X. Then

c1

O^h◦^p^X_n

= [ ˜K(S): ˜K] ·n! ·MA(h) on p_X⁻¹(τ).

The proof is inspired by the proof of [32, Theorem 5.18]. In order to reduce to the toric situation, a key ingredient will be Lemma2.13, showing that affine linear functions on a closed face of a skeleton induce numerically trivial vertical Cartier divisors on a suitable part of the corresponding formal model.

Finally, in Sect.6, we apply Theorem1.1to obtain two regularity results for solutions to the non-archimedean Calabi–Yau problem. For example we will prove in Proposition6.4:

Proposition 1.3 Let X be a smooth projective curve,μa positive Borel meausre on X^anand ϕa solution to the Monge–Ampère equation c1(L, · e^−ϕ)=μ. Letτbe an open face of a skeleton associated to a strictly semistable formal model of X^an on which(L, · )has a formal model. Suppose thatμis supported on that skeleton and is given onτ by f ·d x whered xdenotes the Lebesgue measure onτ. If f ∈C^k(τ)then we haveϕ

τ∈C^k+2(τ).

HereC^k(τ)is the space ofktimes continuously differentiable functions onτ. Proposition1.3 follows from Theorem1.1and regularity of the real Monge–Ampère equation.

TerminologyIn the following,Kdenotes a complete, non-archimedean, non-trivially valued field and K^◦ its corresponding valuation ring with maximal ideal K^◦◦. All schemes are assumed to be locally of finite type.

2 Skeletons, formal models and divisors

In this section we first define formal schemes and their generic and special fibres. For details we refer to [12, II.7, II.8.3]. Then we recall the concept of skeletons associated to strongly nondegenerate strictly polystable formal schemes introduced by Berkovich [3]. To a subdivision of the skeleton, one can associate a formal analytic structure as in [32, Proposition 5.5]. We generalize the subsequent results of [32, Sect. 5] concerning the stratum face correspondence by dropping the condition of algebraically closedness of the base field. Finally we explain how a piecewise affine linear function on the skeleton induces a Cartier divisor on the formal scheme corresponding to a suitable subdivision of the skeleton.

Definition 2.1 LetY be a reduced scheme of locally finite type over a fieldκ. SetY⁽⁰⁾ :=

Y and letY⁽ⁱ⁺¹⁾ be the complement of the set of normal points inY⁽ⁱ⁾. The irreducible

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components ofY⁽ⁱ⁾\Y⁽ⁱ⁺¹⁾are calledstrataofY. There is a partial ordering on the set of strata given byR₁ ≤R₂if and only ifR₁ ⊆R₂. A cycle Z∈ Z(Y)is called astrata cycle if there are strataS₁, . . . ,S_n ofYsuch thatZ=

m_iS_iwithm_i ∈R.

Definition 2.2 A topological ring Ais calledadicif there is an ideala⊆ Asuch that the ideals(aⁿ)_n∈Nform a neighbourhood basis for 0. We callaadefining ideal. LetAbe an adic, complete, separated ring with finitely generated defining ideala. Theaffine formal schemeof Ais the locally topologically ringed space Spf(A)=(X,O_X)whereXandO_Xare defined as follows:Xis the set of all open prime ideals ofA. As a prime ideal is open if and only if it containsa, we may identifyXwith Spec(A/a) ⊆Spec(A)and we endowXwith the topology induced by the Zariski topology on Spec(A). Moreover we define

O_X:=lim

← O_Spec(A/aⁿ₎.

Aformal schemeis a locally topologically ringed space(X,OX)such that for eachx ∈X there is an open neighbourhood Uof x with

U,OX

U

isomorphic to an affine formal scheme.

Now letabe a defining ideal ofK^◦. A topologicalK^◦-algebraAis calledadmissible, if

a∈ Aaⁿ·a=0 for somen∈N

= {0}i.e. Adoes not haveK^◦-torsion and if Ais isomorphic to aK^◦-algebra of the formK^◦ ζ1, ..., ζn/(a1, ...,am)endowed with thea-adic topology. A formalK^◦-schemeXis calledadmissibleif there is a locally finite open cover (Ui)i∈IofXwithUi =Spf(A_i)for admissibleK^◦-algebrasA_i.

LetX= Spf(A)be an admissible formal affineK^◦-scheme. Theanalytic generic fibre ofXis defined asX^an:=M(A⊗K^◦ K), whereM(·)denotes the Berkovich spectrum (cf.

