Pluripotential theory for tropical toric varieties and non-archimedean Monge-Ampére equations
José Ignacio Burgos Gil, Walter Gubler, Philipp Jell, Klaus Künnemann
aa r X i v : . [ m a t h . AG ] F e b PLURIPOTENTIAL THEORY FOR TROPICAL TORIC VARIETIESAND NON-ARCHIMEDEAN MONGE–AMP`ERE EQUATIONS
JOS´E IGNACIO BURGOS GIL, WALTER GUBLER, PHILIPP JELL, AND KLAUS K ¨UNNEMANN
Abstract.
Tropical toric varieties are partial compactifications of finite dimensional realvector spaces associated with rational polyhedral fans. We introduce plurisubharmonicfunctions and a Bedford–Taylor product for Lagerberg currents on open subsets of a trop-ical toric variety. The resulting tropical toric pluripotential theory provides the link to givea canonical correspondence between complex and non-archimedean pluripotential theoriesof invariant plurisubharmonic functions on toric varieties. We will apply this correspon-dence to solve invariant non-archimedean Monge–Amp`ere equations on toric and abelianvarieties over arbitrary non-archimedean fields.
Contents
1. Introduction 12. Forms and currents 73. Plurisubharmonic functions on partial compactifications 134. Bedford Taylor calculus on a partial compactification 205. Semipositive metrics and θ -psh functions 236. Toric Monge–Amp`ere equations 327. Monge–Amp`ere equations on totally degenerate abelian varieties 438. Monge–Amp`ere equations on arbitrary non-archimedean abelian varieties 52Appendix A. Proof of Theorem 6.2.13 61References 631. Introduction
Pluripotential theory is a non-linear complex counterpart of classical potential theory.It is the study of plurisubharmonic (short: psh) functions and the complex Monge–Amp`ereequation. There are rich applications to multidimensional complex analysis, K¨ahler geom-etry, algebraic geometry and to dynamics. A celebrated result was Yau’s solution [Yau78]of the complex Monge–Amp`ere equation for smooth forms on a K¨ahler manifold more thantwenty years after Calabi proved uniqueness. Bedford and Taylor [BT82] showed that onecan go beyond the smooth case as there is a well-defined product for the positive (1 , ϕ are allowed Mathematics Subject Classification.
Primary 32P05; Secondary 32U05, 14T90, 32W20.
Key words and phrases.
Pluripotential theory, tropical toric varieties, tropicalization of abelian varieties,non-archimedean Monge–Amp`ere equations.J. I. Burgos was partially supported by MINISTERIO DE CIENCIA E INNOVACION research projectsPID2019-108936GB-C21 and ICMAT Severo Ochoa project CEX2019-000904-S. W. Gubler, P. Jell andK. K¨unnemann were supported by the collaborative research center SFB 1085
Higher Invariants - Interac-tions between Arithmetic Geometry and Global Analysis funded by the Deutsche Forschungsgemeinschaft. to take the value −∞ and the product is only well-defined under certain restrictions on theunbounded loci ϕ − ( {−∞} ).In non-archimedean geometry, continuous semipositive metrics were introduced by Zhang[Zha95]. Chambert-Loir [CL06] defined corresponding Monge–Amp`ere measures. Kontse-vich and Soibelman [KS06] emphasized the role of non-archimedean geometry for mirrorsymmetry, while Kontsevich and Tschinkel [KT02] proposed a strategy to solve the non-archimedean Monge–Amp`ere equation. Yuan and Zhang [YZ17] showed uniqueness of thesolution up to scaling. Liu [Liu11] solved the non-archimedean Monge–Amp`ere equation fortotally degenerate abelian varieties. In case of a trivially or discretely valued complete field K of residue characteristic zero, Boucksom, Favre and Jonsson [BFJ15] gave a global ap-proach to non-archimedean pluripotential theory and presented a variational method whichgives the existence of a solution for any smooth projective variety over K .The goal of this paper is to introduce a pluripotential theory for certain partial com-pactifications of R n called tropical toric varieties. Noting that convex functions are thenatural real analogue of psh functions, it is clear that the Monge–Amp`ere measure on sucha compactification is closely related to the classical real Monge–Amp`ere operator studiedby Aleksandrov. The classical real Monge–Amp`ere equation has several variants. The cor-responding Dirichlet problem was solved by Aleksandrov [Ale58] and Bakelman [Bak57],the corresponding second boundary problem (i.e. with prescribed gradient image) was doneby Pogorelov [Pog64].Toric geometry is very useful for testing conjectures in algebraic and arithmetic geom-etry. The reason is that invariant objects in toric geometry are completely described bycombinatorial means which comes handy for such tests. In particular, a toric variety withdense split torus T is completely determined by a fan Σ in N R = N ⊗ Z R where N is thecocharacter lattice of T and so we denote the associated toric variety by X Σ . Note that X Σ is naturally defined over Spec Z and by base change, we get a toric variety over any basefield. In the introduction, we consider the associated complex toric variety X Σ , ∞ and theassociated toric variety X Σ ,v for a non-archimedean field K with valuation v of rank one.There is a tropical toric variety N Σ associated to the fan Σ which is a partial compacti-fication of N R . For w ∈ {∞ , v } , it comes with a canonical tropicalization maptrop w : X anΣ ,w −→ N Σ which is a continuous proper surjective map. We exploit trop ∞ and trop v to relate thecomplex space X anΣ , ∞ and the non-archimedean Berkovich space X anΣ ,v .Lagerberg [Lag12] has introduced real valued ( p, q )-forms on N R which form a bigradedsheaf of algebras A · , · N Σ with differential operators d ′ and d ′′ . Lagerbergs construction wasgeneralized by Jell, Shaw and Smacka [JSS19] to polyhedral spaces as N Σ . For an opensubset U of N Σ , the space D p,q ( U ) of Lagerberg currents of bidegree ( p, q ) is the topologicaldual of A p,qc ( U ). In [BGJK20], we have worked out a suitable positivity notion for Lagerbergcurrents of bidegree ( p, p ). For a summary and a comparison to the corresponding notionsin complex and non-archimedean geometry, we refer to Section 2.1.1. Tropical psh functions.
In Section 3, we introduce psh functions on an open subset U of N Σ . These are strongly upper-semicontinuous functions ϕ : U → R ∪ {−∞} whichrestrict to convex functions on U ∩ N R . We will show in Theorem 3.1.4 that every psh func-tion on U is continuous. This is not true for classical psh functions on complex manifolds.We show that psh functions on U share many properties with complex psh functions. InTheorem 3.2.2, we prove that a function ϕ on U is psh if and only if ϕ is a locally inte-grable strongly upper-semicontinuous function such that the associated Lagerberg currentis positive. In Theorem 3.3.4, we will see that any psh function on U is locally a decreasinglimit of smooth psh functions. In Section 4, we introduce a Bedford–Taylor product for pshfunctions on the tropical toric variety N Σ associated to the fan Σ in N R ≃ R n . ROPICAL TORIC PLURIPOTENTIAL THEORY 3
Theorem A.
For an open subset U of N Σ and locally bounded psh functions u , . . . , u q on U , there is a unique Lagerberg current u d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∈ D q,q ( U ) such that this product is determined locally in U , agrees with the product of Lagerberg formsin the smooth case, and is continuous with respect to uniform convergence of psh functions. For u ≡
1, we show that the Bedford–Taylor product d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q is a closedpositive Lagerberg current on U . For q = n , we get a positive Radon measure on U whichwe call the associated Monge–Amp`ere measure . If Σ is a smooth fan, then we allow inthe Bedford–Taylor product in Theorem A additionally a factor given by a closed positive( p, p )-current T (see Theorem 4.1.2). Moreover, we give in Theorem 4.2.2 a Bedford–Taylorproduct for unbounded psh functions if the unbounded loci intersect transversally.The proofs of the above results on the tropical Bedford–Taylor product use that thefunctions v j := u j ◦ trop ∞ are psh functions on the open subset V := trop − ∞ ( U ) of thecomplex toric manifold X anΣ ,v and that T = trop ∞ , ∗ ( S ) for a canonical positive ( p, p )-current S on V by the main theorem in [BGJK20]. Then we define u d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T := trop ∞ , ∗ ( v d ′ d ′′ v ∧ . . . ∧ d ′ d ′′ v q ∧ S )by using the corresponding complex Bedford–Taylor product on V and the claim followsfrom the complex case and local regularization. For a non-smooth fan, we reduce TheoremA to the smooth case by toric resolution of singularities and use a projection formula (seeRemark 4.3.3). In principle, it is possible to prove Theorem A purely by means of realanalysis, but a reduction to the complex case by using tropicalization is very handy andgives the results much quicker. Such comparison principles in the toric case are a permanentguideline in the whole paper.1.2. Tropical θ -psh functions. For a complete fan Σ, we show in Proposition 3.1.9 thatpsh functions on N Σ are constant. To get also an interesting global theory, we proceedsimilarly as in the complex case and introduce θ -psh functions. Here, we fix a closed(1 , θ given by a similar construction as the first Chern current associated with acontinuous metric of a line bundle in the complex or non-archimedean setting. In Section5, we show that our construction makes sense also for non-complete fans and we introduce θ -psh functions on any open subset U of N Σ . Since θ -psh functions can be locally identifiedwith psh functions, most of the properties of psh functions generalize to θ -psh functions.In particular, we have a Bedford–Taylor product for θ -psh functions as in Theorem A.In Theorems 5.2.12 and 5.3.11, we show the following correspondence theorem . We con-sider any non-archimedean field K with valuation v . For w ∈ {∞ , v } , let S w be the maximalcompact torus in the dense torus orbit of X anΣ ,w . We pick θ as above and set θ w := trop ∗ w ( θ ). Theorem B.
Let Σ be a smooth fan and let U be an open subset of N Σ . Then there arecanonical isomorphisms between the following cones:(i) the cone of continuous S ∞ -invariant θ ∞ -psh functions ϕ ∞ on trop − ∞ ( U ) ;(ii) the cone of continuous θ -psh functions on U ;(iii) the cone of continuous S v -invariant θ v -psh functions ϕ ∞ on trop − v ( U ) .For w ∈ {∞ , v } , the correspondence is given by ϕ ϕ w := ϕ ◦ trop w . The above bijective correspondence is true for any fan Σ. For simplicity, we consider inthis paper only psh functions on complex manifolds. There is also a theory of psh functionson singular complex analytic spaces as described in [Dem85]. If π : X Σ ′ → X Σ is a toricdesingularization, then pull-back gives an isomorphism from the space of psh-functions on U to the space of psh-functions on π − ( U ) [Dem85, Theorem 1.7] which can be used toprove Theorem B and many other statements in this paper also in the singular case. J.I.BURGOS GIL, W. GUBLER, P. JELL, AND K. K ¨UNNEMANN
We generalize the correspondences in Theorem B also to singular θ -psh functions. In § θ -psh functions.1.3. Toric Monge–Amp`ere equations.
In the non-archimedean case, we show that thecorrespondence of continuous psh-functions in Theorem B is compatible with the tropicaland non-archimedean Bedford–Taylor products (see Theorem 6.1.1). Let µ be a positiveRadon measure on an open subset U of the n -dimensional tropical toric variety N Σ . Inthe complex case, we assume again that Σ is a smooth fan, then there is a unique S ∞ -invariant Radon measure on µ ∞ on the toric manifold X anΣ , ∞ with trop ∞ ( µ ∞ ) = µ . In thenon-archimedean case, there is a canonical section ι v : N Σ → X anΣ ,v of trop v which allowsto identify N Σ with the canonical skeleton of X anΣ ,v . There is a unique Radon measure µ v on X anΣ ,v supported on the canonical skeleton N Σ and with trop v ( µ v ) = µ . In Proposition6.3.3, we show for w ∈ { v, ∞} that(1.1) ( d ′ d ′′ ϕ + θ ) ∧ n = µ ⇔ ( d ′ d ′′ ϕ w + θ w ) ∧ n = µ w using ϕ w := ϕ ◦ trop w for continuous θ -psh functions ϕ on U as in Theorem B.We assume that Σ is a complete fan and that L is an ample line bundle on the toric vari-ety X Σ . For any non-archimedean field K with valuation v , Boucksom and Jonsson [BJ18,Theorem 6.9] introduce a Monge–Amp`ere measure c ( L an v , k k ) ∧ n for singular semipositivemetrics k k on L an v which are (locally) bounded or more generally of finite energy. The con-struction is based on the continuous case using continuity along decreasing nets. Similarlyas in complex geometry, we can define the non-pluripolar Monge–Amp`ere operator µ k k := lim k →∞ {k k Under the above assumptions, let µ v be a positive Radon measure on X anΣ ,v which is supported on the canonical skeleton N Σ satisfying µ v ( N Σ \ N R ) = 0 and with µ v ( X anΣ ,v ) = deg L ( X Σ ) . Then there is a metric k k v in E ( L an v ) solving the non-archimedeanMonge–Amp`ere equation c ( L an v , k k v ) ∧ n = µ v . If K is a discretely or trivially valued field of residue characteristic 0 and µ v is a measureof finite energy, then the above theorem follows from the solution of the non-archimedeanMonge–Amp`ere equation of Boucksom and Jonsson [BJ18, Theorem B]. However, there areRadon measure µ v as in Theorem C which are not of finite energy and the result holds forany non-archimedean field K . In Remark 6.3.15, we give also a natural generalization ofTheorem C for L semiample and big.In the proof of Theorem C, we show first the corresponding statement for θ -psh functionson N Σ which can be done either by referring to the second boundary problem for thereal Monge–Amp`ere operator (see Theorem 6.2.13) or by referring to the solution of thecorresponding complex Monge–Amp`ere equation due to Boucksom, Eyssidieux, Guedi andZeriahi [BEGZ10]. Then the correspondence results for θ -psh functions yield the claim. Fordetails, we refer to the proof of Theorem 6.3.13.1.4. Abelian varieties. In Section 7, we show that a similar correspondence as in TheoremB is possible between θ -psh functions on polarized complex and non-archimedean totallydegenerate abelian varieties over any non-archimedean field K with non-trivial valuation v .This is a bit surprising as abelian varieties are not defined over Spec( Z ) in contrast to toricvarieties. For simplicity, we assume here in the introduction that K is algebraically closed.An abelian variety A v over K is called totally degenerate if there is a non-archimedeanTate uniformization A an v ≃ T an v / Λ for a multiplicative torus T and a discrete subgroup ROPICAL TORIC PLURIPOTENTIAL THEORY 5 Λ of A ( K ). Let N be the cocharacter lattice of T , then it is additionally required thattrop v : T an v → N R maps Λ isomorphically onto a complete lattice in N R and we will identifyΛ with this lattice. Note that Λ can be intrinsically attached to A an v and hence we get acanonical tropicalization map trop v : A an v −→ N R / Λonto the tropical abelian variety N R / Λ.We consider now a polarization λ on the totally degenerate abelian variety A v over K . This is an isogeny φ : A v → A ∨ v to the dual abelian variety induced by an ampleline bundle L v . Let T ∨ v be the torus with character lattice Λ, then we have the Tateuniformization ( A ∨ v ) an ≃ ( T ∨ v ) an /M for M := Hom( N, Z ) and so the lift of ϕ to the Tateuniformizations gives a homomorphism λ : Λ → M . Using that M is the character latticeof T , set [ γ, m ] := m ( γ ) for γ ∈ Λ and m ∈ M . We define trop v ( A v , φ ) := (Λ , M, [ · , · ] , λ )and we will see in § (i) M and Λ are finitely generated free abelian groups of rank n ; (ii) the paring [ · , · ] extends to a non-degenerate real bilinear form Λ R × M R → R ; (iii) the homomorphism λ : Λ → M has finite cokernel of cardinality d ; (iv) the bilinear form [ · , λ ( · )] on Λ is symmetric and positive definite.A tuple (Λ , M, [ · , · ] , λ ) with these four properties is called a polarized tropical abelian varietyof degree d as studied in [FRSS18]. By (ii) , we identify Λ R = N R and the bilinear form in (iv) extends to an euclidean inner product b on N R . Similarly as in Riemannian geometry, b induces a smooth positive (1 , ω on the underlying tropical abelian variety N R / Λ.For details, we refer to § A v , φ ), we get a canonical smooth(1 , ω v := trop ∗ v ( ω ) on A an v which agrees with the first Chern form of a canonicalmetric of the ample line bundle L v .It is well-known that a complex abelian variety A ∞ has a complex Tate uniformization A an ∞ ∼ = T an ∞ / Λ for a multiplicative torus T ∞ over C . Note that T an ∞ is not simply connected.The torus T and the lattice Λ are not intrinsically associated to A ∞ . This is in contrast tothe classical uniformization A