[1, 1.2]). The special fibre ofXis given byX˜ := Spec(A⊗K^◦ k), wherek := K^◦/K^◦◦

is the residue field ofK. For an admissible formal K^◦-schemeXone obtains the generic and the special fibre by a gluing process. There is a canonical surjective reduction map red:X^an→ ˜X, see [28, Sect. 2.13].

Definition 2.3 Forn∈N>0anda∈K^◦◦we define

X(n,a):=Spf(K^◦ x0, ...,xn/(x0...xn−a)).

For tuplesn=(n0, ...,np)∈N_>0^p+1anda=(a0, ...,ap)∈(K^◦◦)^p+1we defineX(n,a):=

X(n₀,a₀)×K^◦...×K^◦X(n_p,a_p)and form ∈Nwe setX(m):=X(m,1). Astrictly polystable formal scheme overK^◦is an admissible formal schemeXoverK^◦which can be covered by formal open setsUwith étale morphisms

ψ:U→X(n,a,m):=X(n,a)×K^◦X(m)

wheren,aandmmay depend onU. We say thatXisstrongly nondegenerate strictly polystable if alla_ican be chosen nonzero.

To a strongly nondegenerate strictly polystable formal scheme Xover K^◦ Berkovich introduced in [3] a canonical polytopal subset S(X) of X^an called the skeleton. It is a closed subset of X^an which is locally given by canonical polysimplices and can be described as follows. Let ψ : U → X(n,a,m) be an étale morphism as above.

The generic fibre of the right hand side is given as X(n,a,m)^an = M(A) where A=(K T₀^±, ...,T_m^±) T₀₀, ...,T_p,n_p/(T₀₀...T_0,n₀−a₀, ...,T_p0...T_p,n_p−a_p). The elements of Acan be expressed as

μa_μT^μwitha_μ ∈ K T₀^±, ...,T_m^±anda_μ =0 if there is an i∈ {0, ...,p}such thatμi,k ≥1 for allk ∈ {0, ...,ni}. Now to an elementtin the polysimplex

t∈Rⁿ⁺¹_≥₀ t_i0+...+t_{i n}_i = −log(|a_i|),0≤i ≤p

we associate a seminorm on A

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by sending a power series as above to max_μ{|a_μ|exp(−t·μ)}. This gives an embedding of the polysimplex intoM(A)whose image is denoted by. The skeletonS(U)ofUis defined to be(ψân)⁻¹(). One can show thatψâninduces a homeomorphism from(ψân)⁻¹()to ifUhas a unique minimal stratum which maps to the minimal stratum ofX(n,a,m). The skeletonS(X)ofXis the union of allS(U)and is independent of all choices.

To a stratumSofXone can associate a canonical polysimplexS in the skeleton such that the interiors of theTform a disjoint cover ofS(X)whereTranges over all strata ofX˜. In order to do so, we choose a refinement of the cover ofXas described in the Proposition below and chooseUsuch thatSis its distinguished stratum. We then defineS :=S(U).

An admissible formal schemeXis calledstrongly nondegenerate polystableif there exists a strongly nondegenerate strictly polystable formal schemeXand a surjective étale morphism X→X. The skeleton ofXis defined to be the image of the skeleton ofXunder the map X^an→X^an.

One can endow the skeleton with a piecewise linear structure, see [4, Sect. 6]. We will define piecewise affine linear functions on the skeleton of a strongly nondegenerate strictly polystable formal scheme in Definition2.10. There is a canonical continuous retraction map p_X :X^an →S(X)which restricts to the identity onS(X). For details see [3, Sect. 4], [4, Sect. 4] or [32, 5.3].

We have the following stratum face correspondence due to Berkovich:

Proposition 2.4 LetXbe a strongly nondegenerate polystable formal scheme with skeleton . There is a bijective correspondence between the open faces ofand the strata ofX˜ given by

R=red(p⁻_X¹(τ)), τ =p_X(red⁻¹(R)).

Proof [3, Theorem 5.2 (iv), Theorem 5.4].

Proposition 2.5 LetXbe a strongly nondegenerate strictly polystable formal scheme over K^◦. Any formal open covering ofXadmits a refinement{U}by formal open subsetsUas in Definition2.3such that

(i) EveryUis a formal affine open subscheme ofX,

(ii) there is a distinguished stratum S ofX˜ associated toUsuch that for any stratum T of X˜, we have S⊆T if and only ifU˜∩T = ∅,

(iii) ψ˜⁻¹({˜0} ×X(m)) is the stratum ofU˜which is equal toU˜∩ S for the distinguished stratum S associated toU,

(iv) every stratum ofX˜ is the distinguished stratum of a suitableU.

Proof The very same arguments as in [32, Proposition 5.2] apply to our situation.

From now on letXbe a strongly nondegenerate strictly polystable formal scheme overK^◦ and denote by the value group ofK. For the basic notions of convex geometry we refer to [33, Appendix A]. We will work with -rationalpolytopal subdivisionsDofS(X), i.e.

Dis a family of -rational polytopes contained in a canonical polysimplex such that for every stratumSofX˜ the set

∈D⊆S

is a polytopal decomposition ofS. Here apolytopal decompositionmeans a finite family of polytopes coveringS which is closed under taking faces and such that the intersection of two polytopes in the family is a face of both and a -rational polytopemeans a polytope which is defined by inequalities of the form mx+c≥0 withm∈Z^r,c∈ .

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Construction 2.6 LetDbe such a subdivision. We will construct a canonical formal scheme XoverK^◦associated toDtogether with a morphismι:X→Xwhich induces the identity on the generic fibre such that there is a one to one correspondence between the open faces ofDand the strata ofX˜. First of all we choose a covering ofXas in Proposition2.5. LetU be a member of this covering with an étale morphismψ :U→X(n,a,m)and letSbe the distinguished stratum ofU. For∈D∩Swe set

A:=

μ

a_μT^μ∈K((T₀₀, ...,T_p,n_p))∀u∈: limv(a_μ)+u·μ= ∞

andA:=A/(T00...T_0,n₀−a₀, ...,T_p0...T_p,n_p−a_p)and define

A:=

μ

a_μT^μ∈A∀_u∈,μ∈Zn+1 : v(a_μ)+μ·u≥0

andU:=SpfA. If1, 2∈D∩Sthen1∩2is a face of both and by transferring the arguments in [33, Proposition 6.12] to the analytic situation, we obtain that the canonical morphismsU1∩2 → Ui are open immersions. Hence we can glue theUalong this data to obtain a formal scheme which we denote by X(n,a) together with a morphism ι : X(n,a) → X(n,a). Let ψ : U → X(n,a)×X(m) be the base change ofψ with respect toι×Id. The construction ofUdoes not depend on the choice ofψ up to isomorphism: Letρ : U→X(n,a,m)be another étale morphism. Then up to reordering the coordinates,ρ^∗x_i =u_iψ^∗x_ifor someu_i ∈O(U)^×. Then we have canonicalK^◦-algebra isomorphisms:

O(U)ˆ⊗ψ^∗A→O(U)ˆ⊗ρ^∗A, a⊗xi →uia⊗xi,

which yield an isomorphism of theUconstructed withψrespectivelyρ.

We glue theUto obtain our formal schemeX. AlthoughXmight not be admissible, we can define its generic fibre and reduction map in the usual way as the algebrasA⊗K^◦ K are strictlyK-affinoid (see [33, Proposition 6.17]). Thenιinduces the identity on the generic fibres and we set p_X:= p_X. Note thatXis admissible if the vertices of the polytopes in Dare -rational, in particular the base change ofXto the valuation ring of the completion of an algebraic closure ofK is admissible, see [33, Proposition 6.7].

Remark 2.7 IfDis trivial i.e.∈Donly if=S for some stratumSofX˜ then it is an immediate consequence from the construction thatX=X.

We will frequently use the following generalization of [32, Proposition 5.7] which is a stratum face correspondence for theXconstructed above.

Proposition 2.8 LetXbe a strongly nondegenerate strictly polystable formal scheme with skeletonandDa subdivision ofwith associated formal structureX. Then there is a bijective correspondence between the open faces ofDand the strata ofX˜given by

R=red(p⁻¹_X(τ)), τ= p_X(red⁻¹(R)).

Furthermore, in the second equality, R can be replaced by any nonempty subset of R.

Proof We follow the proof of [32, Proposition 5.7] but in order to establish the result for an arbitrary non-archimedean field K (not necessarily algebraically closed), we use [33,

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Proposition 6.22] instead of [31, Proposition 4.4]. Letτ be an open face ofD. We prove first thatR:=red(p_X⁻¹(τ))is a stratum ofX˜. There is a unique stratumSofX˜ such that τ is contained in the interior ofS. LetUbe a formal open subset ofXsuch thatSis the distinguished stratum ofU(Proposition2.5). As strata are compatible with localization we may assumeX = U. Letψ₁ : X → X(n,a)be the base change of the composition of the étale mapψ :X→X(n,a,m)with the projection on the first factorX(n,a). By [33, Proposition 6.22] the first part of the proposition holds forX(n,a). LetTbe the stratum of X(n,a)corresponding toτ, i.e.