an ∞ ≃ V /U where V is the tangent space of A an ∞ at 0 and U isthe complete lattice H ( A an ∞ , Z ) in V .We note that a polarization on a complex abelian variety A ∞ is given by a Riemannform H , i.e. a positive definite sesquilinear form H on V such that the alternate form E = Im( H ) takes values in Z on the lattice U . A symplectic basis for E in U generatesisotropic sublattices U and U of rank n with U = U ⊕ U . We will see in § N ≃ U of the torus T ∞ andwith Λ ≃ U . Similarly as in the non-archimedean case, we can associate to ( A ∞ , H, U , U )a canonical polarized tropical abelian variety trop ∞ ( A ∞ , H, U , U ) = (Λ , M, [ · , · ] , λ ) and acanonical tropicalization map trop ∞ : A an ∞ −→ N R / Λ. For the canonical (1 , ω oftrop ∞ ( A ∞ , H, U , U ), we note that ω ∞ := trop ∗∞ ( ω ) is the canonical K¨ahler form of thepolarization H of A ∞ .For any polarized totally degenerate abelian variety ( A v , φ ) over K , there is a corre-sponding decomposed complex polarized abelian variety ( A ∞ , H, U , U ) withtrop ∞ ( A ∞ , H, U , U ) = (Λ , M, [ · , · ] , λ ) = trop v ( A v , φ ) . We will see this in § w ∈ {∞ , v } , the maximal compact torus S w in T an w acts on A an w through the Tate uniformization. We will prove the following analogue of Theorem B forcorresponding polarized complex, tropical and totally degenerate abelian varieties: Theorem D. For any open subset W of the tropical abelian variety N R / Λ , there are canon-ical isomorphisms between the following cones:(i) the cone of continuous S ∞ -invariant ω ∞ -psh functions ϕ ∞ on trop − ∞ ( W ) ; J.I.BURGOS GIL, W. GUBLER, P. JELL, AND K. K ¨UNNEMANN (ii) the cone of ω -psh functions on W ;(iii) the cone of continuous S v -invariant θ v -psh functions ϕ ∞ on trop − v ( W ) .For w ∈ {∞ , v } , the correspondence is given by ϕ ϕ w := ϕ ◦ trop w . This follows from Theorem B using that W can be locally lifted to an isomorphic opensubset of N R . Let µ be any positive Radon measure on the tropical abelian variety N/ Λwith polarization (Λ , M, [ · , · ] , λ ) of degree d . Assuming the necessary condition µ ( N R / Λ) = n ! · √ d , we will see in Corollary 7.5.6 that the tropical Monge–Amp`ere equation(1.2) ( ω + d ′ d ′′ ϕ ) ∧ n = µ has an ω -psh function ϕ : N R → R as a solution, unique up to adding constants. UsingTheorem B, this follows from the corresponding case in complex K¨ahler geometry.Section 8 is dedicated to solve the invariant Monge–Amp`ere equation for an arbitraryabelian variety A over any non-trivially valued non-archimedean field K with non-trivialvaluation. Using Raynaud extensions, Berkovich [Ber90, § canonicalskeleton as a compact subset of A an which is a strong deformation retraction. Theorem E. Let L be an ample line bundle on any abelian variety A over the non-triviallyvalued non-archimedean field K . Let µ be a positive Radon measure on A an supportedin the canonical skeleton of A and with µ ( A an ) = deg L ( A ) . Then there is a continuoussemipositive metric k k on L an , unique up to scaling, with c ( L, k k ) ∧ dim( A ) = µ. Uniqueness follows from a much more general result of Yuan and Zhang. Existence wasproven for totally degenerate abelian varieties by Liu [Liu11] in case of such a measure µ with smooth density proving that the solution is a smooth metric. If A has good reduction,then the canonical skeleton is a single point and the solution is the canonical metric on L an .In § K and for such fields we give the argument in § § A an along the canonical tropicalization map and a similar approximationargument as in [Gub10] shows semipositivity.1.5. Notation and conventions. The set of natural numbers N includes 0. For any t ∈ R ,let R ≥ t := { r ∈ R | r ≥ t } . We write R ∞ = R ∪ {∞} and R −∞ = R ∪ {−∞} . A lattice is afree Z -module of finite rank. A lattice in a finite dimensional real vector space is a discretesubgroup. For a ring A , the group of invertible elements is denoted by A × . A variety overa field F is an integral scheme which is of finite type and separated over Spec F .In this paper, we denote by K a non-archimedean field which is a field K complete withrespect to a given ultrametric absolute value | | : K → R ≥ . The valuation is v := − log | | and Γ := v ( K × ) is the value group . We have the valuation ring K ◦ := { α ∈ K | v ( α ) ≥ } with maximal ideal K ◦◦ := { α ∈ K | v ( α ) > } and residue field ˜ K := K ◦ /K ◦◦ . If Y is avariety over K , then Y an denotes the analytification of Y as a Berkovich analytic space.Let N be a free abelian group of rank n , M = Hom Z ( N, Z ) its dual and denote by N R resp. M R the respective scalar extensions to R . We also fix a subgroup Γ of R , usually thevalue group of a non-archimedean field. A function f : N R → R is called affine if f = u + c for some u ∈ M R and c ∈ R . We call u the slope of f and integral slope means u ∈ M . Wesay that f is integral Γ -affine if u ∈ M and c ∈ Γ.A polyhedron in N R is a finite intersection of half-planes { f ≤ } for affine functions f on N R . A polytope is a bounded polyhedron. A polyhedron is integral Γ -affine if theaffine functions f can be chosen integral Γ-affine. The relative interior of a polyhedron σ is denoted by relint( σ ). We use ∆ ◦ for the interior of ∆ in N R . A face of a polyhedron ROPICAL TORIC PLURIPOTENTIAL THEORY 7 σ is the intersection of σ with the boundary of a half-space containing σ . We also allow σ and ∅ as faces of σ . We write τ ≺ σ if τ is a face of a polyhedron σ . The open faces of σ are the relative interiors of the faces of σ . A polyhedral complex in N R is a finite set Π ofpolyhedra in N R such that for σ ∈ Π all faces of σ are in Π and for another ρ ∈ Π we havethat σ ∩ ρ is a common face of σ and ρ . The support of Π is defined by | Π | := S σ ∈ Π σ .We mean by a fan a finite polyhedral complex Σ in N R consisting of strictly convexpolyhedral rational cones. We denote by X Σ the corresponding toric variety with densetorus T and by N Σ the corresponding partial compactification of N R . We refer to [BGJK20]for details about N Σ which we often call the tropical toric variety associated to Σ.A piecewise affine function f is usually defined on a finite union P of polyhedra σ suchthat f | σ is affine. There is always a polyhedral complex Π with P = | Π | such that f | σ isaffine for every σ ∈ Π. Then we say that f is piecewise affine with respect to Π. If we maychoose Π as a fan Σ and if f | σ ∈ M R for all σ ∈ Σ, then f is called piecewise linear .We refer to [BGJK20, Appendix A] for our conventions about Radon measures. Werecall the following useful fact for a locally compact Hausdorff space Y with a continuousaction by a compact group G . Let π : Y → X := Y /G be the quotient which is also a locallycompact Hausdorff space. Then for any positive Radon measure µ X on X , there is a unique G -invariant positive Radon measure µ Y on Y with image measure π ( µ Y ) = µ X . This followsby averaging compactly supported continuous functions with respect to the probability Haarmeasure on G and then using the Riesz representation theorem, see [Bou63, chap. VII, § Acknowledgements. We are grateful to Ana Botero, Yanbo Fang, Roberto Gualdi, Mat-tias Jonsson, Yifeng Liu, and C´esar Mart´ınez for their comments on a previous version.2. Forms and currents In this section we fix notation for forms and currents in the complex, tropical and non-archimedean setting. We also introduce and discuss positivity of forms and currents. In thetoric setting, we relate complex (resp. non-archimedean) forms and currents to Lagerbergforms and currents on the associated tropical variety.2.1. The complex situation. Let X be a connected complex manifold of dimension n .We denote by A = A X the sheaf of complex smooth differential forms on X , which is abigraded differential sheaf of complex algebras with respect to the differential operators ∂ and ∂ . We will use also the differential operators d = ∂ + ∂ and d c = πi ( ∂ − ∂ ) on A . Notethat dd c = iπ ∂∂ and our definition of d c follows the convention from pluripotential theoryas in [Dem12]. Observe that our d c is two times the d c -operator used in Arakelov theory[GS90, Sou92].Dually, we have the sheaf of complex currents on X which we denote by D . The bigradingof D p,q is made in such a way that a locally integrable form ω of bidegree ( p, q ) on an opensubset V of X gives rise to an associated current [ ω ] ∈ D p,q ( V ) by[ ω ] : A n − p,n − qc ( V ) −→ R , α [ ω ]( α ) := Z V ω ∧ α where A c ( V ) denotes the space of compactly supported smooth differential forms as usual.A differential form ω on V of bidegree ( p, p ) is called positive , if for some m ∈ N , thereexist non-negative functions f j : V → R and ( p, α j for j = 1 , . . . , m , such that ω = m X j =1 i p f j α j ∧ α j . Here, we differ from the terminology used in [Dem12, III.1.A], where the weakly positiveforms from [BGJK20] are called positive. We observe that weakly positive forms are alwayspositive if p ∈ { , , n − , n } and refer to [BGJK20, Proposition 2.3.5] for an explanation, J.I.BURGOS GIL, W. GUBLER, P. JELL, AND K. K ¨UNNEMANN why our positivity notation is more suitable for comparing with Lagerberg forms. A current T ∈ D p,p ( V ) is called positive , if T ( ω ) ≥ ω ∈ A n − p,n − pc ( V ).A function ϕ : V → R −∞ is called strongly upper semicontinuous if for every null set E ⊂ V and every a ∈ V , we have(2.1) ϕ ( a ) = lim sup x → a, x/ ∈ E ϕ ( x ) . We refer to [Dem12, I.5] for the definition and the study of plurisubharmonic functions (psh functions for short). For V connected, a function ϕ : V → R −∞ , not identically −∞ ,is psh if and only if ϕ is strongly usc, locally integrable and dd c [ ϕ ] is a positive current[Lel68, Th´eor`eme II.3]. Bedford–Taylor theory shows that for locally bounded psh functions v , . . . , v q on V and a closed positive current T ∈ D p,p ( V ) on V , there is a unique way todefine a positive current(2.2) v dd c v ∧ . . . ∧ dd c v q ∧ T ∈ D p + q − ,p + q − ( V )and a closed positive current(2.3) dd c v ∧ . . . ∧ dd c v q ∧ T ∈ D p + q,p + q ( V )such that the products (2.2) and (2.3) agree with the usual products in the case of smoothpsh functions v , . . . , v q and such that these products are continuous along monotone de-creasing sequences with respect to weak convergence of currents. For details and a gener-alization to the unbounded case, we refer to [Dem12, Sections III.3-4].2.2. The tropical situation. We recall some facts from [BGJK20]. Let N be a a freeabelian group of rank n with dual M = Hom Z ( N, Z ) and let N R := N ⊗ Z R . We con-sider a fan Σ in N R and the associated partial compactification N Σ of N R . The partialcompactification N Σ is also called tropical toric variety and has a natural stratification N Σ = a σ ∈ Σ N ( σ )with N ( σ ) := N R / h σ i R [BGJK20, Section 3.1]. The toric variety X Σ comes with a contin-uous tropicalization map trop ∞ : X anΣ , ∞ −→ N Σ where X anΣ , ∞ is the complex analytification of X Σ . We denote by A = A N Σ the sheaf ofsmooth Lagerberg forms on N Σ . It is a bigraded sheaf of real algebras with differentials d ′ , d ′′ . We have the Lagerberg involution J which sends A p,q to A q,p . Similarly as in thecomplex case, we have a bigraded sheaf D of Lagerberg currents on N Σ with differentials d ′ , d ′′ [BGJK20, Section 3.2]. By integration again, any locally integrable Lagerberg form ω has an associated Lagerberg current [ ω ]. Note that integration on N R depends on theunderlying lattice N in N R . Sections of A , are called smooth functions .A Lagerberg form ω of bidegree ( p, p ) on the open subset U of N Σ is called positive , if forsome m ∈ N there exist functions f j : U → R ≥ and ( p, α j on U for j = 1 , . . . , m such that ω = m X j =1 ( − p ( p − f j α j ∧ J ( α j ) . We call T ∈ D p,p ( U ) symmetric if T ( J ω ) = ( − q T ( ω ) for all ω ∈ A q,qc ( U ) where q := n − p .A positive Lagerberg current is a symmetric T ∈ D p,p ( U ) with T ( ω ) ≥ ω ∈ A q,qc ( U ). Remark 2.2.1. Let ( p, q ) be with max( p, q ) = n and let U ⊂ N Σ be open. By definitionof the forms on U , they are constant towards the boundary and hence any ω ∈ A p,q ( U )has support in U ∩ N R . In particular, the inclusion A p,qc ( U ∩ N R ) → A p,qc ( U ) is an iso-morphism. We conclude that the dual restriction map D n − p,n − q ( U ) → D n − p,n − q ( U ∩ N R ) ROPICAL TORIC PLURIPOTENTIAL THEORY 9 is an isomorphism. For a locally integrable form η ∈ A n − p,n − q ( U ∩ N R ), we get [ η ] ∈ D n − p,n − q ( U ∩ N R ) = D n − p,n − q ( U ). In particular, a locally integrable function f on U ∩ N R defines a current [ f ] ∈ D , ( U ) on the whole U . Remark 2.2.2. The partial compactification N Σ is a special case of a polyhedral space . Werefer to [JSS19, § 2] for definitions and for an introduction of the sheaf A of smooth Lagerbergforms on polyhedral spaces. Integration of compactly supported top dimensional forms iswell-defined on rational weighted polyhedral spaces. A special case of a rational weightedpolyhedral space is a tropical space where the weights satisfy a balancing condition. Remark 2.2.3. Let N ′ be a free abelian group of finite rank and let Σ ′ be a fan in N ′ R .Recall that a morphism ψ : X Σ ′ → X Σ is called toric if ψ restricts to a homomorphism onthe dense tori and if ψ is equivariant with respect to the torus actions. Toric morphisms X Σ ′ → X Σ are in bijective correspondence to group homomorphisms N ′ → N , which mapevery cone of Σ ′ into a cone of Σ [CLS11, Theorem 3.3.4].More generally, let L : N ′ → N be a homomorphism of cocharacter lattices inducing thehomomorphism ρ : T ′ → T of associated split complex tori. We recall from [BPS14, § ρ -equivariant morphism ψ : X Σ ′ → X Σ of T ′ -(resp. T -)toric varieties is given by thecomposition of a translation by an element in T ′ with a toric morphism to a stratum of X Σ . Clearly, ρ -equivariant toric morphisms ψ : X Σ ′ → X Σ are in bijective correspondenceto L -equivariant morphisms of tropical toric varieties E : N ′ Σ ′ → N Σ as defined below. Definition 2.2.4. Let N ′ be a free abelian group of finite rank and let Σ ′ be a fan in N ′ R .Let L : N ′ → N be a homomorphism of abelian groups. A map E : N ′ Σ ′ → N Σ is called L -equivariant morphism of tropical toric varieties (or of partial compactifications) , if themap E is continuous and L R -equivariant. Remark 2.2.5. Let E : N ′ Σ ′ → N Σ be an L -equivariant morphism of tropical toric varieties. (i) The map E is a morphism of polyhedral spaces by [Jel16, § E ∗ : A p,q ( U ′ ) → A p,q ( U ) for every open subset U of N Σ and U ′ := E − ( U ). (ii) The map E is proper (in the sense of topological spaces), if and only if the preimageof | Σ | with respect to L R : N ′ R → N R equals | Σ ′ | . Observe that this is also equivalentto properness of the corresponding ρ -equivariant morphism ψ : X Σ ′ → X Σ fromRemark 2.2.3 [CLS11, Theorem 3.4.11]. (iii) Assume that E is proper. Then we get a linear map E ∗ : A p,qc ( U ) → A p,qc ( U ′ ) whichis continuous with respect to the locally convex topologies defined in [BGJK20,3.2.4]. By duality, we get a linear map E ∗ : D p,q ( U ′ ) → D p + m − n,q + m − n ( U ) which iscontinuous with respect to the weak topology of currents. (iv) Pull back of forms and push forward of currents along E respect positivity. We assume now that Σ is a smooth fan. Let U be an open subset of N Σ and let V := trop − ∞ ( U ) be the corresponding S -invariant open subset of the complex toric manifold X anΣ , ∞ where S = S ∞ denotes the maximal compact torus in the dense orbit T an ∞ of X anΣ , ∞ .There is a unique homomorphism [BGJK20, Proposition 4.1.7]trop ∗∞ : A ( U ) −→ A ( V )with trop ∗∞ ϕ = ϕ ◦ trop ∞ for all ϕ ∈ A , ( U ) and which satisfies(2.4) trop ∗∞ ◦ d ′ = π − / ∂ ◦ trop ∗∞ , trop ∗∞ ◦ d ′′ = π − / i ¯ ∂ ◦ trop ∗∞ . This homomorphism respects the bigrading, positivity of forms and integration of top di-mensional forms. On the S -invariant forms in A ( V ), we have defined an antilinear involu-tion F . Let A ( V ) S ,F be the real subalgebra of S - and F -invariant forms. Then we havetrop ∗∞ A ( U ) ⊂ A ( V ) S ,F with equality if U ⊂ N R . By equation (2.4), we obtain(2.5) trop ∗∞ ◦ d ′ d ′′ = dd c ◦ trop ∗∞ . Let D ( V ) S ,F be the space of S - and F -invariant currents on V . By duality, we get alinear map trop ∞ , ∗ : D p,p ( V ) S ,F −→ D p,p ( U ) . By [BGJK20, Theorem 7.1.5], trop ∞ , ∗ maps the cone of closed positive currents inside of D p,p ( V ) S ,F isomorphically onto the cone of closed positive Lagerberg currents in D p,p ( U ). Proposition 2.2.7. Let f be a function on an open subset U of N Σ with values in R ∪{±∞} .Assume that the function f ◦ trop ∞ on the complex