τ =p_X(n,a)(red⁻¹(T)) (2.1) and

T =red(p⁻¹_X(n,a)(τ)) (2.2) wherep_X(n,a) :X(n,a)^an →S is the retraction map. We proveR= ˜ψ₁⁻¹(T). First we observe that

red((ψ^an₁ )⁻¹(p⁻_X(¹_n_,_a₎(τ)))= ˜ψ₁⁻¹(red(p⁻_X(¹_n_,_a₎(τ))).

The inclusion⊆is clear because red◦ψ^an1 = ˜ψ₁ ◦red. The other inclusion follows from this fact and an application of [33, Proposition 6.22]. For details we refer to the proof of [32, Proposition 5.7]. We conclude

R=red(p⁻¹_X(τ))=red((ψ^an1)⁻¹(p_X(n,a)⁻¹ (τ)))= ˜ψ₁⁻¹(red(p⁻¹_X(n,a)(τ)))⁽2.2)

= ˜ψ₁⁻¹(T).

By [3, Lemma 2.2]Ris a strata subset. To see thatRis indeed a stratum it is enough to show thatRis irreducible. But this follows from

ψ˜₁⁻¹(T)=(T× ˜X(m))×_X(_˜ _n_,_a_,_m₎X˜

∼=(T× ˜X(m))×_{˜_{0}× ˜}_X(m)ψ˜⁻¹({˜0} × ˜X(m))∼=T×S,

where the latter is irreducible by [27, Corollaire 4.5.8 (i)]. As the open faces ofDcover, every stratum ofX˜is obtained this way. It remains to prove that we can recoverτfromR.

First note that

p_X((ψ^an₁ )⁻¹(red⁻¹(T)))= p_X(_n_,_a₎(red⁻¹(T)).

The inclusion⊆is clear because p_X = p_X(n,a)◦ψ^an1 . For the other inclusion, letx ∈ p_X(n,a)(red⁻¹(T))=τ. As the sets red⁻¹(T)withTvarying over the strata ofX(n,˜ a) coverX(n,a)^an and using [33, Proposition 6.22] and the fact the p_X(n,a) restricts to the identity onwe deducex ∈red⁻¹(T). Hencex is an element of the left hand side which proves the equality claimed in the display. Now the rest is an easy calculation:

p_X(red⁻¹(R))= p_X(red⁻¹(ψ˜₁⁻¹(T)))

= p_X((ψ^an₁ )⁻¹(red⁻¹(T)))

= p_X(n,a)(red⁻¹(T))

(2=.1)τ.

Finally we want to show thatR may be replaced by a nonempty subsetY of R. Clearly, the arguments in [32, Proposition 5.7] generalize to the polystable situation, so we presume the claim forK algebraically closed and show how to drop this assumption. LetCK be the

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completion of an algebraic closure ofK. We denote byπ :X_C_K →X the base change ofX toC^◦_K. Let R be the union of the strata ofX˜_C_K lying over R. Then π induces a surjectionp_X

CK(red⁻¹(R)) p_X(red⁻¹(R))as the strata inRcorrespond to open faces lying overτ. LetYbe a lift ofYinR. By [32, Proposition 5.7] we havep_X

CK(red⁻¹(Y))= p_X

CK(red⁻¹(R)). Clearlyp_X(red⁻¹(Y))⊆ p_X(red⁻¹(R))and hence it is enough to show that the restriction ofπto p_X

CK(red⁻¹(Y))factors through p_X(red⁻¹(Y)). We have the following commutative diagram:

S(X_C

K)

π

p_X

CK(red⁻¹(Y)) red⁻¹(Y)

p_X CK

π

red Y

π

S(X) ^p^X red⁻¹(Y) ^red Y

Let x ∈ p_X

CK(red⁻¹(Y)) and y ∈ red⁻¹(Y) with p_X

CK(y) = x then π(x) = π(p_X_CK(y))= p_X(π(y))∈ p_X(red⁻¹(Y)). This proves the claim.

Corollary 2.9 Let R be a stratum ofX˜corresponding to the open faceτofD.

(a) dim(τ)=codim(R,X˜).

(b) S:= ˜ι(R)is a stratum ofX˜.