manifold V := trop − ∞ ( U ) is locallyintegrable. Then we have trop ∞ , ∗ [ f ◦ trop ∞ ] = [ f ] . Proof. Note first that local integrability of f ◦ trop ∞ implies local integrability of f | U ∩ N R .Hence the Lagerberg current [ f ] is well defined by Remark 2.2.1. It is enough to show that(2.6) Z V ( f ◦ trop ∞ ) · trop ∗∞ ( ω ) = Z U f · ω for all ω ∈ A n,nc ( U ∩ N R ). By [BGJK20, Corollary 5.1.8], the identity (2.6) holds for smoothfunctions f . Note that integration against the forms ω and trop ∗∞ ( ω ) give Radon measureson U and V , respectively, with compact supports. Since we can approximate a continuousfunction uniformly by smooth functions in a neighbourhood of supp( ω ), we conclude that(2.6) holds for all continuous functions f on U . Since a Radon measure with compactsupport in U ∩ N R is induced by a unique signed Borel measure, it follows that (2.6) is truefor all locally integrable functions f on U ∩ N R . (cid:3) The non-archimedean situation. Let K be a non-archimedean complete field withvaluation v = − log | | . First, we recall the construction of forms and currents of Chambert-Loir and Ducros on analytic spaces over K . For more algebraic approaches, we refer to[Gub16], [GK19] and [Jel19]. Then we will study forms and currents in the toric setting. Let X be an n -dimensional connected good K -analytic space whose topology isHausdorff (see [Ber90] and [Ber93, § X and we refer to [CD12] for details about the following facts.The sheaf A = A X of smooth differential forms on X has similar properties as in thecomplex case. It is a bigraded differential sheaf of real algebras with respect to naturaldifferential operators d ′ and d ′′ . Its construction is based on tropical charts and Lagerbergforms as follows: A moment map on an open subset W of X is a morphism ϕ : W → ( G r m ) an leading to the tropicalization map trop ϕ := trop v ◦ ϕ : W → R r . There is a bigradeddifferential homomorphism trop ∗ ϕ from the sheaf of smooth Lagerberg forms on R r to thesheaf of differential forms on W . If trop ϕ ( W ) is contained in the compact support | C | of apolyhedral complex C in R r and α, β ∈ A p,q ( N R ) have the same restriction to polyhedra in C , then trop ∗ ϕ ( α ) = trop ∗ ϕ ( β ) in A p,q ( W ). For more details about the construction of A werefer to [CD12, § J on A which sends A p,q to A q,p and which is compatiblewith the Lagerberg involution on tropical charts. Smooth differential forms of bidegree(0 , 0) are functions which we call smooth functions .A differential form ω ∈ A p,p ( V ) on an open subset V of X is called positive if there exist m ∈ N , smooth functions f j : V → R ≥ and forms α j ∈ A p, ( V ) for j = 1 , . . . , m such that ω = m X j =1 ( − p ( p − f j α j ∧ J ( α j ) . We assume now that X has no boundary in the sense of [Ber90, § § ROPICAL TORIC PLURIPOTENTIAL THEORY 11 over K . By duality there is also a bigraded sheaf D of currents on X with differentials d ′ , d ′′ .Moreover, one can integrate compactly supported forms on X of bidegree ( n, n ). For everyopen subset V of X , we get that any ω ∈ A p,q ( V ) has an associated current [ ω ] ∈ D p,q ( V ).Symmetric and positive currents on V are defined similarly as the corresponding notions inthe tropical case. Let N be a free abelian group of rank n with dual M and let T be the multiplicativetorus with character lattice M . From now on, we consider the T -toric variety X Σ associatedto a fan Σ in N R . Its Berkovich analytification over K is denoted by X anΣ ,v . The Kajiwara–Payne tropicalization map trop v : X anΣ ,v → N Σ is the unique continuous map given by theusual tropicalization map on T an v . The latter is characterized by h trop v ( x ) , u i = v ◦ χ u ( x )for x ∈ T an v and any u ∈ M with associated character χ u . Note that trop v is a proper mapof topological spaces. For details, see [Kaj08] and [Pay09].We have a canonical section ι : N Σ → X anΣ ,v of trop v . Since ι is a homeomorphism onto aclosed subset of X anΣ , we may view N Σ as a closed subset of X anΣ ,v which we call the canonicalskeleton . For details, we refer to [BPS14, § Let S = S v denote the maximal affinoid torus in T an v acting on X anΣ ,v by the morphism m : S × X anΣ ,v −→ X anΣ ,v . Let p the second projection of S × X anΣ ,v , where we take products in the category of K -analytic spaces. We consider an S -invariant open subset V of X anΣ ,v , i.e. we require that m − ( V ) = p − ( V ). A form α ∈ A p,q ( V ) with m ∗ α = p ∗ α is called S -invariant . Similarly, afunction f on V is said to be S -invariant , if f ◦ m = f ◦ p .Note that S is a K -analytic group. However the underlying topological space of S doesnot inherit a group structure. Nevertheless there are well-defined orbits [Ber90, Chapter 5].The S -orbits of X anΣ ,v are precisely the fibers of the map trop v , such that trop v : X anΣ ,v → N Σ is the quotient by S in the sense of topological spaces [BPS14, Proposition 4.2.15].Using the identity on T an v as a moment map, we get from 2.3.1 a canonical homomorphism(2.7) trop ∗ v : A N R −→ (trop v ) ∗ ( A T an v )of bigraded differential sheaves of algebras with respect to the differentials d ′ , d ′′ . Proposition 2.3.4. The homomorphism in (2.7) extends uniquely to a homomorphism (2.8) trop ∗ v : A N Σ −→ (trop v ) ∗ ( A X anΣ ,v ) of bigraded sheaves of algebras compatible with the differentials d ′ , d ′′ . For any open U in N Σ and α ∈ A p,q ( U ) , we have that trop ∗ v ( α ) is an S -invariant form on trop − v ( U ) .Proof. By [Jel19], one can construct the sheaf of smooth forms on X anΣ ,v using toric chartswith boundaries. The existence of the morphism (2.8) is a natural consequence of thisconstruction. For the convenience of the reader, we include a direct proof.Complex forms and Lagerberg forms are determined by their restriction to the densestratum. This proves uniqueness. It remains to define trop ∗ v ( α ) ∈ A p,q (trop − v ( U )) S for anyopen subset U of N Σ and any α ∈ A p,q ( U ). It is enough to give the definition locally arounda given point u ∈ U . There is a unique σ ∈ Σ with u ∈ N ( σ ). Let π σ : N R → N ( σ ) be thequotient map. By definition of Lagerberg forms on U , there is an open neighbourhood U u of u in U such that α = π ∗ σ α u on U u for some α u ∈ A p,q ( U u ∩ N ( σ )). We note that π σ ◦ trop v is the tropicalization of a moment map and hence we can use 2.3.1 to define ω := trop ∗ v ( α )on V u := trop − v ( U u ). Since ω | V u ∩ T an v agrees with the definition in (2.7) proving existence. Note that V u is an S -invariant open subset of V := trop − v ( U ). On S × V u , we have π σ ◦ trop v ◦ p = π σ ◦ trop v ◦ m. These are tropicalizations of moment maps with p ∗ ω = ( π σ ◦ trop v ◦ p ) ∗ ( α u ) and m ∗ ω =( π σ ◦ trop v ◦ m ) ∗ ( α u ) on S × V u . This proves S -invariance of ω on V u and hence on V . (cid:3) In the complex situation (see 2.2.6),the analogue of the morphism (2.7) is an isomorphism onto the subsheaf of S - and F -invariant forms of (trop v ) ∗ ( A T an v ). Since F is only related to the complex structure, onemight hope in the non-archimedean case that (2.7) gives an isomorphism onto the subsheafof S -invariant forms. The following is a counterexample. Example 2.3.5. Let N := Z and Σ := { (0) } with X Σ = T = G m and N Σ = R . We claimthat the function f := min(0 , trop v ) is an S -invariant smooth function on T an v which is not contained in trop ∗ v ( A , ( R )).Indeed, the function min(0 , u ) is not smooth on R and hence f trop ∗ v ( A , ( R )). Since f factors through trop v , it is S -invariant. To show f ∈ A , ( T an v ), note first that f ( x ) = ( − log | T ( x ) | )( − log | ( T − x ) | )for x ∈ T an v where T is the toric coordinate on T . We consider the closed embedding ϕ : T → T , x ( x, x − u , u denote the coordinates on trop v (( T an v ) ) = R . Then u · u ∈ A , (trop ϕ ( T an v )). We conclude that trop ∗ ϕ ( g ) = f ∈ A , ( T an v ) proving the claim. Proposition 2.3.6. Let X Σ be a toric variety of dimension n . Let ω be a ( p, q ) -form onan open subset U of N Σ with max( p, q ) = n . Then supp(trop ∗ v ( ω )) = ι (supp( ω )) , where ι : N Σ → X anΣ ,v is the embedding of the canonical skeleton as in 2.3.2.Proof. We have seen supp( ω ) ⊂ N R in Remark 2.2.1 and similarly supp(trop ∗ v ω ) ⊂ T an v by [CD12, Lemme 3.2.5], so we may assume X Σ = T . Let x ∈ T an v \ ι ( N R ) and let y = ι (trop v ( x )). Then there exists a regular function f on T such that | f ( x ) | 6 = | f ( y ) | . By thefundamental theorem in tropical geometry, the tropicalization trop v ( f ) is a piecewise affinefunction on N R whose singularity locus V (trop v ( f )) is an ( n − N R equal to trop v ( V ( f ) an v ) for the zero set V ( f ) of f (see [MS15, § ϕ : T −→ T × A , z ( z, f ( z )) , we see that trop ϕ ( x ) = trop ϕ ( y ) and trop ϕ ( x ) / ∈ trop ϕ ( ι ( N R ))). The first projection restrictsto a map π : trop ϕ ( T an v ) → trop v ( T an v ) = N R which is injective away from V (trop v ( f )). Let W := trop − ϕ (trop ϕ ( T an v ) \ trop ϕ ( ι ( N R ))) . Since trop v is a proper map and since the canonical skeleton ι ( N R ) is closed in T an v , the set W is an open neighborhood of x in T an v . For any x ′ ∈ W and y ′ := ι (trop v ( x ′ )), we have π (trop ϕ ( x ′ )) = trop v ( x ′ ) = trop v ( y ′ ) = π (trop ϕ ( y ′ )) . Since x ′ ∈ W , we have x ′ = y ′ and hence the above shows that trop v ( x ′ ) ∈ V (trop v ( f )).This yields trop v ( W ) ⊂ V (trop v ( f )). Since we have max( p, q ) = n > dim( V (trop v ( f ))), wehave ω | V (trop v ( f )) = 0 and hence trop ∗ v ( ω ) | W = 0. This shows x / ∈ supp(trop ∗ v ( ω )) and since x was arbitrary in T an v \ ι ( N R ), we have supp(trop ∗ v ( ω )) ⊂ ι ( N R ). By [CD12, Corollaire3.2.3], we have trop v (supp(trop ∗ v ( ω ))) = supp( ω ) , and the claim follows. (cid:3) Let U be an open subset of N Σ and let V := trop − v ( U ). Then we have a linear maptrop v, ∗ : D p,q ( V ) −→ D p,q ( U ) , T T ◦ trop ∗ v . For α ∈ A r,s ( U ) and T ∈ D p,q ( V ), the projection formula(2.9) trop v, ∗ (trop ∗ v ( α ) ∧ T ) = α ∧ trop v, ∗ ( T ) . is an easy formal consequence of our definitions. ROPICAL TORIC PLURIPOTENTIAL THEORY 13 Plurisubharmonic functions on partial compactifications Let N be a free abelian group of finite rank n . We always consider R −∞ := R ∪ {−∞} as an ordered additive monoid equipped with the unique topology which turns the naturalbijection exp : R −∞ ∼ → [0 , ∞ ) ⊂ R into a homeomorphism. We fix a fan Σ in N R withassociated partial compactification N Σ . We denote by | Σ | the support of the fan Σ.3.1. Definition and basic properties of plurisubharmonic functions.Definition 3.1.1. Let U be an open subset of N R . A function ϕ : U → R −∞ is called convex if ϕ (cid:0) (1 − t ) x + ty (cid:1) ≤ (1 − t ) ϕ ( x ) + tϕ ( y )holds for all t ∈ [0 , 1] and all x, y ∈ U such that the segment [ x, y ] := { (1 − t ) x + ty | t ∈ [0 , } is contained in U . A function ϕ : U → R −∞ is called finite if ϕ ( U ) ⊂ R .For p ∈ N R and v ∈ | Σ | , we have shown in [BGJK20, Lemma 3.1.4] that the limit p + ∞ v := lim µ →∞ p + µv exists in N Σ and p + ∞ v lies in the stratum N ( σ ) for the uniquecone σ ∈ Σ containing v in its relative interior. We will also use the compactified half line [ p, p + ∞ v ] := { p + µv | µ ∈ [0 , ∞ ] } . Definition 3.1.2. Let U ⊂ N Σ be an open subset. A function ϕ : U → R −∞ is called plurisubharmonic (for short psh ) if (i) the function ϕ is upper-semicontinuous (for short usc ), (ii) the function ϕ | U ∩ N R is convex (in the sense of Definition 3.1.1), (iii) for any p ∈ N R and v ∈ | Σ | with [ p, p + ∞ v ] ⊂ U , we have ϕ ( p + ∞ v ) ≤ ϕ ( p ). Proposition 3.1.3. Let U be open subset in N Σ and let ( ϕ i ) i ∈ I be psh functions on U .(i) Plurisubharmonicity is a local property.(ii) Let ϕ ( z ) = sup i ∈ I ϕ i ( z ) ∈ [ −∞ , ∞ ] . If we have ϕ ( z ) < ∞ for all z ∈ U and if ϕ : U → R −∞ is usc (both conditions hold automatically if I is finite), then ϕ ispsh.(iii) If ϕ i +1 ( z ) ≤ ϕ i ( z ) for all i ∈ N and all z ∈ U , then the limit ϕ ( z ) = lim i ϕ i ( z ) exists pointwise and the function ϕ : U → R −∞ is psh.Proof. This is shown easily using standard properties of convex functions. (cid:3) Let U ⊂ N Σ be an open subset. We call a subset E ⊂ U a null set if E ∩ N R is a null setwith respect to the Lebesgue measure on N R . A function ϕ : U → R −∞ is called stronglyupper semicontinuous if for every null set E ⊂ U and every a ∈ U , we have(3.1) ϕ ( a ) = lim sup x → a, x/ ∈ E ϕ ( x ) . Theorem 3.1.4. Let U ⊂ N Σ be an open subset. For a function ϕ : U → R −∞ , thefollowing conditions are equivalent:(i) The function ϕ is psh.(ii) The function ϕ is continuous and ϕ | U ∩ N R is convex.(iii) The function ϕ is strongly upper semicontinuous and ϕ | U ∩ N R is convex.(iv) The function ϕ | U ∩ N R is convex and for every point x ∈ U \ N R we have (3.2) ϕ ( x ) = lim sup y ∈ U ∩ N R y → x ϕ ( y ) . The proof will be given after a series of lemmata. Lemma 3.1.5. Let ϕ : ( M, ∞ ) → R be a convex function for some M ∈ R such that lim x →∞ ϕ ( x ) < ∞ . Then ϕ is monotone decreasing.Proof. This is an easy exercise and will be left to the reader. (cid:3) Lemma 3.1.6. Let ϕ : U → R −∞ be a psh function on an open subset U of N Σ and let σ ∈ Σ . We consider p ∈ N ( σ ) ∩ U , v ∈ relint( σ ) and q ∈ U ∩ N R such that p = q + ∞ v and [ q, q + ∞ v ] ⊂ U . Then the function ϕ | [ q,q + ∞ v ] is continuous.Proof. We equip [ q, p ] with the subspace topology from U . By convexity, ϕ | [ q,p ) is contin-uous [Roc70, Section 10]. Therefore we only need to show the continuity at the point p .Combining upper semicontinuity and condition (iii) in Definition 3.1.2 we deduce that ϕ ( p ) ≥ lim µ →∞ ϕ ( q + µv ) ≥ ϕ ( p )proving continuity. (cid:3) Lemma 3.1.7. Let ϕ be a psh function on an open subset U of N Σ and let σ ∈ Σ such that U ∩ N ( σ ) is connected. Then ϕ | U ∩ N ( σ ) is either identically −∞ or a finite convex function.Proof. We start by showing that ϕ | U ∩ N ( σ ) is convex. Let p, q, r ∈ U ∩ N ( σ ) and λ ∈ (0 , q = λp + (1 − λ ) r and [ p, r ] ⊂ U . We choose v ∈ relint( σ ) and points p ′ , q ′ and r ′ in U ∩ N R such that q ′ = λp ′ + (1 − λ ) r ′ , p = p ′ + ∞ v, q = q ′ + ∞ v, r = r ′ + ∞ v and for every µ ≥ p + µv, r + µv ] ⊂ U holds. This can easily be achievedby the openness of U and the compactness of [ p, r ]. Then ϕ ( q ′ + µv ) ≤ λϕ ( p ′ + µv ) + (1 − λ ) ϕ ( r ′ + µv )for all µ ≥ 0. Therefore, using Lemma 3.1.6, we get ϕ ( q ) = lim µ →∞ ϕ ( q ′ + µv ) ≤ lim µ →∞ (cid:0) λϕ ( p ′ + µv ) + (1 − λ ) ϕ ( r ′ + µv ) (cid:1) ≤ λϕ ( p ) + (1 − λ ) ϕ ( r )showing that ϕ | U ∩ N ( σ ) is convex. The claim follows from the fact that a convex functionon a connected open set with values in R −∞ is either finite or identically −∞ . (cid:3) Proof of Theorem 3.1.4. We recall first some useful facts. Fix p ∈ N Σ , then there is aunique σ ∈ Σ with p ∈ N ( σ ) and we denote by π σ : N R → N ( σ ) the quotient map. For p ∈ N R with π σ ( p ) = p and an open bounded convex neighbourhood Ω of p in N R , set(3.3) U ( σ, Ω , p ) := a τ ≺ σ π τ (Ω + σ ) ⊂ N Σ . It follows from the second description of the topology of N Σ in [BGJK20, Remark 3.1.2]that the sets U ( σ, Ω , p ) form a basis of open neighbourhoods of p in N Σ . Note that thestrata π τ (Ω + σ ) of the basic open sets U ( σ, Ω , p ) are open convex subsets of N ( τ ). Wewill use often that [ u, u + ∞ v ] ⊂ U ( σ, Ω , p ) for any u ∈ U ( σ, Ω , p ) ∩ N R and any v ∈ σ .To prove (i) ⇒ (ii) , we have to show continuity of a psh function ϕ : U → R −∞ in p ∈ U .We choose an open neighbourhood U ( σ, Ω , p ) of p in U as above, fix v ∈ relint( σ ) and let ε > 0. For µ ∈ [0 , ∞ ], let ϕ µ : Ω → R −∞ , q ϕ ( q + µv ). By Lemmata 3.1.5 and 3.1.6, weget a decreasing net ( ϕ µ ) µ ∈ R + of convex functions on Ω which converges pointwise to ϕ ∞ .Since π σ (Ω) is an open convex neighbourhood of p = p + ∞ v in N ( σ ), Lemma 3.1.7 showsthat ϕ ∞ = ϕ ◦ ( π σ | Ω ) is either a finite convex function or identically −∞ .We assume first ϕ ( p ) > −∞ and hence ϕ ∞ is a finite function on Ω. By [Roc70, Theorem10.8], the convergence of ϕ µ to ϕ ∞ is uniform on compact subsets of Ω. Thus we can finda compact neighborhood W of p in Ω and M ≫ ϕ ( p ) − ε = ϕ ∞ ( p ) − ε ≤ ϕ µ ( w ) = ϕ ( w + µv ) ≤ ϕ ∞ ( p ) + ε = ϕ ( p ) + ε ROPICAL TORIC PLURIPOTENTIAL THEORY 15 for all µ ∈ [ M, ∞ ] and all w ∈ W . We pick an open convex neighbourhood Ω of p contained in W and set V := U ( σ, Ω + M v, p + M v ). Since π σ ( p + µv ) = p , we see that V is a basic open neighbourhood of p in U . Continuity in p follows from the claim(3.5) − ε + ϕ ( p ) ≤ ϕ ( q ) ≤ ϕ ( p ) + ε for any q ∈ V . There is a unique τ ≺ σ with q ∈ N ( τ ). Using (3.3), there is q ∈ Ω and s ∈ σ such that q = π τ ( q ′ ) for q ′ := q + M v + s . We pick v ′ ∈ relint( τ ). Then q = q ′ + ∞ v ′ and [ q ′ , q ′ + ∞ v ′ ] ⊂ V = U ( σ, Ω + M v, p + M v ). By Lemma 3.1.6, the restriction of ϕ to [ q ′ , q ′ + ∞ v ′ ] is continuous and hence it is enough to prove (3.5) for q ∈ V ∩ N R .Then we have q = q + M v + s . We have [ q, q + ∞ v ] ⊂ V and [ q M , q M + ∞ s ] ⊂ V for q M := q + M v ∈ V . By Lemmata 3.1.5 and 3.1.6, the restrictions of ϕ to [ q M , q M + ∞ s ]and to [ q, q + ∞ v ] are continuous and decreasing. Together with (3.4) for w = q , this shows ϕ ( p ) − ε ≤ ϕ ∞ ( q ) = ϕ ( q + ∞ v ) ≤ ϕ ( q ) = ϕ ( q + M v + s ) ≤ ϕ ( q M ) ≤ ϕ ( p ) + ε. We assume now ϕ ( p ) = −∞ . By Lemma 3.1.6, there is M ≫ ϕ M ( p ) = ϕ ( p + M v ) < − /ε . Since ϕ M is convex, it is continuous. There is an open convexneighbourhood Ω of p in Ω such that(3.6) ϕ M ( w ) = ϕ ( w + M v ) < − /ε for all w ∈ Ω . Let q ∈ V := U ( σ, Ω + M v, p + M V ). Then q = π τ ( q ′ ) for q ′ := q M + s , q = q ′ + ∞ v ′ and [ q ′ , q ′ + ∞ v ′ ] ⊂ V as in the first case. The same arguments show that therestrictions of ϕ to [ q ′ , q ′ + ∞ v ′ ] and to [ q M , q M + ∞ s ] are continuous and decreasing, hence ϕ ( q ) = ϕ ( q ′ + ∞ v ′ ) ≤ ϕ ( q ′ ) = ϕ ( q M + s ) ≤ ϕ ( q M ) = ϕ ( q + M v ) < − /ε by using (3.6) for w = q ∈ Ω on the right. This shows continuity of ϕ in p .Obviously, (ii) implies (iii) and (iii) implies (iv) . It remains to prove that (iv) yields (i) and so we assume that ϕ | U ∩ N R is convex and (3.2) is satisfied. Note that (3.2) easily yieldsthat ϕ is usc. It remains to check (iii) of Definition 3.1.2. We argue by contradiction andassume that there is p ′ ∈ N R and v ∈ relint( σ ) for some σ ∈ Σ such that [ p ′ , p ′ + ∞ v ] ⊂ U and ϕ ( p ′ + ∞ v ) > ϕ ( p ′ ).The function g : R ≥ → R , µ ϕ ( p ′ + µv ) is convex. Using (3.2) and ϕ ( p ′ + ∞ v ) = ∞ ,we get lim µ →∞ g ( µ ) < ∞ . Lemma 3.1.5 yields that g is monotone decreasing. Note that p := p ′ + ∞ v = π σ ( p ′ ) ∈ N ( σ ). There is p ∈ N R with π σ ( p ) = p and a bounded openconvex neighbourhood Ω of p such that U ( σ, Ω , p ) ⊂ U . There is M ≫ p ′ M := p ′ + M v ∈ U ( σ, Ω , p ). By assumption, there is ε > ϕ ( p ′ ) < ϕ ( p ) − ε .Since ϕ ( p ′ M ) ≤ ϕ ( p ′ ) and ϕ | U ∩ N R continuous, there is a bounded open convex neighbourhoodΩ M of p ′ M in U ( σ, Ω , p ) ∩ N R such that ϕ ( w ) < ϕ ( p ) − ε for all w ∈ Ω M . We claim that(3.7) ϕ ( q ) < ϕ ( p ) − ε for all q in the open neighbourhood V := U ( σ, Ω M , p ′ M ) of p in U . This contradicts (3.2).To prove (3.7), let q ∈ V . Then there is τ ≺ σ , q M ∈ Ω M and s ∈ σ such that q = π τ ( q ′ )for q ′ := q M + s . Note that [ q M , q M + ∞ s ] ⊂ V . For v ′ ∈ relint( τ ), we have q = q ′ + ∞ v ′ and[ q ′ , q ′ + ∞ v ′ ] ⊂ V . By (3.2) and Lemma 3.1.5 again, the restrictions of ϕ to [ q ′ , q ′ + ∞ v ′ ]and to [ q M , q M + ∞ s ] are decreasing. Using q M ∈ Ω M , we get ϕ ( q ) ≤ ϕ ( q ′ ) = ϕ ( q M + s ) ≤ ϕ ( q M ) < ϕ ( p ) − ε proving (3.7). This finishes the proof of (iv) ⇒ (i) . (cid:3) We give now two results which hold similarly for complex psh functions. Corollary 3.1.8. Let U be an open subset of N Σ and let ϕ be a psh function on U ∩ N R .If ϕ is locally bounded from above near the boundary U \ U ∩ N R , then ϕ extends uniquelyto a psh function on U . Proof. We define the extension of ϕ to a ∈ U \ N R by(3.8) ϕ ( a ) = lim sup x ∈ U ∩ N R x → a ϕ ( x ) . Using that ϕ is locally bounded near the boundary, we see that ϕ ( a ) < ∞ . Since ϕ is pshon U ∩ N R , it is convex on U ∩ N R . By Theorem 3.1.4, we deduce that ϕ is psh on U . (cid:3) Proposition 3.1.9. If Σ is a complete fan, then any psh function on N Σ is constant.Proof. Let ϕ : N Σ → R −∞ be a psh function. For any line ℓ in N R , the restriction of ϕ isa convex function. We may write ℓ = p + R v for any p ∈ ℓ and a non-zero v ∈ N R . Bycondition (iii) in Definition 3.1.2, we have lim t →∞ f ( p ± tv ) < ∞ . By Lemma 3.1.5, theconvex functions ϕ ( p ± tv ) are decreasing in t ≥ (cid:3) Proposition 3.1.10. Let L : N ′ → N be a homomorphism of free abelian groups of finiterank, let E : N ′ Σ ′ → N Σ be an L -equivariant morphism of tropical toric varieties as inDefinition 2.2.4 and let ϕ : U → R −∞ be a function on an open subset U of N Σ .(i) If ϕ is psh on U , then ϕ ◦ E is psh on E − ( U ) .(ii) If E is a surjective proper map and if ϕ ◦ E is psh on E − ( U ) , then ϕ is psh on U .Proof. We set U ′ := E − ( U ). Let ϕ be psh on U . Note that E ( N ′ R ) ⊂ N ( σ ) for a unique σ ∈ Σ and hence it follows from Lemma 3.1.7 that ϕ ◦ E is convex on U ′ ∩ N ′ R . By Theorem3.1.4, ϕ is continuous. It follows that ϕ ◦ E is continuous and hence psh. This proves (i) .To prove (ii) , we observe that surjectivity of E yields that E | N ′ R : N ′ R → N R is affine andsurjective. Since ϕ ◦ E is psh , the restriction ϕ ◦ E | U ′ ∩ N ′ R is convex. Therefore ϕ | N R ∩ U isalso convex. Using that E is proper and surjective, it is clear that ϕ is continuous if andonly if ϕ ◦ E is continuous. By Theorem 3.1.4, we get (ii) . (cid:3) Relations to Lagerberg currents and complex geometry. For Proposition 3.2.1below, we assume that the fan Σ is smooth to obtain a characterization of psh functionson N Σ in terms of classical psh functions on the complex toric manifold X anΣ , ∞ using thecomplex tropicalization map trop ∞ : X anΣ , ∞ → N Σ . Proposition 3.2.1. We assume that Σ is a smooth fan in N R . Let U be an open subset of N Σ . Then ϕ : U → R −∞ is psh if and only if ϕ ◦ trop ∞ is a psh function on trop − ∞ ( U ) .Proof. We note first that the statement holds for an open subset U of N R . Indeed, if ϕ ◦ trop ∞ is psh, then [Dem12, Theorem I.5.13] shows that ϕ is psh. The converse implicationfollows by approximating ϕ locally and uniformly by smooth convex functions.Now we allow U to hit the boundary. Assume that ϕ ◦ trop ∞ is psh. The above yieldsthat ϕ | N R ∩ U is convex. By definition of psh functions in complex geometry, the function ϕ ◦ trop ∞ is usc. Using that trop ∞ is surjective, we get { x ∈ U | ϕ ( x ) ≥ t } = trop (cid:0) { y ∈ trop − ∞ ( U ) | ϕ ◦ trop( y ) ≥ t } (cid:1) for all t ∈ R −∞ and these sets are closed as the map trop ∞ is proper. Therefore ϕ is usc.We next prove that (iii) in Definition 3.1.2 is satisfied. Let p ∈ N R and v ∈ | Σ | with[ p, p + ∞ v ] ⊂ U . There is a unique cone σ ∈ Σ with v ∈ relint( σ ). Choose a vector v ∈ N with v ∈ relint( σ ). This can always be done because the fan is rational. The descriptionof the topology of N Σ given in [BGJK20, Remark 3.1.2] and the proof of [BGJK20, Lemma3.1.4] show that there is µ ≥ p + µ v + µv ∈ U for all µ ≥ p + ∞ v = p + µ v + ∞ v . We have seen that ϕ | N R ∩ U is convex, ϕ is usc and does not takethe value ∞ , hence Lemma 3.1.5 shows that ϕ ( p ) ≥ ϕ ( p + µ v ). Thus is enough to prove ROPICAL TORIC PLURIPOTENTIAL THEORY 17 that ϕ ( p + µ v ) ≥ ϕ ( p + ∞ v ). We denote by E the complex unit disc and let E ∗ := E \ { } .We choose a point x ∈ X anΣ , ∞ with trop ∞ ( x ) = p + µ v and consider the map h : E ∗ −→ X anΣ , ∞ , t v ( t ) · x where we view N as the group of 1-parameter subgroups of the torus T . This map extendsto a holomorphic curve e h : E → X anΣ , ∞ with trop ∞ ( e h (0)) = p + ∞ v and e h ( E ) ⊂ trop − ∞ ( U ).The restriction of ϕ ◦ trop to E is a psh function. Therefore ϕ ( p + ∞ v ) = ϕ (trop ∞ ( e h (0))) ≤ Z ϕ (trop ∞ ( e h ( e πiθ ))) dθ = ϕ (trop ∞ ( e h (1))) = ϕ ( p + µ v ) . Conversely, assume that ϕ is psh. The beginning of the proof yields that the restrictionof ϕ ◦ trop ∞ to trop − ∞ ( U ) ∩ T an ∞ is psh. Using that ϕ is continuous by Theorem 3.1.4, wededuce from [Dem12, Theorem I.5.24] that ϕ ◦ trop ∞ is psh. (cid:3) Recall from Remark 2.2.1 that a function ϕ on an open subset of N Σ with ϕ | U ∩ N R locallyintegrable yields a Lagerberg current [ ϕ ] on U . This applies if ϕ | U ∩ N R is finite and convex. Theorem 3.2.2. Let Σ be a fan and let ϕ : U → R −∞ be a function on a connected opensubset U ⊂ N Σ such that ϕ 6≡ −∞ . Then ϕ is psh if and only if(i) ϕ is strongly upper semicontinuous;(ii) ϕ | U ∩ N R is locally integrable and d ′ d ′′ [ ϕ ] is a positive Lagerberg current on U .Proof. Let ϕ be a psh function. We show that ϕ satisfies (i) and (ii) . We first considerthe case where the fan Σ is smooth. Assume that ϕ is psh. By Theorem 3.1.4 the function ϕ is strongly upper semicontinuous and the restriction ϕ | U ∩ N R is locally integrable. ByProposition 3.2.1, the function trop ∗∞ ( ϕ ) := ϕ ◦ trop ∞ is psh on V := trop − ∞ ( U ) and hence dd c [trop ∗∞ ( ϕ )] is a positive current on V . Note that(3.9) trop ∞ , ∗ [trop ∗∞ ( ϕ )] = [ ϕ ]by Proposition 2.2.7. Using 2.2.6 and (3.9), we conclude that(3.10) d ′ d ′′ [ ϕ ] = trop ∞ , ∗ dd c [trop ∗∞ ( ϕ )]is a positive current on U .Now we consider the case of a general fan Σ. Assume that ϕ is psh. We choose asmooth subdivison Σ ′ of Σ in N R . There is a unique proper surjective morphism of tropicaltoric varieties g : N Σ ′ → N Σ which extends the identity on N R . By Proposition 3.1.10 thefunction ϕ ′ := ϕ ◦ r is psh on r − ( U ). As Σ ′ is smooth, ϕ ′ satisfies conditions (i) and (ii) .Then ϕ is strongly usc as well and ϕ | U ∩ N R = ϕ ′ | U ∩ N R is locally integrable. From [ ϕ ] = r ∗ [ ϕ ′ ]we conclude that d ′ d ′′ [ ϕ ] = r ∗ d ′ d ′′ [ ϕ ′ ] is positive as well.For the converse implication, we need only that ϕ satisfies (i) and the weaker condition (ii’) ϕ | U ∩ N R is locally integrable and d ′ d ′′ [ ϕ ] is a positive Lagerberg current on U ∩ N R . As in the proof of [Lag12, Proposition 2.5], we construct from ϕ | U ∩ N R a sequence ( ϕ ǫ ) ǫ ofconvex smooth functions on U ∩ N R such that ϕ ǫ converges weakly to a convex function g on U ∩ N R . From [ ϕ | U ∩ N R ] = [ g ], we get that ϕ | U ∩ N R = g outside of a null set. As ϕ isstrongly usc, this shows convexity of ϕ | U ∩ N R = g . Theorem 3.1.4 implies that ϕ is psh. (cid:3) Regularisation. Recall that Σ is a fan and U is an open subset of the partial com-pactification N Σ . For a face τ of σ ∈ Σ, the canonical map N ( τ ) → N ( σ ) is denoted by π σ,τ . We say that a function ϕ : U → R is constant towards the boundary if for each σ ∈ Σ and each p ∈ U σ := U ∩ N ( σ ) there is a neighborhood V of p such that for all τ ≺ σ wehave V τ = ( π σ,τ | V τ ) − ( V σ ) and ϕ | V τ = ( π σ,τ | V τ ) ∗ ( ϕ | V σ ).(3.11)By definition, smooth functions on U are constant towards the boundary.We recall that a smoothing kernel in N R is a non-negative smooth function η : N R → R with compact support and R N R η ( y ) dy = 1. Here dy denotes the Haar measure on N R suchthat the lattice N has covolume one. Lemma 3.3.1. Let ϕ : U → R be a continuous function which is constant towards theboundary and let U ′ be a relatively compact open subset of U . Then there exists a smoothingkernel η with compact support in N R such that the convolution ϕ ⋆ η : U ′ ∩ N R −→ R , ϕ ⋆ η ( x ) = Z N R ϕ ( x − y ) η ( y ) dy is defined and extends uniquely to a smooth function ϕ ⋆ η : U ′ → R . We call such an η a smoothing kernel for ϕ and U ′ . Proof. Since ϕ is constant towards the boundary, we can pick for any p ∈ U an openneighbourhood V ( p ) such that ϕ satisfies (3.11) for V = V ( p ). We choose an open neigh-bourhood V ′ ( p ) of p which is relatively compact in V ( p ) and hence there is a relativelycompact convex subset W ( p ) ⊂ N R , symmetric with respect to zero and such that(3.12) V ′ ( p ) ∩ N R + W ( p ) ⊂ V ( p ) ∩ N R . Since the closure of U ′ is compact in U , it can be covered by open subsets V ′ ( p ) with p ranging over a finite I ⊂ U . Then W := T p ∈ I W ( p ) is a relatively compact convex andsymmetric subset of N R with U ′ ∩ N R + W ⊂ U ∩ N R by (3.12). For a smoothing kernel η with compact support in W , we conclude that the convolution ϕ ⋆ η is well defined on U ′ ∩ N R . Since η is smooth, it is clear that ϕ ∗ η is smooth on U ′ ∩ N R .We have to check that ϕ ∗ η can be extended to a smooth function on U ′ . We can checkthis locally. Hence it is enough to check for any p ∈ I that ϕ ⋆ η extends from V ′ ( p ) ∩ N R to a smooth function on V ′ ( p ). There is a unique σ ∈ Σ with p ∈ N ( σ ). Let π σ : N R → N σ be the quotient map. For x, x ′ ∈ V ′ ( p ) ∩ N R with π σ ( x ) = π σ ( x ′ ) and any y ∈ W ( p ), wehave π σ ( x − y ) = π σ ( x ′ − y ). Since ϕ satisfies (3.11) on V ( p ), we get ϕ ( x − y ) = ϕ ( x ′ − y )and hence ϕ ⋆ η ( x ) = Z N R ϕ ( x − y ) η ( y ) dy = Z N R ϕ ( x ′ − y ) η ( y ) dy = ϕ ⋆ η ( x ′ ) . Since V ′ ( p ) is dense, we deduce that ϕ ⋆ η has a unique extension from V ′ ( p ) ∩ N R to afunction on V ′ ( p ) satisfying (3.11). By definition, this extension is smooth.Uniqueness is clear by density of U ′ ∩ N R in U ′ and continuity. (cid:3) Lemma 3.3.2. Let ϕ be a continuous real function on U which is constant towards theboundary. Let η be a smoothing kernel for ϕ and a relatively compact open subset U ′ of U .If ϕ is psh in U , then ϕ ⋆ η is psh in U ′ and ϕ ⋆ η ≥ ϕ in U ′ .Proof. If ϕ is psh, then ϕ | U ∩ N R is convex. This readily implies that ϕ ⋆ η ≥ ϕ and that ϕ ⋆ η is convex on U ′ ∩ N R . By Lemma 3.3.1, we know that ϕ ⋆ η is continuous on U ′ and henceTheorem 3.1.4 shows that it is psh. (cid:3) Lemma 3.3.3. Let ϕ be a continuous real function on U and let U ′ be a relatively compactopen subset of U . Then there exists a sequence of smoothing kernels ( η k ) k ≥ for ϕ and U ′ such that ( ϕ ⋆ η k ) k ≥ converges pointwise to ϕ on U ′ . If ϕ is psh, then the convergence ismonotone decreasing. ROPICAL TORIC PLURIPOTENTIAL THEORY 19 Proof. Let η and W be as in the proof of Lemma 3.3.1. For k ≥ 1, we consider thesmoothing kernels η k ( x ) = k n η ( kx ). Since W is convex and symmetric, the support of η k is also contained in W . On N R , the measures η k dx converge weakly to the Dirac deltameasure δ centered at zero. Since ϕ is constant towards the boundary, ( ϕ⋆η k ) k ≥ convergespointwise to ϕ on U ′ . If ϕ is psh, then Lemma 3.3.2 shows that ϕ ⋆ η k is psh and ≥ ϕ ,hence ϕ ⋆ η k and ϕ restrict to convex functions on U ′ ∩ N R and we get( ϕ ⋆ η k )( x ) = Z R n ϕ ( x − y ) k n η ( ky ) dy = Z R n ϕ (cid:16) x − yk (cid:17) η ( y ) dy ≤ Z R n (cid:18) k ϕ ( x ) + k − k ϕ (cid:16) x − yk − (cid:17)(cid:19) η ( y ) dy = 1 k ϕ ( x ) + k − k ( ϕ ⋆ η k − )( x ) ≤ ( ϕ ⋆ η k − )( x )for x ∈ U ′ ∩ N R , k ≥ 2. Hence the convergence is monotonically decreasing on U ′ . (cid:3) The next Theorem gives local regularization of psh functions. In § global regularization of θ -psh functions. Theorem 3.3.4. Let Σ be a fan and let U be an open subset of N Σ . A function ϕ : U → R −∞ is psh if and only if ϕ is locally a decreasing limit of smooth psh functions.Proof. By Proposition 3.1.3, any decreasing limit of psh functions is psh. Conversely, weassume that ϕ is psh. The claim is local and so we may assume U connected. The claimis obvious for ϕ ≡ −∞ and so we may assume that ϕ is not identically equal to −∞ . Thedifficulty is that our definition of a smooth function on U means that it is a smooth functionon U ∩ N R which is constant towards the boundary, therefore a function ϕ with ϕ ◦ tropsmooth is not necessarily smooth on U .By Proposition 3.1.3 the function ϕ can be approximated monotonically decreasing bythe finite psh functions ϕ M := max( ϕ, − M ) for M ∈ N .Now we show that a finite function ϕ is locally a decreasing limit of continuous pshfunctions which are constant towards the boundary. So let us check this in p ∈ U ∩ N ( σ ).Passing to an open neighbourhood, we may assume that Σ contains a single maximal cone σ and that U satisfies the first condition in (3.11) for V ( p ) = U . Fix ε > ε σ := ε .For each τ ∈ Σ with τ ≺ σ , we fix ε τ > ε τ < ε τ whenever τ ≺ τ . Since weare assuming that Σ contains a single maximal cone, there is a unique continuous extension π τ : N Σ → N ( τ ) of the quotient map N R → N ( τ ). Using that π τ ( U ) = U ∩ N ( τ ), we define g : U −→ R , x max τ ≺ σ ϕ ( π τ ( x )) + ε τ . Since π τ is an equivariant morphism of tropical toric varieties and ϕ is psh, Proposition3.1.10 shows that ϕ ◦ π τ is psh and hence g is psh by Proposition 3.1.3. Using that ϕ isdecreasing towards the boundary, we have | ϕ − g | ≤ ε . Since ϕ is a finite function which iscontinuous by Theorem 3.1.4, the choice of the family ε τ shows that g is constant towardsthe boundary. Choosing ε k and ε σ,k as above converging monotonically to zero for k → ∞ ,we obtain a sequence of continuous psh functions which are constant towards the boundaryand converge monotonically decreasing to ϕ .Now assume that ϕ is a finite psh function which is constant towards the boundary. Weprove the claim for