(c) R→^˜^ι S is a fibre bundle with fibre T where T is thedim(R)−dim(S)dimensional torus orbit from the proof of Proposition2.8.

(d) Every stratum ofX˜is smooth.

(e) The closure R is the union of all strata of¯ X˜corresponding to open facesσofDwith τ ⊆ ¯σ.

(f) For an irreducible component Y ofX˜, letζY be the unique point ofX^anwith reduction equal to the generic point of Y . Then Y → ζY is a bijection between the irreducible components ofX˜and the vertices ofD.

Proof The statements can be proven the same way as in [32, Corollary 5.9]. In order to bypass the algebraically closedness of the base field one can use [33, Proposition 6.22] instead of [31, Proposition 4.4] for (a), [33, Proposition 6.22] instead of [31, Remark 4.8] for (e) and [33, Proposition 6.14] instead of [31, Proposition 4.7] for (f).

Definition 2.10 Letbe a skeleton associated to a strongly nondegenerate strictly polystable formal schemeXoverK^◦. A continuous functionh : →Ris calledpiecewise affine linearif there exists a -rational polytopal subdivisionDofsuch that for any canonical polysimplexSof, any formal open subsetψ:U→X(n,a,m)ofXwhose distinguished stratum isSand any∈Dwith⊆S, there existm∈Zⁿ⁺¹andα ∈K^×such that h

=(m·x+v(α))◦ψ^an

(see Definition2.3for the notation and setting).

Proposition 2.11 LetXbe a strongly nondegenerate strictly polystable formal scheme over K^◦ with associated skeleton S(X)and h a piecewise affine linear function on S(X). Let Dbe a -rational polytopal subdivision of S(X)suitable for h as in Definition2.10and ι:X→Xbe the canonical formal scheme overXassociated toD(see Construction2.6).

Then h induces a canonical Cartier divisor D onXwhich is trivial on the generic fibre. If

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Xis admissible, then D has the property that1O(D)=e⁻^h^◦^p^X where · O(D)is the formal metric onO_X^angiven by the formal modelO(D)ofO_X^an(see Definition3.1).

Proof As in Construction2.6, we coverXby étale mapsψ:U→X(n,a,m)=X(n,a)× X(m)and for eachUand∈Dwith⊆U^anwe obtain the affine formal schemeU. We writeψ :U →Ufor the base change with respect toψand obtain a cover ofX. On ∈D,his given bymx+v(α)withm∈Zⁿ⁺¹,α∈K^×. We defineDlocally onUby ψ^∗(α·x^m). ThenDis indeed a Cartier Divisor onXas forU1,U2, 1, 2as above and U:=U1∩U2we haveα1·x^m¹/α2·x^m² ∈O(U1∩2)^×sincem1x+v(α1)=m2x+v(α2) on1∩2. Hence

ψ₁^∗(α1·x^m¹)/ψ₂^∗(α2·x^m²)

U 1∩2

=ψ^∗(α1·x^m¹/α2·x^m²)∈O(U₁_∩₂)^× and thereforeψ₁^∗(α1x^m¹)/ψ₂^∗(α2x^m²)∈O(U₁_,₁ ∩U₂_,₂)^×. FurthermoreDis trivial on

the generic fibre, asα·x^m ∈O(X(n,a)^an)^×.

Remark 2.12 Note that we can ensure thatXis admissible and hence a formal model by performing base change to the completion of an algebraic closure ofK(see Construction2.6) which will be enough for our purposes.

Lemma 2.13 In the situation of Proposition2.11letτbe an open face of the skeletonof dimension equal to the dimension ofX^an and assume that h is affine linear onτ¯. Let D be the induced Cartier divisor onXand Y a proper curve inX˜with Y ⊆red_X(p_X⁻¹(τ))e.g.

if Y lies inside an irreducible component ofX˜corresponding to a vertex u∈τofD. Then deg(D.Y)=0.

Proof Note that we do not assumeτ¯ ∈ D. But by passing to the formal open subscheme ofXconsisting of the formal open subsetsUwithS(U)= ¯τ, we may assume= ¯τ and then the polytopal subdivisionDconsisting the polytopeτ¯ and its faces is suitable forh.

The corresponding formal scheme isX. Let Dbe the Cartier divisor onXinduced byh as in Proposition2.11. Notice that by construction we have D=ι^∗D. Nowιis proper by [41, Corollary 4.4] (the result requiresXto be admissible but by [27, Proposition 2.7.1] it is enough to check properness after base change to the completion of an algebraic closure of K, after whichXis always admissible, see Construction2.6). Hence the projection formula yields deg(D.Y)=deg(D.ι∗Y). Now

ι(Y)⊆ι(red_X(p⁻¹_X(τ)))=red_X(p⁻¹_X(τ)),

where the latter is the stratum inX˜corresponding toτand hence a point. ThereforeD.ι∗Y=

0.