ϕ locally in p ∈ U . We pick a relatively compact open neighbourhood U ′ of p in U . Then applying the sequence of smoothing kernels η k from Lemma 3.3.3 wesee that ϕ is a decreasing limit of smooth psh functions on U ′ .Finally, applying a standard diagonal argument based on Dini’s theorem, we deduce fromthe above steps that locally any psh function ϕ is a limit of a decreasing sequence of smoothpsh functions. (cid:3) Bedford Taylor calculus on a partial compactification In this section, Σ denotes a smooth fan in N R for a free abelian group N of rank n . Thegoal is to develop a Bedford–Taylor calculus on the partial compactification N Σ using theBedford Taylor calculus on the complex manifold X anΣ . At the end, functoriality will helpus to construct Bedford–Taylor products for psh functions also for non-smooth fans.4.1. Locally bounded case. Let U be an open subset of N Σ . Let D p,p cl , + ( U ) denote thespace of positive Lagerberg currents on U of bidegree ( p, p ) which are closed with respectto the differentials d ′ and d ′′ . Let u , . . . , u q be smooth psh functions on U and let T ∈ D p,p ( U ) cl , + with p + q ≤ n . From the calculus of smooth Lagerberg forms and currents on U (see § u d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T ∈ D p + q − ,p + q − ( U ) , (4.1) d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T ∈ D p + q,p + q cl , + ( U ) . (4.2)The second type of currents is called Monge-Amp`ere currents . We want to extend theconstruction of the products (4.1) and (4.2) to locally bounded psh functions. By Theorem3.1.4, a psh function is locally bounded if and only if it is finite continuous. Remark 4.1.1. For smooth psh functions u , . . . , u q on U and a current T as above, wewrite v j = trop ∗∞ ( u j ) for j = 1 , . . . , q . By Proposition 3.2.1, the functions v j are psh on V = trop − ∞ ( U ). By the main theorem in [BGJK20] recalled in 2.2.6, there is a unique closedpositive current S ∈ D p,p ( V ) S ,F with trop ∞ , ∗ ( S ) = T . The projection formula [BGJK20,Proposition 5.1.7] and (2.5) imply u d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T = trop ∞ , ∗ ( v dd c v ∧ . . . ∧ dd c v q ∧ S ) ,d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T = trop ∞ , ∗ ( dd c v ∧ . . . ∧ dd c v q ∧ S ) . Theorem 4.1.2. For any open subset U of N Σ , locally bounded psh functions u , . . . , u q on U and T ∈ D p,p ( U ) cl , + , there are unique Lagerberg currents u d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T ∈ D p + q − ,p + q − ( U ) ,d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T ∈ D p + q,p + q cl , + ( U ) such that these products are given locally in U and satisfy the following two properties:(i) If u , . . . , u q are smooth, then the product currents agree with (4.1) and (4.2) .(ii) If there are psh functions { u j,k } k ≥ converging locally uniformly to u j for j =1 , . . . , q and if there are closed positive Lagerberg currents { T k } k ≥ converging weaklyto T , then we have the weak convergence of Lagerberg currents u ,k d ′ d ′′ u ,k ∧ . . . ∧ d ′ d ′′ u q,k ∧ T k k →∞ −−−→ u d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T,d ′ d ′′ u ,k ∧ . . . ∧ d ′ d ′′ u q,k ∧ T k k →∞ −−−→ d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T. Proof. For j = 1 , . . . , q , Proposition 3.2.1 shows that v j := trop ∗∞ ( u j ) is psh on V :=trop − ∞ ( U ). As seen in Remark 4.1.1, there is a unique closed positive S ∈ D p,p ( V ) S ,F withtrop ∞ , ∗ ( S ) = T . In [Dem12, § III.3] there is an inductive definition of v dd c v ∧ . . . ∧ dd c v q ∧ S ∈ D p + q − ,p + q − ( V ) and dd c v ∧ . . . ∧ dd c v q ∧ S ∈ D p + q,p + q cl , + ( V ). Then we define u d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T := trop ∞ , ∗ ( v dd c v ∧ . . . ∧ dd c v q ∧ S ) d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T := trop ∞ , ∗ ( dd c v ∧ . . . dd c v q ∧ S ) . This definition is local in U as the complex construction is local in V . By Remark 4.1.1,we get (i) . By Theorem 3.1.4, any psh function on U is continuous and hence (ii) followsfrom the corresponding fact on complex manifolds given in [Dem12, Corollary III.3.6]. It ROPICAL TORIC PLURIPOTENTIAL THEORY 21 follows from the regularization in Theorem 3.3.4 that these two properties and locality in U characterize the product currents uniquely. (cid:3) The unbounded case. We will now consider a version of the previous theorem forunbounded psh-functions. Similarly as in [Dem12, § T k ) by one current T . Additionally,the loci of unboundedness of the psh functions and supp( T ) have to intersect properly.We still assume that Σ is a smooth fan in N R and that U is an open subset of N Σ . A stratum of U is a connected component of U ∩ N ( σ ) for some σ ∈ Σ. A strata subset of U is a union of strata of U . Given a psh function u on U , we will write L ( u ) := { x ∈ U | u ( x ) = −∞} . By Theorem 3.1.4 and Lemma 3.1.7, the set L ( u ) is a closed strata subset of U .In the following definition, the closed strata subsets D , . . . , D q should morally be viewedas supports of divisors. Definition 4.2.1. Let B be a closed strata subset of U and let D , . . . , D q be closed stratasubsets of U of codimension at least one. For d ∈ N , we say that D , . . . , D q intersect Bd -properly if for any subset I ⊂ { , . . . , q } we have(4.3) dim B ∩ \ j ∈ I D j ≤ d − | I | Note that for I = ∅ , this gives that the dimension of B is bounded by d . Theorem 4.2.2. Let D , . . . , D q be closed strata subsets of U of codimension at least onewhich intersect the closed strata subset B of U ( n − p ) -properly for some p ∈ N . For j ∈ { , . . . , q } , let u j be a psh function on U with L ( u j ) ⊂ D j and let T ∈ D p,p cl , + ( U ) with supp( T ) ⊂ B . Then there are product currents u d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T ∈ D p + q − ,p + q − ( U ) ,d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T ∈ D p + q,p + q cl , + ( U ) that agree with the ones in Theorem 4.1.2 when the functions u j are locally bounded. Theformation of the product currents is local in U . If there are decreasing sequences { u j,k } k ≥ of psh functions converging pointwise to u j for j = 1 , . . . , q , then we have u ,k d ′ d ′′ u ,k ∧ . . . ∧ d ′ d ′′ u q,k ∧ T k →∞ −−−→ u d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T,d ′ d ′′ u ,k ∧ . . . ∧ d ′ d ′′ u q,k ∧ T k →∞ −−−→ d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T as weak convergence of currents. These conditions determine the product currents uniquely.Proof. Let V , v j and S be as in the proof of Theorem 4.1.2. For every I ⊂ { , . . . , q } , theset B ∩ T j ∈ I D j is a closed stata subset of U . If N ( σ ) is a stratum of N Σ , thendim R (trop − ( N ( σ ))) = 2 dim C (trop − ( N ( σ ))) = 2 dim R ( N ( σ ))and hence our Assumption (4.3) yieldsdim R trop − B ∩ \ j ∈ I D j ≤ n − p − | I | < n − p − | I | + 1 . Therefore, the functions v j and the current S satisfy the hypotheses of [Dem12, TheoremIII.4.5], where the role of p and n − p is interchanged. It follows that the currents v dd c v ∧ . . . ∧ dd c v q ∧ S ∈ D p + q − ,p + q − ( V ) ,dd c v ∧ . . . ∧ dd c v q ∧ S ∈ D p + q,p + q cl , + ( V )are well defined. As in Theorem 4.1.2, we define the product currents on U as the directimage of the corresponding currents in V with respect to trop : V → U . The continuityresult is a consequence of the weak continuity of trop ∗ and [Dem12, Theorem III.4.5].Uniqueness of the product currents follows again from the regularization in Theorem 3.3.4and from locality in U . Since the current dd c v ∧ . . . ∧ dd c v q ∧ S is closed and positive, thesame is true for d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T by [BGJK20, Proposition 5.1.13]. (cid:3) Remark 4.2.3. In the above proof, we have seen that the tropical Bedford–Taylor productsof the psh functions u , . . . , u q and T ∈ D p,p cl , + ( U ) satisfying the hypotheses in Theorem 4.2.2are compatible with the Bedford–Taylor products of the psh functions v j := u j ◦ trop ∞ onthe complex manifold V = trop − ∞ ( U ) and the unique closed positive S ∈ D p,p ( V ) S ,F withtrop ∞ , ∗ ( S ) = T in the sense that we have u d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T = trop ∞ , ∗ ( v dd c v ∧ . . . ∧ dd c v q ∧ S ) d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T = trop ∞ , ∗ ( dd c v ∧ . . . dd c v q ∧ S ) . Functoriality. Let N, N ′ be free abelian groups of finite rank n and n ′ respectively. Asin Definition 2.2.4, we consider a homomorphism L : N ′ → N and a proper L -equivariantmorphism E : N ′ Σ ′ → N Σ of tropical toric varieties for fans Σ and Σ ′ in N R and N ′ R ,respectively. Let U be an open subset of N Σ and U ′ := E − ( U ). Lemma 4.3.1. For T ∈ D r,s ( U ′ ) and α ∈ A p,q ( U ) we have α ∧ E ∗ T = E ∗ ( E ∗ α ∧ T ) ∈ D r + n − n ′ + p,s + n − n ′ + q ( U ) . Proof. This follows directly from testing against a form β ∈ A n ′ − r − p,n ′ − s − qc ( U ). (cid:3) By Proposition 3.1.10, the pull-back of a psh function on U with respect to E is psh. Proposition 4.3.2. In the above setting, assume that Σ , Σ ′ are smooth fans. Let T be aclosed positive current on U ′ and let u , . . . , u q be locally bounded psh-functions on U . Then E ∗ (cid:0) E ∗ ( u ) d ′ d ′′ E ∗ ( u ) ∧ . . . ∧ E ∗ ( u q ) ∧ T (cid:1) = u d ′ d ′′ u ∧ . . . ∧ u q ∧ E ∗ T Proof. When the u i are smooth, this follows from Lemma 4.3.1. In general, we use Theorem3.3.4 to approximate the u i locally by decreasing sequences of smooth psh functions. Thenwe use locality and continuity of the product in Theorem 4.1.2 and of E ∗ . (cid:3) Remark 4.3.3. In the same setting, the projection formula in Proposition 4.3.2 holdsmore generally for unbounded psh-functions u , . . . , u q on U if we assume L ( u j ) ⊂ D j andsupp( T ) ⊂ B for strata subsets D j of codimension at least one in U and a strata subset B of U ′ such that E − ( D ) , . . . , E − ( D q ) intersect B ( m − p )-properly.Indeed, then E ∗ ( T ) ∈ D n − m + p,n − m + p ( U ) and D , . . . , D q intersect the closed stata subset E ( B ) ( m − p )-properly. The projection formula follows as above by using Theorem 4.2.2. Remark 4.3.4. We can use functoriality to define Bedford–Taylor products also in thecase of a non-smooth fan Σ in N R . Let u , . . . , u q be locally bounded psh functions on theopen subset U of N Σ . By toric resolution of singularities, there is a smooth fan Σ ′ in N R refining Σ. The associated morphism E : N Σ ′ → N Σ is a proper id N -equivariant morphism ROPICAL TORIC PLURIPOTENTIAL THEORY 23 of tropical toric varieties and we define the product currents by u d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q := E ∗ (cid:0) E ∗ ( u ) d ′ d ′′ E ∗ ( u ) ∧ . . . ∧ E ∗ ( u q ) (cid:1) ∈ D q − ,q − ( U ) ,d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q := E ∗ (cid:0) E ∗ ( u ) d ′ d ′′ E ∗ ( u ) ∧ . . . ∧ E ∗ ( u q ) (cid:1) ∈ D q,q cl , + ( U ) . It follows from Proposition 4.3.2 that the definition of the product currents does not de-pend on the choice of Σ ′ . Obviously, the above product currents are still local in U andare continuous along monotonically decreasing sequences of psh functions. Moreover, theprojection formula in Proposition 4.3.2 holds also for not necessarily smooth fans.5. Semipositive metrics and θ -psh functions We recall properties of semipositive metrics on line bundles over complex manifolds ornon-archimedean analytic spaces and discuss the equivalent concept of θ -psh functions fora closed first Chern current θ of the line bundle. We also introduce θ -psh functions ontropical toric varieties. For toric varieties, tropical θ -psh functions serve as a link betweeninvariant θ -psh functions in the complex and the non-archimedean situation. We provecorrespondence theorems which will later be applied to solve Monge–Amp`ere equations.5.1. The complex setting. Let L be a holomorphic line bundle on a complex manifold X . We first introduce singular psh metrics on L and θ -psh functions in this general setting.Then we restrict to the toric setting. A continuous metric k k on L is given by continuous functions − log k s k : U → R for all frames (i.e. nowhere vanishing sections) s ∈ H ( U, L ) on open subsets U of X suchthat for frames s on U and s ′ on U ′ we have log k s k − log k s ′ k = log | s/s ′ | on U ∩ U ′ .A singular metric of L is defined similarly, but without assuming continuity and withallowing the function − log k s k to take the value −∞ as well. Definition 5.1.2. A singular metric k k of L is called psh if for all local frames s of L onany open subset U of X , the function − log k s k : U → R −∞ is psh. Remark 5.1.3. If X is connected, then a singular metric k k of L is psh if and only ifeither the function − log k s k is identically −∞ or the function − log k s k is strongly usc asdefined in (2.1), locally integrable and the first Chern current c ( L, k k ) := dd c [ − log k s k ]is positive for all local frames s of L [Lel68, Th´eor`eme II.3]. Note that a smooth metricis psh if and only if the first Chern form is a positive form. Some authors would call thelatter form semipositive, but we use here the positivity notions of forms and currents givenin [Dem12, § III.1.A, § III.1.B] following also our conventions from Section 2. Definition 5.1.4. We fix a continuous reference metric k k on L and we set θ := c ( L, k k )for the associated first Chern current. A θ -psh function ϕ on X is a function ϕ : X → R −∞ such that for every connected open subset U of X , we have either ϕ | U ≡ −∞ or ϕ | U isstrongly usc, locally integrable and dd c [ ϕ | U ] + θ is a positive current on U .Observe that if k k is a smooth metric, then θ is a smooth (1 , X . Remark 5.1.5. Note that every singular metric k k on L induces a function ϕ := − log( k k / k k ) := − log( k s k / k s k ) : X → R −∞ independent of the choice of a local frame s of L . Obviously, the singular metric k k is pshif and only if the function ϕ is θ -psh. Remark 5.1.6. For a line bundle L on a complex smooth proper variety Y , there is anothersemipositivity notion for metrics on L introduced by Zhang [Zha95]: Let Y an be the complexanalytification of Y . A metric k k on L an is called semipositive , if k k is a uniform limitof smooth psh metrics k k k on L an , i.e. the Chern forms c ( h ∗ L an , h ∗ k k k ) are assumed to be positive forms. Obviously, a semipositive metric is continuous. It is easy to see that forevery semipositive metric k k on L an , the first Chern current c ( L, k k ) is a positive currenton X an and hence k k is a psh metric as in Definition 5.1.2. The converse requires existenceof global regularization and holds for L ample [Mai00, Theorem 4.6.1]. Let N be a lattice with dual lattice M . Let Σ be a smooth fan in N R with associatedsmooth complex toric variety X Σ . Recall from [BPS14, Definition 3.3.4], that a toric linebundle L on X Σ is given by a line bundle L on X Σ together with a fixed trivialization ofthe fiber of L over the origin of the generic torus T of X Σ . A meromorphic section s of atoric line bundle is called toric if it is regular and nowhere vanishing on T and the fixedtrivialization is induced by s . The choice of a toric section of a toric line bundle correspondsprecisely to the choice of a T -linearization of the toric line bundle [BPS14, Remark 3.3.6].Let s be a toric meromorphic section of a toric line bundle L on X Σ . Associated with L and s there is a piecewise linear function Ψ = Ψ( L, s ) : Σ → R , whose constructionwe now recall: The invariant Cartier divisor of s is given on the affine open subset U σ =Spec C [ σ ∨ ∩ M ] associated to the cone σ ∈ Σ by a character χ − m and then Ψ is given on σ by the corresponding linear form m ∈ M .For a continuous metric k k on L an , we call g := − log k s k the associated Green functionfor div( s ). A metric on L an is called toric if it is invariant under pull-back with respect tothe action of the compact torus S in T an .If Σ is complete, then a result of Burgos, Philippon and Sombra [BPS14, Theorem 4.8.1]shows that for a fixed toric meromorphic section s of L the invariant continuous semipositivetoric metrics on L an are in bijection to the concave functions ψ on N R with ψ = Ψ + O (1).The correspondence is given by(5.1) ψ ◦ trop = log k s k . We will generalize this correspondence below. The canonical metric on a nef toric linebundle L on the proper toric variety X Σ corresponds to the choice ψ = Ψ. This canonicalmetric does not depend on the choice of s and is continuous but not necessarily smooth.5.2. The Lagerberg setting. Let Σ be a fan in N R for a lattice N of rank n . Our goalis to transfer the notions from the previous subsection to an open subset U of the partialcompactification N Σ . As usual, we set M := Hom Z ( N, Z ).In the tropical world, there is no perfect analogue of line bundles and so we prefer toestablish the analogue of θ -psh functions. According to 5.1.7, a good way to replace theholomorphic line bundle is to fix a function Ψ : | Σ | → R which is piecewise linear withrespect to the fan Σ which means that Ψ is linear on each cone σ of Σ. We also requirethat the slope of Ψ | σ is integral, i.e. Ψ | σ = m σ for some m σ ∈ M . This makes sure thatthere is an associated line bundle L on the complex toric variety X Σ by the reverse of theconstruction in 5.1.7. This will be important for comparing to semipositive metrics in thecomplex setting. Note that Ψ might be seen as a tropical Cartier divisor on N Σ and linearequivalence is given by adding linear functions from M [AR10].The role of continuous metrics is played in the following by Green functions. Definition 5.2.1. A continuous Green function for Ψ is a function g : U ∩ N R → R suchthat for all σ ∈ Σ and all m σ ∈ M with Ψ | σ = m σ | σ , there exists an open neighbourhoodΩ σ of U ∩ N ( σ ) in U and a continuous function h σ : Ω σ → R which extends the function g + m σ | Ω σ ∩ N R . We say that g is a smooth Green function for Ψ if we may choose h σ smoothfor all σ ∈ Σ.A continuous singular Green function for Ψ is defined as above, but allowing the contin-uous function h σ to take the value −∞ . Note that the function g is still assumed to havefinite values on U ∩ N R . Therefore h σ can only take the value −∞ on U \ N R . ROPICAL TORIC PLURIPOTENTIAL THEORY 25 Remark 5.2.2. A partition of unity argument shows that there exists a smooth Greenfunction for Ψ. In case of a complete fan, a canonical continuous Green function is givenby g := − Ψ | U ∩ N R .We use in the following that a real valued continuous or more generally locally integrablefunction f on U ∩ N R induces a Lagerberg current [ f ] ∈ D , ( U ), see Remark 2.2.1. Definition 5.2.3. Let g be a continuous singular Green function for Ψ. For a cone σ ∈ Σ,we pick an open subset Ω σ and m σ ∈ M as in Definition 5.2.1 and define the first Cherncurrent c (Ψ , g ) on Ω σ by c (Ψ , g ) := d ′ d ′′ [ g + m σ ]. Since the open sets Ω σ cover U andsince the first Chern currents agree on overlappings, we get a well-defined Lagerberg current c (Ψ , g ) ∈ D , ( U ).It is easy to see that c (Ψ , g ) is a d ′ - and d ′′ -closed symmetric Lagerberg current in D , ( U ) and does not depend on the choice of the linear functions m σ .We fix a reference continuous Green function g for Ψ and we set θ := c (Ψ , g ) ∈ D , ( U ). Definition 5.2.4. A function ϕ : U → R −∞ is called θ -psh if it is strongly upper semicon-tinuous (see (3.1)) and for any connected component W of U , the function ϕ either restrictsto a locally integrable function W ∩ N R → R and d ′ d ′′ [ ϕ ] + θ is a positive Lagerberg currenton W or ϕ is identically −∞ on W .The following characterization of θ -psh functions uses the open sets Ω σ and m σ ∈ M from Definition 5.2.1 for the reference continuous Green function g instead of g . Proposition 5.2.5. A function ϕ : U → R −∞ is θ -psh if and only if ϕ + g + m σ | Ω σ ∩ N R extends to a psh function on Ω σ for all σ ∈ Σ . Hence any θ -psh function is continuous.Proof. We may assume that U is connected and that ϕ is not identically −∞ on U . ByDefinition 5.2.1, the function g + m σ extends to a continuous function h σ : Ω σ → R for any σ ∈ Σ. On Ω σ , we have d ′ d ′′ [ ϕ + h σ ] = d ′ d ′′ [ ϕ + g + m σ ] = d ′ d ′′ [ ϕ ] + θ. Since the open subsets Ω σ cover U , it follows from Theorem 3.2.2 that ϕ is θ -psh if andonly if ϕ + h σ is psh on Ω σ for every σ ∈ Σ. Now Theorem 3.1.4 yields continuity. (cid:3) Proposition 5.2.6. A function ϕ : U → R −∞ is θ -psh if and only if the restriction of ϕ + g to U ∩ N R is convex and for every point x ∈ U \ N R we have ϕ ( x ) = lim sup y ∈ U ∩ N R y → x ϕ ( y ) . Proof. Follows from Theorem 3.1.4, Theorem 3.2.2 and Proposition 5.2.5. (cid:3) Next, we deal with functoriality of θ -psh functions. Remark 5.2.7. Let L : N ′ → N be a homomorphism of free abelian groups of finite rank.Let E : N ′ Σ ′ → N Σ be an L -equivariant morphism of tropical toric varieties as in Definition2.2.4. Then Ψ ′ := Ψ ◦ L is a function on | Σ ′ | which is piecewise linear with respect to Σ ′ .There exists a unique cone σ ∈ Σ such that E ( N ′ R ) ⊂ N ( σ ). Adding to Ψ a linear functionin M , we may assume that Ψ | σ = 0. Then g ′ := g ◦ E is a continuous Green function forΨ ′ on U ′ := E − ( U ) and we set θ ′ := c (Ψ ′ , g ′ ) = E ∗ ( θ ). If we use the characterization ofpsh functions in Proposition 5.2.5, we can conclude from Proposition 3.1.10 the followingproperties of functions ϕ : U → R −∞ : (i) If ϕ is θ -psh on U , then ϕ ◦ E is θ ′ -psh on U ′ . (ii) If E is a surjective proper map and if ϕ ◦ E is θ ′ -psh on U ′ , then ϕ is θ -psh on U . Proposition 5.2.8. Let E : N Σ ′ → N Σ be a proper surjective L -equivariant morphism oftropical toric varieties, θ ′ := E ∗ ( θ ) and U ′ := E − ( U ) as above. If ϕ ′ : U ′ → R −∞ is a θ ′ -psh function, then there is a unique θ -psh function ϕ : U → R −∞ with ϕ ′ = ϕ ◦ E .Proof. Let x ∈ U . Since E is L -equivariant, the fiber E − ( x ) is isomorphic to a partialcompactification N Σ ′′ . Using that E is proper, we conclude that the fiber is compact andhence Σ ′′ is a complete fan. Let σ ∈ Σ be the unique cone with x ∈ N ( σ ). It follows fromProposition 5.2.5 that ϕ ′ + g ◦ E + m σ ◦ E extends to a psh function ψ on neighbourhoodof E − ( x ). By Proposition 3.1.10, the pull-back of ψ to N Σ ′′ is psh and hence constantby Proposition 3.1.9. We conclude that ϕ ′ is constant on each fiber and hence there is aunique ϕ : U → R −∞ with ϕ ′ = ϕ ◦ E . By Remark 5.2.7 (ii) , the function ϕ is θ -psh. (cid:3) Remark 5.2.9. If Σ is not a smooth fan, then we can apply toric resolution of singularities[CLS11, Theorem 11.1.9] to get a smooth fan Σ ′ which is a subdivision of Σ. The inducedmorphism E : N Σ ′ → N Σ of tropical toric varieties is id N -equivariant and proper. ByRemark 5.2.7, the function ϕ : U → R −∞ is θ -psh if and only if the function ϕ ′ := ϕ ◦ E is E ∗ ( θ )-psh on U ′ := E − ( U ). Moreover, we can use Proposition 5.2.8 to descend a E ∗ ( θ )-pshfunction on U ′ to a θ -psh function on U .Bedford–Taylor theory extends to θ -psh functions as follows. Similarly as in Theorem4.2.2, we consider closed strata subsets D , . . . , D q of U of codimension at least one inter-secting a closed strata subset B of U ( n − p )-properly for some p ∈ N . Theorem 5.2.10. Assume that Σ is smooth. For j = 1 , . . . , q , let g j be a continuousGreen function for the piecewise linear function Ψ j with respect to the fan Σ and let θ j := c (Ψ j , g j ) . For θ j -psh functions ϕ j with L ( ϕ j ) ⊂ D j and any T ∈ D p,p cl , + ( U ) with supp( T ) ⊂ B , there are unique product currents ( ϕ + g )( d ′ d ′′ ϕ + θ ) ∧ . . . ∧ ( d ′ d ′′ ϕ q + θ q ) ∧ T ∈ D p + q − ,p + q − ( U ) , ( d ′ d ′′ ϕ + θ ) ∧ . . . ∧ ( d ′ d ′′ ϕ q + θ q ) ∧ T ∈ D p + q,p + q cl , + ( U ) such that the products are local in U , depend only on the factors and not on the choice of ϕ j , Ψ j , g j , θ j , and agree with the product currents in Theorem 4.2.2 in case that all θ j = 0 .All properties of product currents for psh functions seen in Section 4 extend to the aboveproduct currents.Proof. Let σ ∈ Σ and for j = 1 , . . . , q let m σ,j ∈ M such that Ψ j | σ = m σ,j | σ . There is anopen neighbourhood Ω σ of U ∩ N ( σ ) in U such that for j = 1 , . . . , q we have a continuousmap h j,σ : Ω σ → R extending g j + m σ,j | Ω σ ∩ N R . By definition of θ j -psh, the function u j := ϕ j + h j,σ is psh on Ω σ . Using Theorem 4.2.2, we define product currents on Ω σ by( ϕ + g )( d ′ d ′′ ϕ + θ ) ∧ . . . ∧ ( d ′ d ′′ ϕ q + θ q ) ∧ T := u d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T, ( d ′ d ′′ ϕ + θ ) ∧ . . . ∧ ( d ′ d ′′ ϕ q + θ q ) ∧ T := d ′ d ′′ u ∧ . . . ∧ d ′ d ′′ u q ∧ T. Since a continuous function on Ω σ satisfies d ′ d ′′ [ f ] = 0 if and only if f is an affine boundedfunction on each connected component of Ω σ , we deduce that the above definition of theproduct currents on Ω σ depends only on the factors and not on ϕ j , g j and not on thechoice of m σ,j . Since the products in Theorem 4.2.2 are local and since the Ω σ cover U ,we get globally defined product currents. All the required properties are obvious from thecorresponding properties for products of psh functions in Section 4 as they can be checkedlocally. Uniqueness is clear by construction. (cid:3) Remark 5.2.11. If Σ is not necessarily smooth and Ψ j , g j , θ j are as in Theorem 5.2.10,then for locally bounded θ j -psh functions ϕ j on U , there are unique product currents( ϕ + g )( d ′ d ′′ ϕ + θ ) ∧ . . . ∧ ( d ′ d ′′ ϕ q + θ q ) ∈ D q − ,q − ( U ) , ( d ′ d ′′ ϕ + θ ) ∧ . . . ∧ ( d ′ d ′′ ϕ q + θ q ) ∈ D q,q cl , + ( U ) ROPICAL TORIC PLURIPOTENTIAL THEORY 27 such that the products are local in U , depend only on the factors and not on ϕ j and θ j ,and agree with the product currents in Remark 4.3.4 in case that all θ j = 0. This followsagain by applying locally Remark 4.3.4. All the properties of the product currents of pshfunctions seen in Section 4 extend to the products above.Now we want to compare the Lagerberg setting with the complex situation. We assumethat Σ is a smooth fan. By [BPS14, § L on X Σ together with a toric meromorphic section s on X Σ . Let V := trop − ∞ ( U ) ⊂ X anΣ . The reference continuous Green function g gives rise to a unique continuous metric k k on L an | V such that(5.2) k s k := exp ◦ ( − g ) . Let θ ∞ := c ( L, k k ) ∈ D , ( V ) be the associated first Chern current . Choosing m σ ∈ M with Ψ | σ = m σ | σ , the function g + m σ extends from U ∩ N R to a continuous real function h σ on a neighbourhood Ω σ of U ∩ N ( σ ) for any σ ∈ Σ. This corresponds to a change to anowhere vanishing section over W σ := trop − ∞ (Ω σ ) and hence θ ∞ is given on W σ by(5.3) θ ∞ = dd c [ h σ ◦ trop ∞ ] = dd c [( g + m σ ) ◦ trop ∞ ] . For θ := c (Ψ , g ), Proposition 2.2.7 and (2.5) yield that(5.4) trop ∞ , ∗ ( θ ∞ ) = θ. Theorem 5.2.12. Let Σ be a smooth fan, U an open subset of N Σ and V := trop − ∞ ( U ) .In the above notation, the following conditions are equivalent for a function ϕ : U → R −∞ :(i) The function ϕ is θ -psh on U .(ii) The function ϕ ◦ trop ∞ is θ ∞ -psh on V .(iii) The singular metric k k ϕ := e − ϕ ◦ trop ∞ k k on L an | V is psh.This induces bijections between the set of θ -psh functions on U , the set of S -invariant θ ∞ -psh functions on V and the set of singular psh toric metrics of L an | V .Proof. By Remark 5.1.5, the equivalence of (ii) and (iii) is obvious. The equivalence of (i) and (ii) can be checked locally in U . Using Proposition 5.2.5, we easily reduce the claim tothe case of psh functions. Then (i) is equivalent to (ii) by Proposition 3.2.1.By [BGJK20, Remark 3.1.3], the tropicalization map induces the identification X anΣ / S = N Σ and hence the S -invariant functions V → R −∞ correspond bijectively to the functions U → R −∞ . This makes the remaining claims obvious. (cid:3) Remark 5.2.13. Clearly, a function ϕ : U → R −∞ is continuous real valued if and onlyif ϕ ◦ trop ∞ is continuous real valued on V and the latter holds if and only if k k ϕ isa continuous metric on L an | V . It follows that Theorem 5.2.12 is a generalization of thearchimedean part of a result of Burgos, Philippon and Sombra [BPS14, Theorem 4.8.1]where the continuous case for a complete fan Σ and for U = N Σ was handled. Remark 5.2.14. We give a canonical map from the set of θ ∞ -psh functions on V to the setof θ -psh functions on U , which induces the inverse of the construction in Theorem 5.2.12:Let Φ : V → R −∞ be any θ ∞ -psh function on V := trop − ∞ ( U ). We denote by Φ av the S -invariant function on V obtained from Φ averaging over the fibers of trop ∞ with respectto the probability Haar measure on S . Since θ ∞ is S -invariant by the construction in (5.2),we conclude that Φ av is an S -invariant θ ∞ -psh function on V . Hence, there is a function ϕ : U → R −∞ with Φ av = ϕ ◦ trop ∞ . By Theorem 5.2.12, the function ϕ is θ -psh. Remark 5.2.15. Suppose that we have chosen a smooth reference Green function g andhence θ is a smooth Lagerberg form. Then our normalizations of trop ∗∞ yield thattrop ∗∞ ( θ ) = θ ∞ . Note that if ϕ is a smooth θ -psh function on U , then ϕ ◦ trop ∞ is a smooth θ ∞ -psh functionon V and hence k k ϕ is a smooth psh metric on L an | V . However, the converse is not true,i.e. a smooth psh toric metric on L an | V is not necessarily of this type as the smooth functionson U have to be constant towards the boundary. Remark 5.2.16. We have assumed through Subsection 5.2 that the piecewise linear func-tion Ψ on the fan Σ has integral slopes. The reason was to get an induced line bundle L on X Σ . If one is willing to abandon this geometric point of view working exclusivelywith θ ∞ -psh functions on V for θ ∞ defined by (5.3), then the equivalence of (i) and (ii) inTheorem 5.2.12 holds also without the assumption that the slopes of Ψ are integral.5.3. The non-archimedean setting. We consider a non-archimedean field K with val-uation v := − log | | . We will study psh metrics and θ -psh functions in non-archimedeangeometry. Let X be a boundaryless separated K -analytic space of pure dimension n . Ob-serve that X is then a good K -analytic space whose topology is Hausdorff [Tem15, 4.2.4.2].We define continuous metrics and singular metrics on a line bundle L over X as in thecomplex case in 5.1.1. We will use forms and currents on X introduced in § As in complex geometry, a continuous metric on L is given by continuous functions − log k s k : U → R for all frames s on open subsets U of X such that for frames s on U and s ′ on U ′ we have log k s k − log k s ′ k = log | s/s ′ | on U ∩ U ′ . A singular metric is definedsimilarly, but without assuming continuity and with allowing the function − log k s k to takethe value −∞ . A continuous singular metric is a singular metric such that the functions − log k s k : U → R −∞ are continuous for all frames s of L on open subsets U of X . A Borel measurable function f : X → R ∪ {±∞} is called locally integrable , if R X | f | η < ∞ for all positive forms η ∈ A n,nc ( X ). Hence a locally integrable function f yields a well-defined current [ f ] on X . A singular metric k k on L is called locally integrable if the functions − log k s k : U → R −∞ are locally integrable for all frames s of L on opensubsets U of X .For a locally integrable singular metric k k on L , the first Chern current c ( L, k k ) ∈ D , ( X ) is defined similarly as in complex geometry. Locally for a frame s of L over U , it isgiven by c ( L, k k ) | U = d ′ d ′′ [ − log k s k ]. For a smooth metric, we get a smooth Chern form.Pluripotential theory in non-archimedean geometry is not yet as well developed as incomplex geometry. In [CD12, § −∞ . We will apply this generalization inthe toric setting and there we expect that invariant psh functions are always of this formfor all reasonable notions of psh functions. We refer to [BFJ16] for a global approach to θ -psh functions in case of a discretely valued field of residue characteristic zero where thefunctions are only assumed to be usc. Definition 5.3.3. We fix a continuous reference metric k k on L and set θ := c ( L, k k )as a current on X . Let U ⊂ X be an open set. A continuous function f : U → R −∞ iscalled θ -psh on U if for every connected component C of U , the restriction f | C is eitheridentically −∞ or locally integrable and d ′ d ′′ [ f | C ]+ θ is a positive current on C .We say that a function f : X → R −∞ is psh if it is θ -psh for θ = 0. A continuous singular metric k k of L is called psh if for any frame s of L on anopen U ⊂ X the function − log k s k is psh on U . A smooth metric k k is psh if and onlyif c ( L, k k ) is a positive form. Similarly to Remark 5.1.5, our choice of a reference metric k k leads to a bijection of the set of continuous singular psh metrics of L onto the set ofcontinuous θ -psh functions X → R −∞ . ROPICAL TORIC PLURIPOTENTIAL THEORY 29 Remark 5.3.5. Following [CD12, § L is called locally pshapproximable if it is locally a uniform limit of smooth psh metrics on L . It is obvious thata locally psh approximable metric is psh in the sense of 5.3.4. Remark 5.3.6. If L is a line bundle on a proper variety Y over K , then Zhang [Zha95]introduced semipositive metrics of L as uniform limits of metrics of L induced by semipos-itive model metrics (see [GM19] for details). Here, we call k k a model metric on L if thereis a k ∈ N \ { } such that