3 Metrics

In this section we introduce metrics on line bundles on strictly K-analytic spaces. This includes piecewise linear, algebraic and formal metrics. We will see that under certain con- ditions they are all the same. The main reference is [25].

Definition 3.1 LetX be a strictlyK-analytic space andLa line bundle on X, i.e. a locally free sheaf of rank 1 on the G-topology. Acontinuous metric · onLis a function which asserts to any admissible open subsetU ⊆ Xand any sections ∈ (U,L)a continuous (with respect to the Berkovich topology) functions(·) :U →R_≥0such that:

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(i) For an admissible open subsetV⊆U we haves

V(·)= s(·)

V, (ii) for f ∈ (U,OX)we havef s(·) = |f(·)|s(·),

(iii) forp∈Uwe haves(p) =0 if and only ifs(p)=0.

Given a formal model(X,L)of(X,L)one can define an associated so calledformal metric · _LonLin the following way: Ifsis a local frame ofLon a formal open subsetU⊆X we definef s(·)_L= |f(·)|onUânfor any f ∈ (Uân,Oân_X). As this is independent of the choice ofsandXânis covered by such sets, this gives a well-defined metric onL.

Remark 3.2 We will work with paracompact (i.e. Hausdorff and every open cover has a locally finite refinement) strictlyK-analytic spaces. As discussed in [25, 2.2] the category of these spaces is equivalent to the category of quasiseparated rigid analytic varieties overK with a strictlyK-affinoid G-covering of finite type ([2, 1.6]). This allows us to apply Raynaud’s theorem ([12, Theorem 8.4.3]) which shows that formalK^◦-models of paracompact strictly K-analytic spaces exist and that the set of isomorphism classes of formal K^◦-models is directed.

Proposition 3.3 Let X be a paracompact strictly K -analytic space, L a line bundle on X and W a compact strictly K -analytic domain of X . Then every formal metric on L

W extends to a formal metric on L.

Proof [25, Proposition 2.7].

Definition 3.4 Let X be a proper scheme overK andL a line bundle onX. An algebraic K^◦-model ofXis a proper flat schemeX overK^◦with a fixed isomorphism from the generic fibreX_ηtoX. An algebraicK^◦-model of(X,L)is a pair(X,L)whereX is an algebraic K^◦-model ofX andL is a line bundle onX with a fixed isomorphism fromL

X toL.

An algebraicK^◦-model of(X,L)gives rise to a formalK^◦-model of(Xân,Lân)by formal completion. Hence by the above, an algebraic model of(X,L)induces a formal metric on Lân. We call such metricsalgebraic metrics.

Proposition 3.5 Let X be a proper scheme over K and L a line bundle on X . Then a formal metric on L^anis the same as an algebraic metric.

Proof [23, Proposition 8.13], see also [25, Remark 2.6].

Definition 3.6 LetXbe a strictlyK-analytic space andLa line bundle onX. A metric · onLis calledpiecewise linearif there is a G-covering(V_i)i∈I and framess_i ofLoverV_i for everyi ∈I such thatsi(·) =1 onVi.

Proposition 3.7 Let X be a strictly K -analytic space and L a line bundle on X . Then (i) the isometry classes of piecewise linear metrics on line bundles on X form an abelian

group with respect to⊗.

(ii) the pull-back f^∗ · of a piecewise linear metric · on L with respect to a morphism f :Y→X of strictly K -analytic spaces is a piecewise linear metric on f^∗L.

(iii) the minimum and the maximum of two piecewise linear metrics on L are again piecewise linear metrics on L.

Proof [25, Proposition 2.12] (The proof does not use paracompactness).

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Proposition 3.8 Let X be a paracompact strictly K -analytic space and L a line bundle on X . Then a piecewise linear metric on L is the same as a formal metric.

Definition 3.9 LetX be a strictlyK-analytic space andLa line bundle onX. A piecewise linear metric on L is calledsemipositivein x ∈ X if there exists a compact strictly K- analytic domainWwhich is a neighbourhood ofxsuch that there is a formal model(W,L) of

W,L

W

inducing the metric onWand satisfying deg_L(C)≥0 for every proper closed curveC in the special fibre ofW. The metric onLis called semipositive in a subsetV⊆X if it is semipositive in everyx∈V. It is called semipositive if it is semipositive inX.