k k ⊗ k is a metric of L ⊗ k induced by a line bundle L on a properscheme Y over the valuation ring K ◦ such that( Y, L ⊗ k ) = ( Y ⊗ K ◦ K, L | Y ⊗ K ◦ K ) . The model metric is called semipositive if L is nef. By construction, semipositive metricsare continuous on Y an where the latter denotes the analytification of Y as a Berkovichspace. We note that Y an is a good K -analytic space whose topology is Hausdorff. Let X Σ be a toric variety over K induced by a fan Σ in N R for the lattice N . In 2.3.2,we have defined the tropicalization map trop v : X anΣ → N Σ in the non-archimedean setting.Again, a toric line bundle L with toric meromorphic section s is given by a piecewise linearfunction Ψ on the fan Σ such that Ψ has integral slopes. Here and in the following, wealways refer to [BPS14] for more details about the toric setting.Let T be the dense torus of X anΣ with character lattice M = Hom Z ( N, Z ). Then we havethe compact torus S := S v := { x ∈ T an | | χ m ( x ) | = 1 ∀ m ∈ M } in T an . We note that N Σ is the canonical skeleton of X anΣ given as follows. The maptrop v : X anΣ → N Σ has a canonical section ι : N Σ → X anΣ which identifies N Σ homeomorphi-cally with a closed subset of X anΣ and then trop v is a strong deformation retraction.For any open subset U of N Σ and V := trop − ( U ), we get from 2.3.3 that(5.5) trop ∗ v : C ( U ) −→ C ( V ) S , f f ◦ trop v is an isomorphism from the space of continuous functions on U onto the space of S -invariantcontinuous functions on V and the inverse is given by restriction to U = V ∩ N Σ . Obviously,the same holds for the spaces of usc functions on U and V .If Σ is a complete fan and if the valuation on K is non-trivial, then the S -invariantsemipositive toric metrics on L are in bijection to the concave functions ψ on N R with ψ = Ψ + O (1) by a result of Burgos, Philippon and Sombra (see [BPS14, Theorem 4.8.1]for discrete valuations and [GH17, Theorem II] for algebraically closed non-trivially val-ued non-archimedean fields which implies the general case by [GM19, Lemma 3.3]). Thecorrespondence is given by ψ ◦ trop v = log k s k . (5.6)The canonical metric on a nef toric line bundle L on the proper toric variety X Σ correspondsto the choice ψ := Ψ and again this metric is not necessarily smooth. Remark 5.3.8. Let Σ be any fan in N R . For an open subset U of N Σ , we set V :=trop − v ( U ). It follows from the definition of integration of top dimensional forms on V that(5.7) trop v, ∗ ( δ V ) = δ U for the currents of integration over V and over U . If ω is any smooth Lagerberg form of type( p, p ) on U , then ω is positive on U if and only if trop ∗ v ( ω ) is positive on V . To prove this,we note ω positive obviously implies trop ∗ v ( ω ). On the other hand, assume that trop ∗ v ( ω ) ispositive. By autoduality of positivity [BGJK20, Corollary 2.2.5], it is enough to show thatthe current [ ω ] is positive which follows from the other direction and (5.7). In the following, we fix a fan Σ in N R for a lattice N of rank n . Let ( L, s ) be atoric line bundle on X Σ inducing the piecewise linear function Ψ as in 5.3.7. Let U be anopen subset of N Σ and let V := trop − v ( U ). As before Definition 5.2.4, we fix a referencecontinuous Green function g for Ψ and set θ := c (Ψ , g ) ∈ D , ( U ). Our choices alsodetermine a continuous reference metric k k on L an | V by requiring that − log k s k = g .Then we set θ v := c ( L, k k ) as the corresponding first Chern current on V . Proposition 5.3.10. Under the hypotheses 5.3.9, let ϕ v be a θ v -psh function on V and let ϕ be the restriction of ϕ v to the canonical skeleton V ∩ N Σ = U .(i) We have trop v, ∗ ( θ v ) = θ .(ii) If k k is a smooth metric, then θ ∈ A , ( U ) and trop ∗ v ( θ ) = θ v .(iii) The function ϕ is θ -psh on U .(iv) If ϕ v | C 6≡ −∞ for any connected component C of V , then [ ϕ ] = trop v, ∗ ([ ϕ v ]) .Proof. If the metric k k is smooth, then (ii) follows easily from the definitions and in thisspecial case we deduce (i) from (ii) and (5.7). In general, θ has locally a finite continuouspotential ρ with d ′ d ′′ [ ρ ] = θ . By [BGJK20, Corollary 3.2.13], we have locally a uniformapproximation of ρ by smooth functions and so we deduce (i) from the smooth case.To prove the remaining claims, we may assume that ϕ v is not identically −∞ on anyconnected component of V . Since ϕ is the restriction of ϕ v , it is clear that ϕ is continuous.For α ∈ A n,nc ( U ), the ( n, n )-form trop ∗ v ( α ) has support in the canonical skeleton V ∩ N Σ byProposition 2.3.6 and hence the associated measure also has support in V ∩ N Σ . We get Z V ϕ v trop ∗ v ( α ) = Z V ∩ N Σ ϕ trop ∗ v ( α ) = Z V ∩ N Σ trop ∗ v ( ϕα ) = Z V trop ∗ v ( ϕα ) = Z U ϕα using (5.7) as above for the last equality. We get [ ϕ ] = trop ∗ ([ ϕ v ]) proving (iv) . Note that d ′ d ′′ [ ϕ ] + θ = d ′ d ′′ trop v, ∗ [ ϕ v ] + trop v, ∗ ( θ v ) = trop v, ∗ ( d ′ d ′′ [ ϕ v ] + θ v ) . Since trop v, ∗ maps positive currents to positive Lagerberg currents (use duality in Remark5.3.8), Theorem 3.2.2 yields that d ′ d ′′ [ ϕ ] + θ is positive and hence ϕ is θ -psh. (cid:3) Theorem 5.3.11. Under the hypotheses 5.3.9, the following conditions are equivalent fora function ϕ : U → R −∞ .(i) The function ϕ is θ -psh on U .(ii) The function ϕ ◦ trop v is a continuous θ v -psh function V → R −∞ .(iii) k k ϕ := e − ϕ ◦ trop v k k is a continuous singular psh metric on L an | V .This induces bijections between the set of θ -psh functions U → R −∞ , the set of S -invariantcontinuous θ v -psh functions V → R and the set of toric continuous singular psh metrics of L an | V . Moreover, the following are equivalent:(i’) The function ϕ is finite and θ -psh on U .(ii’) The function ϕ ◦ trop v is finite, θ v -psh and continuous on V .(iii’) The metric k k ϕ := e − ϕ ◦ trop v k k on L an | V is a continuous psh metric.(iv’) The metric k k ϕ is locally psh approximable on L an | V .If the valuation on K is non-trivial, Σ is complete and U = N Σ (hence V = X anΣ ), then(i’)–(iv’) are also equivalent to:(v’) k k ϕ is a continuous semipositive metric on L an .Proof. It follows from Remark 5.3.4 that our choice of a continuous reference metric k k on L an induces a bijection from the set of continuous θ v -psh functions V → R −∞ onto theset of continuous singular psh metrics on L an | V proving the equivalence of (ii) and (iii) aswell as the equivalence of (ii’) and (iii’) . We will show now the equivalence of (i) and (ii) which also proves the equivalence of (i’) and (ii’) . ROPICAL TORIC PLURIPOTENTIAL THEORY 31 Assume first that ϕ ◦ trop v is a continuous θ v -psh function V → R −∞ . Then Proposition5.3.10 shows that ϕ = ϕ ◦ trop v | V ∩ N Σ is θ -psh on U = V ∩ N Σ . Conversely, we assumethat ϕ is θ -psh on U . We want to prove that ϕ ◦ trop v is θ v -psh on V . Similarly as in theproof of Theorem 5.2.12, this can be checked locally and again we just have to prove that ϕ ◦ trop v is psh on V for any psh function ϕ on U . Since ϕ is continuous by Theorem 3.1.4and since trop v is continuous, it is clear that ϕ ◦ trop v is continuous. We may assume that U is connected and that ϕ is not identically −∞ . Then ϕ is a finite continuous functionon U ∩ N R . Since any ( n, n )-form on V has support in T an ∩ V as the boundary is oflower dimension (see [CD12, Lemme 3.2.5] or [Gub16, Corollary 5.12]), the continuity ofthe function ϕ ◦ trop v : T an ∩ V → R yields that ϕ ◦ trop v is locally integrable (see [GK17,Proposition 6.13]). As we may argue locally in U , the regularization in Theorem 3.3.4 showsthat we may even assume that the function ϕ is a decreasing limit of smooth psh functions ϕ n on U . Since trop ∗ v ( d ′ d ′′ ( ϕ n )) = d ′ d ′′ ( ϕ n ◦ trop v ) and since the pull-back of positiveLagerberg forms with respect to trop v is positive (see Remark 5.3.8), we conclude that ϕ n ◦ trop v is psh. By the dominated convergence theorem, we deduce that d ′ d ′′ [ ϕ ◦ trop v ]is a positive current on V . This proves the equivalence of (i) and (ii) .If ϕ is a finite continuous function in the above argument, then Dini’s theorem showsthat ϕ is locally the uniform limit of the smooth psh functions ϕ n which shows that (i’) yields (iv’) . We have noted in Remark 5.3.5 that (iv’) yields (iii’) . We conclude that (i’) to (iv’) are all equivalent.The functions ϕ ◦ trop v are S -invariant which in turn is equivalent for the metrics k k ϕ to be toric. Conversely, it follows from 5.3.7 that any S -invariant continuous function ϕ v : V → R −∞ is of the form ϕ v = ϕ ◦ trop v for a unique continuous function ϕ : U → R −∞ .By Proposition 5.3.10 again, it follows that ϕ is θ -psh. We get a bijection from the set of θ -psh functions on U onto the set of S -invariant continuous θ v -psh functions V → R −∞ .We assume that the valuation v is non-trivial, Σ is complete and U = N Σ . It followsfrom the above that continuity of ϕ and k k ϕ are equivalent. For a continuous function ϕ : U → R , we deduce from the result of Burgos, Philippon and Sombra mentioned in 5.3.7that k k ϕ is semipositive if and only if ψ = log k s k ϕ = − g − ϕ is concave on N R . The lattercondition is equivalent for ϕ to be θ -psh by Proposition 5.2.6. This proves the equivalenceof (i’) and (v’) . (cid:3) Remark 5.3.12. Similarly as in Remark 5.2.16, the equivalences of (i) with (ii) and of (i’) with (ii’) hold more generally for any piecewise linear function Ψ on the fan Σ withoutassuming integral slopes.5.4. Global regularization. Our goal is to show a global regularization result for θ -pshfunctions. In this subsection, we fix a piecewise linear function Ψ on the fan Σ of N R withintegral slopes. So far, we have used Green functions only as a reference to define θ . We willshow first that the concept of θ -psh functions is equivalent to considering convex singularGreen functions for Ψ. The latter have the advantage that we can omit a reference Greenfunction and the corresponding θ which means that we do not have to impose regularityconditions on θ later during the regularization. Proposition 5.4.1. Let g be a continuous Green function for Ψ on a connected opensubset U of N Σ and let θ := c (Ψ , g ) . Then the map ϕ → ϕ + g is an isomorphism fromthe cone { ϕ : U → R −∞ | ϕ θ -psh, ϕ 6≡ −∞} onto the cone of continuous singular Greenfunctions g : U ∩ N R → R for Ψ that are convex.Proof. Since g is a continuous Green function for Ψ, it is clear for a function ϕ : U → R −∞ with ϕ 6≡ −∞ that ϕ is continuous if and only if ϕ + g is a continuous singular Greenfunction for Ψ. If such a ϕ : U → R −∞ is continuous, then it follows from Proposition 5.2.6that ϕ is a θ -psh function if and only if ( ϕ + g ) | U ∩ N R is convex. (cid:3) Definition 5.4.2. A rational piecewise affine function f on N R is given by a finite poly-hedral complex Π with support equal to N R and with vertices in N Q such that for every τ ∈ Π, there are m τ ∈ M Q and γ τ ∈ Q with f = m τ + γ τ on τ , where M := Hom Z ( N, Z ).We start with global regularization by rational piecewise affine functions. Proposition 5.4.3. Let Ψ be a piecewise linear concave function on the complete fan Σ withintegral slopes. Then every convex continuous singular Green function for Ψ is the point-wise limit of a decreasing sequence of rational piecewise affine convex continuous Greenfunctions for Ψ .Proof. By definition of a Green function and compactness of N Σ , the function g + Ψ isbounded from above on N R . For k ∈ N , let g k := max( g, − Ψ − k ). Since g is a convexcontinuous singular Green function for Ψ and − Ψ − k is a convex continuous Green functionfor Ψ, it is clear that g k is a convex continuous Green function for Ψ with g k + Ψ bounded.Since g k + Ψ is bounded, it follows from [BPS14, Propositions 2.5.23, 2.5.24] that theconvex function g k is the uniform limit of a sequence of convex rational piecewise affinefunctions ( g k,j ) j ∈ N on N R . Obviously, we have g k,j + Ψ bounded for all k, j and hence g k,j is a continuous Green function for Ψ. As ( g k ) k ∈ N is a decreasing sequence convergingpointwise to g , and ( g k,j ) j ∈ N is a sequence converging uniformly to g k , one can find j k ∈ N and ε k ∈ R > such that the sequence ( g k,j k + ε k ) k ∈ N of rational piecewise affine continuousGreen functions for Ψ is decreasing and converges pointwise to g . (cid:3) Now we obtain our main global regularization result by smooth Green functions for Ψ. Theorem 5.4.4. Let Ψ be a piecewise linear concave function on the complete fan Σ of N R with integral slopes and let g be a convex singular Green function for Ψ . Then g is thepointwise limit of a decreasing sequence of convex smooth Green functions for Ψ .Proof. Using piecewise linear regularization in Proposition 5.4.3, it is enough to show thatevery piecewise affine convex continuous Green function g for Ψ is a uniform limit of convexsmooth Green functions. Such a function g can be written as g = max( a , . . . , a p )for affine functions a , . . . , a p on N R . For any ε > 0, we replace max by the regularizedmaximum M ε from [Dem12, Lemma I.5.18] to obtain a function g ε . The properties of theregularized maximum show that g ε is a smooth convex function on N R with | g − g ε | ≤ ε . Thisshows that g ε is a convex continuous Green function for Ψ. Since g is piecewise affine anda continuous Green function for Ψ, it is clear that g + Ψ is constant towards the boundarywhich implies that g ε is a smooth Green function in the sense of Definition 5.2.1. (cid:3) Corollary 5.4.5. Let Ψ be a piecewise linear concave function on the complete fan Σ withintegral slopes and let g be a smooth Green function for Ψ with θ := c (Ψ , g ) . Thenany θ -psh function on N Σ is the pointwise limit of a decreasing sequence of smooth θ -pshfunctions.Proof. This follows from Proposition 5.4.1 and Theorem 5.4.4. (cid:3) Toric Monge–Amp`ere equations In this section, we compare the tropical Bedford–Taylor product with the Bedford–Taylorproduct defined by Chambert–Loir and Ducros on the corresponding toric variety over anynon-archimedean field K with valuation v := − log | | . Using Bedford–Taylor products onecan define the Monge–Amp`ere operators and look at the Monge–Amp`ere equations. We willextend the correspondence theorems to Monge–Amp`ere equations comparing the complexsolutions with the tropical solutions and with the non-archimedean solutions. ROPICAL TORIC PLURIPOTENTIAL THEORY 33 Bedford–Taylor theory on a non-archimedean toric variety. Chambert-Loirand Ducros have introduced a wedge product of first Chern currents of locally psh approx-imable metrized line bundles on a boundaryless separated equidimenional Berkovich space,similar to the Bedford–Taylor product in complex differential geometry. The product isconstructed first in the case of smooth metrics where it is the wedge product of smooth firstChern forms. Psh approximable metrics are locally uniform limits of smooth psh metricsand then the Bedford–Taylor product is obtained locally as a weak limit of currents fromthe smooth case (see [CD12, § § X Σ is any toric varietyover a non-archimedean field K with associated fan Σ in N R , also U is an open subset of N Σ and V := trop − v ( U ). For j = 1 , . . . , p , we consider a toric line bundle L j on X Σ withnon-trivial toric meromorphic section s j and corresponding piecewise linear function Ψ j onΣ. We fix a continuous Green function g j, : U ∩ N R → R for Ψ j (see Definition 5.2.1) anddenote by θ j = c (Ψ j , ψ j, ) the first Chern current from Definition 5.2.3. Theorem 6.1.1. For j = 1 , . . . , q , let k k j be the locally psh approximable toric metric on L an j corresponding to the continuous θ j -psh function ϕ j : U → R by Theorem 5.3.11. Then trop v, ∗ (cid:0) c ( L , k k ) ∧ . . . ∧ c ( L q , k k q ) (cid:1) = ( d ′ d ′′ [ ϕ ] + θ ) ∧ . . . ∧ ( d ′ d ′′ [ ϕ q ] + θ q ) as positive Lagerberg currents on U . On the left hand side, we use the Bedford–Taylorproduct on the Berkovich space V introduced by Chambert–Loir and Ducros and on theright hand side, we use the product introduced in Remark 5.2.11.Proof. The claim is local in the target U of trop v , hence, by using Proposition 5.2.5 andthe multilinearity of the Bedford-Taylor product we may assume that θ = 0 and that thefunctions ϕ i are finite continuous psh functions on U . It remains to show that the identity(6.1) trop v, ∗ (cid:0) d ′ d ′′ [ ϕ ◦ trop v ] ∧ . . . ∧ d ′ d ′′ [ ϕ q ◦ trop v ] (cid:1) = d ′ d ′′ [ ϕ ] ∧ . . . ∧ d ′ d ′′ [ ϕ q ]of Lagerberg currents on U holds in this case. Arguing locally on U , the regularizationin Theorem 3.3.4 shows that we may assume that the continuous psh functions ϕ j aredecreasing limits of smooth psh functions on U . By Dini’s theorem and arguing