Proposition 3.10 Let X be a paracompact strictly K -analytic space and L a line bundle on X . A formal metric · on L is semipositive in every x∈X if and only if there exists a nef formal K^◦-modelLof L inducing · . In particular we regain the original global definition of semipositivity by Zhang[47].

Proof This is proved in [25, Proposition 3.11] under the additional assumption that X is separable, which was necessary in order to be able to use [15, Lemme 6.5.1]. Replacing this with CorollaryA.4, the same proof applies to the more general case.

Proposition 3.11 Let X be a proper scheme over K and L a line bundle on X . Let·1,·2

be two piecewise linear metrics on L^anwhich are semipositive in x ∈ X^an. Then · :=

min( · 1, · 2)is semipositive in x.

Definition 3.12 LetXbe a strictlyK-analytic space andLa line bundle onX. A metric · onLis calledpiecewiseQ-linearif for everyx ∈ Xthere is an open neighbourhoodW of xand a non-zeron∈Nsuch that · ^⊗n

Wis a piecewise linear metric onL^⊗n

W. A piecewiseQ-linear metric onLis calledsemipositiveinx ∈Xif in the above · ^⊗n is semipositive inx. W

Proposition 3.13 Let X be a paracompact strictly K -analytic space and L a line bundle on X . Any continuous metric on L can be uniformly approximated by piecewiseQ-linear metrics on L.

Proof [25, Theorem 2.17].

4 Measures

We recall the real Monge–Ampère operator which associates to a convex function a positive Borel measure. Then we introduce the Chambert-Loir measure on the generic fibres of admissible formal schemes and on paracompact strictlyK-analytic spaces. Chambert-Loir introduced these measures in [16] on the analytificationX^anof a proper variety XoverK under the assumption thatK has a countable dense subfield and associates to a family of semipositive metrized line bundles a positive Radon measure. This was later extended by Gubler to the case of an algebraically closed base field in [31]. Using the local approach to metrics from Sect.3, it is now possible to define Monge–Ampère measures locally. Note

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that there is also a local approach by Chambert-Loir and Ducros in [15] which associates a measure to a metric which is locally psh-approximable. However it is not known whether a semipositive metric is locally psh-approximable. In this section we assume that the non- archimedean complete base fieldK is algebraically closed which is no restriction as one can always reduce to this case by base change (see Remark4.16).

Definition 4.1 Let ⊆ Rⁿ be bounded, open and convex and denote byλthe standard Lebesgue measure onRⁿand by ·,·the standard scalar product onRⁿ. Lethbe a convex function onandx₀∈. We define thegradient imageofx₀underhto be

∇h(x₀):=

p∈Rⁿ∀x ∈ : h(x₀)+ x−x₀,p ≤h(x) and forE⊆

∇h(E):=

x₀∈E

∇h(x₀).

Note that ifEis a Borel set, the same is true for∇h(E). Finally we define the Monge–Ampère measure associated tohby

MA(h)(E):=λ(∇h(E))

for all Borel sets E ⊆ . It is indeed a measure on the Borelσ-algebra, for details see [39, Sect. 2]. The real Monge–Ampère operator is continuous in the sense that if(u_n)n∈N

is a sequence of convex functions onconverging pointwise to a convex functionuthen (MA(u_n))n∈Nconverges weakly to MA(u). Ifhis two times continuously differentiable then MA(h)=detD²h·λ.

For convex functionsh1, ...,hnonwe define MA(h₁, ...,h_n):= 1

n! n k=1

(−1)^n−k·

1≤i₁<···<i_k≤n

MA(h_i₁+...+h_i_k)

and call it the mixed Monge–Ampère measure ofh1, ...,hn. It is multilinear, symmetric and satisfies MA(h, ...,h)=MA(h)(for details see [38, Sect. 5]).

Definition 4.2 In [17, Definition 2.2.2] Conrad defined the notion of irreducibility for analytic spaces which we recall here. LetXbe a paracompact strictlyK-analytic space andp: ˜X→X the normalization ofX([17, 2.1]). Then the irreducible components ofXare defined to be the setsXi:= p(X˜i)whereX˜iare the connected components ofX. The space˜ Xis said to be irreducible if it has a unique irreducible component. By [17, Lemma 2.2.3]Xis irreducible if and only if it can not non trivially be written as a union of two closed strictlyK-analytic subsets.

LetY be an irreducible component of X andV = M(A)an affinoid domain withY ∩ V = ∅. Then by [17, Corollary 2.2.9] there is an irreducible componentYofV which is contained inV∩Y. ThenYcorresponds to a minimal prime idealpofA and hence to an irreducible component of Spec(A). We define themultiplicity of Y to be the multiplicity of this component. Note that this does not depend on the choice ofVandY: IfV=M(B)⊆V andpis a minimal prime ideal ofBlying overpthenB/pBis reduced by [8, Corollary 7.3.2/10] as it induces an affinoid domain inM(A/p)which is reduced. Hence alsoB_p/pB_p is reduced and sinceB_p is a local ring of dimension 0, this impliespB_p =pB_p. Hence by [21, Lemma A.4.1] the multiplicity of the irreducible component corresponding topis equal to that of the irreducible component corresponding top.

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Letϕ:X→Ybe a proper surjective morphism of irreducible and reduced strictlyK-analytic spaces. If dim(Y) <dim(X)we set deg(ϕ)=0. Otherwiseϕis a finite morphism outside a lower dimensional analytic subsetW ofY. Let_M(A)be an affinoid domain inY\W, V an irreducible component of Spec(A)andM(A) :=ϕ⁻¹(M(A))then Spec(A) → Spec(A) is finite and we define deg(ϕ)to be the sum of the degrees of the irreducible components of Spec(A)overV. As explained in [29, 2.6] this again does not depend on the choices.

4.3 Monge–Ampère measure for line bundles on admissible formal schemes

LetXbe an admissible formal scheme overK^◦of dimensionn+1 with generic fibreX. Our goal is to introduce a Monge–Ampère measure onXfor formal line bundlesL1, ...,LnonX. We assume first thatXis irreducible and reduced and that the special fibre ofXis reduced.

Then the non-archimedean Monge–Ampère measure onX with respect to these metrized line bundles is defined as

c1(L1)∧ · · · ∧c1(Ln):=

Y∈irr(X)˜ Yproper

deg_L₁_,...,L_n(Y)·δ_ζ_Y,

whereδ_ζ_Y denotes the Dirac-measure at the unique pointζY which is mapped to the generic point of the proper irreducible componentY under the reduction map (cf. [1, Proposition 2.4.4]).

IfXhas irreducible and reduced generic fibre but no longer reduced special fibre, there is a canonical admissible formal modelXofXwith reduced special fibre together with a finite morphismι:X→Xwhich restricts to the identity onXwhich can be constructed as follows (cf. [29, Definition 3.10]). Choose a cover(Ui =Spf(Ai))i∈IofXby affine formal subschemes. DefineAi := A⊗K^◦ K. If Spf(B) ⊆ Spf(A_i)is a formal open subscheme for somei ∈ I then A_i →Binduces a morphismA_i^◦ →B^◦ forB :=B⊗K^◦ K. Hence by standard arguments we can glue the Spf(A_i^◦)to obtainXand the canonical morphisms Ai →A_i^◦induce the morphismX→X. We then define

c1(L1)∧ · · · ∧c1(Ln):=(ι^an)∗(c1(ι^∗L1)∧ · · · ∧c1(ι^∗Ln)).

In the general case, letX=

jmjXjbe the decomposition of the generic fibre into prime cycles. By [29, Proposition 3.3] the closureX_j ofX_j inXis an admissible formal scheme with irreducible and reduced generic fibreX_j. We define

c₁(L1)∧ · · · ∧c₁(Ln):=

j

m_j·c₁

L1

Xj

∧ · · · ∧c₁

Ln

Xj

as a measure onX.

Remark 4.4 There is a close connection of the Monge–Ampère measure with the intersection product on formal schemes as defined in [29]: Assume thatXhas irreducible, reduced and boundaryless generic fibre and reduced special fibre. In addition toL1, ...,Ln letL0 be a formal line bundle onXwhich is trivial on the generic fibre and set f := −log1where· is the formal metric induced byL0. Suppose that f has compact support and letD:=div(1) be the Cartier divisor onXinduced by 1 as in [29, Remark 3.1]. We examine the Weil divisor cyc(D)associated toDas defined in [29, Sect. 3]. SinceL0is trivial on the generic fibre, the horizontal part of cyc(D)is zero while the vertical part is by definition ([29, 3.8]) given

by

Y∈irr(X)˜ f(ζY)·Y. Now sinceX^anhas no boundary, every irreducible component ofX˜