again locallyin U , we may assume that the limit is uniform. Since the Bedford–Taylor type productof Chambert–Loir and Ducros on the left hand side of (6.1) is continuous with respect touniform convergence of psh metrics and weak limits of currents [CD12, Corollaire 5.6.5]and since Theorem 4.1.2 gives a similar kind of continuity on the right hand side of (6.1),it is enough to prove the identity (6.1) for smooth psh functions ϕ j on U . Then the claimfollows from the compatibility of trop ∗ v with d , d ′ and the product of smooth forms, theprojection formula (2.9) and equation (5.7). (cid:3) The Monge-Amp`ere equation. In this subsection, we recall some facts about thecomplex, the non-archimedean, and the real Monge-Amp`ere equation. Remark 6.2.1. Let X be either a complex manifold or a boundaryless separated Berkovichanalytic space of pure dimension n and V ⊂ X an open subset. Then to each top degreepositive current T ∈ D n,n ( V ) we can associate a positive Radon measure λ ( T ) on U and con-versely every positive Radon measure defines a positive current [CD12, Proposition 5.4.6].To avoid cumbersome notation, we will identify the spaces of positive Radon measures andthat of top degree positive currents. Similarly, if Σ is a fan, N Σ is the corresponding partialcompactification and U ⊂ N Σ is an open subset, then we can identify the space of positiveLagerberg currents on U with the space of positive Radon measures on U . We will introduce the Monge–Amp`ere measure in the three situations mentioned above.First, we do it from the local perspective restricing our attention to continuous metrics.Then we will also allow certain singular metrics in a global compact setting. Let L be a holomorphic line bundle on a complex manifold X of pure dimension n .Let k k be a continuous psh metric on L . Then Bedford–Taylor theory gives c ( L, k k ) ∧ n as a positive ( n, n ) current. We view it as a positive Radon measure on X which we callthe associated Monge–Amp`ere measure .Fix a continuous reference metric k k on L with first Chern current θ := c ( L, k k ).Using the Bedford–Taylor products for locally bounded psh functions from (2.3), we definethe Monge–Amp`ere measure(6.2) ( dd c ϕ + θ ) ∧ n := c ( L, k k ϕ ) ∧ n for any continuous or more generally bounded θ -psh function as the corresponding boundedmetric k k ϕ := e − ϕ k k on L is psh. Let L be a line bundle on a boundaryless separated Berkovich analytic space X ofpure dimension n over a non-archimedean field K with valuation v := − log | | . For a locallypsh-approximable metric k k on L , the Monge–Amp`ere measure c ( L, k k ) ∧ n is defined asa positive Radon measure by using Theorem 6.1.1.Fixing a continuous reference metric k k on L with associated first Chern current θ , wedefine ( d ′ d ′′ ϕ + θ ) ∧ n for a θ -psh function ϕ corresponding to a locally psh-approximablemetric k k ϕ similarly as in (6.2). Let Σ be a fan in N R for a free abelian group N of rank n and let U be an opensubset of N Σ . In this tropical toric situation, we stick to θ -psh functions, where Ψ is apiecewise linear function on Σ with integral slopes and θ := c ( g , Ψ) for a fixed continuousGreen function g for Ψ (see § θ -psh function ϕ : U → R , Remark 5.2.11 gives( d ′ d ′′ ϕ + θ ) ∧ n as a positive Lagerberg current of type ( n, n ) on U which we identify againwith a positive Radon measure on U called the Monge–Amp`ere measure of ϕ . Remark 6.2.5. The constructions of the Monge–Amp`ere measures in 6.2.2–6.2.4 are con-tinuous along uniformly converging sequences ([Dem12, Corollary I.3.6], [CD12, Corollaire5.6.5], Theorem 4.1.2 and Remark 5.2.11) and local in X resp. U . Remark 6.2.6. If U is an open subset of N Σ for a smooth fan Σ in N R and if ϕ is a θ -pshfunction on U as in 6.2.4, then ϕ ∞ := ϕ ◦ trop ∞ is a θ ∞ -psh function on trop − ∞ ( U ) byTheorem 5.2.12 and we have the compatiblitytrop ∞ , ∗ ( dd c ϕ ∞ + θ ∞ ) ∧ n = ( dd c ϕ + θ ) ∧ n of Monge–Amp`ere measures using Remark 4.2.3 and Theorem 5.2.10. If v is the valuationof a non-archimedean field K , then the same compatiblity holds replacing ∞ by v . This isbased on Theorem 5.3.11 and Theorem 6.1.1, and does not need that Σ is smooth.We now discuss the global Monge-Amp`ere equation starting with the complex case. Let ( X, ω ) be a connected compact K¨ahler manifold of dimension n . Recall that an ω -psh function is a strongly usc function u : X → R −∞ which is either identically −∞ oris locally integrable with dd c [ u ] + ω a positive current. Let µ be a positive Radon measureon X such that µ ( X ) = Z X ω ∧ n . We are interested in solutions of the Monge-Amp`ere equation (6.3) ( ω + dd c u ) ∧ n = µ ROPICAL TORIC PLURIPOTENTIAL THEORY 35 with u an ω -psh function. First, we will specify the class of Radon measures and the classof ω -psh functions that are allowed in that equation.Locally, ω is of the form dd c w for a smooth real function w and hence 6.2.2 gives awell-defined Monge–Amp`ere measure ( ω + dd c u ) ∧ n for any bounded ω -psh function u on X . Guedj and Zeriahi [GZ07] introduced the space E ( X, ω ) of ω -psh functions u with fullmass R X ω n on X \ { u = −∞} . More precisely, for an ω -psh function u on X one definesthe non-pluripolar Monge–Amp`ere operator by a strong convergence of measures(6.4) µ u := lim k →∞ { u> − k } (cid:0) ω + dd c (max( u, − k )) (cid:1) ∧ n using that max( u, − k ) is bounded. By Stokes’s theorem on X , one has µ u ( X ) ≤ R X ω n .Hence µ u is a positive Radon measure on X . Guedj and Zeriahi define E ( X, ω ) := n u : X → R −∞ (cid:12)(cid:12)(cid:12) u is ω -psh and µ u ( X ) = Z X ω n o . For u ∈ E ( X, ω ), the Monge-Amp`ere measure is defined by ( dd c u + ω ) ∧ n := µ u . Thecontinuous and more generally the (locally) bounded ω -psh functions on X are includedin E ( X, ω ) and the above Monge–Amp`ere measure agrees with the previously constructedMonge–Amp`ere measures on these subspaces.Guedj and Zeriahi introduced also the subspace E ( X, ω ) of E ( X, ω ) of ω -psh functionsof finite energy. It is given by all u ∈ E ( X, ω ) such that u ∈ L (( ω + dd c u ) ∧ n ). The mainresult of [GZ07] is the existence of solutions of the Monge–Amp`ere equation in the space E ( X, ω ) and uniqueness up to adding constants in the subspace E ( X, ω ). Later, Dinewproved in [Din09] uniqueness of the solution up to adding constants in E ( X, ω ). Theorem 6.2.8 ([GZ07, Din09]) . Let X be a complex K¨ahler manifold and ω a K¨ahlerform on X . Let µ be a positive Radon measure on X that does not charge any pluripolarset and satisfies µ ( X ) = R X ω n . Then there exists a function u ∈ E ( X, ω ) such that ( ω + dd c u ) ∧ n = µ. Moreover, if u and u are two solutions in E ( X, ω ) , then u − u is constant. In the case of a non-archimedean field K , we stick to a projective geometricallyintegral K -variety Y of dimension n with an ample line bundle L . We follow the globalapproach to pluripotential theory on Y an by Boucksom, Favre and Jonsson in [BFJ16] and[BFJ15] with generalizations in [BJ18]. We refer to [BJ18, Definition 2.1] for the definitionof a Fubini–Study–metric . In the non-trivially valued case, the metric k k of L is a Fubini–Study metric if and only if k k ⊗ m is a model metric associated to a globally generated modelof L , see [BE18, Theorem 5.14]. Fubini–Study metrics are globally decreasing (uniform)limits of smooth psh metrics (see [CD12, Corollaire 6.3.4], [BJ18, Lemma 2.9]) and there isalways a Fubini–Study metric of L . In the following, we fix a Fubini–Study metric k k of L as a reference metric.A possibly singular metric k k on L is called semipositive if it is the pointwise limit ofan increasing net of Fubini–Study metrics of L . As in [BJ18, Definition 5.1], we excludethe metric which is identically equal to ∞ outside the zero section (note that in loc.cit. alogarithmic notion of metrics is used). It follows from [BE18, Theorem 7.8] that for a non-trivially valued field and a for a continuous metric on L an , the above semipositivity notionagrees with the one introduced by Zhang (see Remark 5.3.6).The Monge–Amp`ere measure of a Fubini–Study metric k k is defined by 6.2.3. It isshown in [BJ18, § L ( Y ) (note that themeasures were normalized in [BJ18] to obtain probablity measures). The Monge–Amp`ere measure c ( L, k k ) ∧ n for (locally) bounded metrics k k on L satisfiesthe locality principle from [BFJ16, Theorem 5.1]. In loc.cit. , this is shown for a discretelyvalued field K of residue characteristic zero, but the arguments extend to the general caseusing that the Monge–Amp`ere operator for Fubini–Study metrics is local in the analytictopology. For any semipositive metric k k on L , we define the non-pluripolar Monge–Amp`ereoperator (6.5) µ k k := lim k →∞ {k k Through the identification of Remark 6.2.1, the Monge–Amp`ere operatorfor convex functions is given by MA( ϕ ) = ( d ′ d ′′ ϕ ) ∧ n . ROPICAL TORIC PLURIPOTENTIAL THEORY 37 This is a direct computation in the case of a C function and follows in general from regu-larization using the continuity of both sides for decreasing sequences of convex functions.Each convex function ϕ on N R has a stability set ∆( ϕ ) ⊂ M R defined as the convex set∆( ϕ ) = { x ∈ M R | the map u ϕ ( u ) − h u, x i is bounded below } . The following result solves the second boundary problem. Theorem 6.2.13. Let ∆ ⊂ M R be a convex body , i.e. a compact convex set with nonempty interior ∆ ◦ . Let µ be a positive Radon measure on N R such that µ ( N R ) = n !vol(∆) .Then there exists a convex function ϕ : N R → R such that MA( ϕ ) = µ and ∆ ◦ ⊂ ∆( ϕ ) ⊂ ∆ . Moreover, if ϕ and ϕ are two such solutions, then ϕ − ϕ is constant. The second boundary problem was originally solved by Pogorelov [Pog64]. In Bakelman’sbook [Bak94, Theorem 17.1], this was shown for positive Radon measures µ which areabsolutely continuous with respect to the Lebesgue measure. Existence of a solution in themore general situation of Theorem 6.2.13 was shown by Berman and Berndtsson [BB13,Theorem 2.19]. We add a proof for uniqueness in Appendix A.The next result will come in handy when we want to understand the solutions of thesecond boundary value problem as singular psh metrics on toric varieties.Recall that given a compact convex set ∆ ⊂ M R , its convex support function is thefunction Φ ∆ : N R → R defined as Φ ∆ ( u ) := sup x ∈ ∆ h x, u i . Lemma 6.2.14. Let ∆ ⊂ M R be a compact convex set and ϕ : N R → R a convex function.Then ∆( ϕ ) ⊂ ∆ if and only if ϕ − Φ ∆ is bounded from above.Proof. Let ϕ ∗ : ∆( ϕ ) → R be the Legendre dual to ϕ given by ϕ ∗ ( x ) = sup u ∈ N R h x, u i− ϕ ( u ).Then ϕ ∗ is a convex function and ϕ is the Legendre dual of ϕ ∗ [Roc70, Corollary 12.2.1]: ϕ ( u ) = sup x ∈ ∆( ϕ ) h x, u i − ϕ ∗ ( x ) . In particular, we have ϕ (0) = sup x ∈ ∆( ϕ ) ( − ϕ ∗ ( x )) ∈ R . Assume that ∆( ϕ ) ⊂ ∆. Then ϕ ( u ) = sup x ∈ ∆( ϕ ) h x, u i − ϕ ∗ ( x ) ≤ sup x ∈ ∆( ϕ ) h x, u i + ϕ (0) ≤ sup x ∈ ∆ h x, u i + ϕ (0) = Ψ ∆ ( u ) + ϕ (0) . Therefore ϕ ( u ) − Φ ∆ ( u ) ≤ ϕ (0) for any u ∈ N R . Conversely, if ϕ − Φ ∆ is bounded fromabove and x ∈ ∆( ϕ ), then ϕ ( u ) − h u, x i is bounded below. HenceΦ ∆ ( u ) − h u, x i = ( ϕ ( u ) − h u, x i ) − ( ϕ ( u ) − Φ ∆ ( u ))is also bounded below. So x ∈ ∆(Φ ∆ ) = ∆. (cid:3) Toric Monge-Amp`ere equations. In this subsection we will compare the real, thecomplex and the non-archimedean Monge–Amp`ere equations for toric varieties. First, wewill compare Monge-Amp`ere equations in the local case and then in the global case. Thelocal case will be applied in the next sections to solve Monge–Amp`ere equations on abelianvarieties over non-archimedean fields and the global case leads to a solution of toric Monge–Amp`ere equations over non-archimedean fields at the end of this subsection.Let Σ be a fan in N R for a lattice N of rank n . Let K be a non-archimedean field.Let us denote the valuation of K by v . We denote by X anΣ , ∞ (resp. X anΣ ,v ) the complex(resp. Berkovich) analytification of the associated toric variety over C (resp. over K ) withtropicalization map trop ∞ (resp. trop v ) onto N Σ . In the complex case, we assume for simplicity that Σ is smooth in order to have X anΣ , ∞ as a complex manifold. Let U be anopen subset of N Σ and let V ∞ := trop − ∞ ( U ) (resp. V v := trop − v ( U )).Let Ψ be a piecewise linear function on Σ with integral slopes, which comes as in 5.1.7(resp. 5.3.7) from a toric line bundle L an ∞ on X anΣ , ∞ (resp. L an v on X anΣ ,v ). We choose acontinuous Green function g : U ∩ N R → R for Ψ with Lagerberg current θ = c (Ψ , g ) on U as in § g induces a reference metric on L an ∞ (resp. L an v ) and we denote the corresponding first Chern current by θ ∞ (resp. θ v ).The Monge–Amp`ere measures introduced in 6.2.2, 6.2.3 and 6.2.4 are related as follows. Proposition 6.3.1. Let ϕ : U → R be a θ -psh function and let k k ∞ (resp. k k v ) be thecorresponding toric continuous psh metric on L an ∞ | V ∞ (resp. L an v | V v ) (see Theorems 5.2.12and 5.3.11). Then we have the identities (trop ∞ ) ∗ ( c ( L an ∞ | V ∞ , k k ∞ ) ∧ n ) = ( d ′ d ′′ ϕ + θ ) ∧ n = (trop v ) ∗ ( c ( L an v | V v , k k v ) ∧ n ) of positive Radon measures on U . Observe that for the left-hand equality we are assumingthat the fan Σ is smooth whereas for the right-hand equality we are not.Proof. The complex case follows from the projection formula for θ -psh functions with re-spect to trop ∞ , see Remark 4.2.3 and Theorem 5.2.10. The non-archimedean case followsfrom Theorem 6.1.1 and the definition of the Bedford–Taylor product on U . (cid:3) Recall that the S ∞ -invariant positive Radon measures on V ∞ are in canonical bi-jection with the positive Radon measures on U [BGJK20, Corollary 5.1.17]. We considera positive Radon measure on U and we denote the corresponding S ∞ -invariant positiveRadon measure on V ∞ by µ ∞ , i.e. we have(6.8) trop ∗ ( µ ∞ ) = µ. In the non-archimedean situation, we have seen in Remark 5.3.7 that we may view N Σ asthe canonical skeleton in X anΣ ,v and hence U may be viewed as a subset of V v . Let µ v be theimage measure of µ to V v with respect to this inclusion. Since the inclusion is proper as asection of the proper map trop v , we deduce that µ v is a positive Radon measure. Proposition 6.3.3. Under the above hypotheses, let ϕ : U → R be a θ -psh function. Let k k ∞ be the corresponding continuous toric psh metric on L an ∞ | V ∞ and let ϕ ∞ := ϕ ◦ trop ∞ be the corresponding θ ∞ -psh function on V ∞ from Theorem 5.2.12. Similarly, we picka corresponding continuous toric psh metric k k v on L an v | V v and a corresponding θ v -pshfunction on V v from Theorem 5.3.11. For the measures µ , µ ∞ and µ v from 6.3.2, thefollowing are equivalent:(i) The complex Monge–Amp`ere equation c ( L an ∞ | V ∞ , k k ∞ ) ∧ n = µ ∞ is satisfied on V ∞ .(ii) The complex Monge–Amp`ere equation ( dd c ϕ ∞ + θ ∞ ) ∧ n = µ ∞ is satisfied on V ∞ .(iii) The tropical Monge–Amp`ere equation ( d ′ d ′′ ϕ + θ ) ∧ n = µ is satisfied on U .(iv) The non-arch. Monge–Amp`ere equation c ( L an v | V v , k k v ) ∧ n = µ v is satisfied on V v .(v) The non-arch. Monge–Amp`ere equation ( dd c ϕ v + θ v ) ∧ n = µ v is satisfied on V v .The equivalences of (iii), (iv) and (v) do not require the fan Σ to be smooth.Proof. Clearly, (i) is equivalent to (ii) and (iv) is equivalent to (v) . Proposition 6.3.1 yields(trop ∞ ) ∗ (cid:0) c ( L an ∞ | V ∞ , k k ∞ ) ∧ n (cid:1) = ( d ′ d ′′ ϕ + θ ) ∧ n . Therefore (iii) follows from (i) by (6.8). Conversely, if (iii) is satisfied, then, equation (i) issatisfied after applying trop ∗ . Since the metric k k ∞ is toric, the current c ( L an ∞ | V ∞ , k k ∞ ) ∧ n is S ∞ -invariant. Since µ ∞ is also S ∞ -invariant, (i) is satisfied by [BGJK20, Corollary 5.1.17].Again by Proposition 6.3.1, we have(trop v ) ∗ (cid:0) c ( L an v | V v , k k v ) ∧ n (cid:1) = ( d ′ d ′′ ϕ + θ ) ∧ n . ROPICAL TORIC PLURIPOTENTIAL THEORY 39 Note that µ v agrees with µ on the canonical skeleton N Σ . As trop v is the identity on thecanonical skeleton, we have (trop v ) ∗ ( µ v ) = µ . We conclude that (iv) yields (iii) .Now we show that (iii) implies (iv) . Arguing as above, it is enough to show that thenon-archimedean Monge–Amp`ere measure c ( L an v | V v , k k v ) ∧ n has support in the canonicalskeleton V v ∩ N Σ = U . We may argue locally and so, by Lemma 5.2.5, after replacing ϕ by ϕ + g + m σ , we may assume that ϕ is psh on U . As a consequence the continuousfunction ϕ is by Theorem 3.3.4 a decreasing limit of smooth psh-functions ϕ j on U . ByDini’s theorem, the limit is uniform and so by construction the Bedford–Taylor product c ( L an v | V v , k k v ) ∧ n on V v is the weak limit of the currents associated to the smooth Lagerbergforms trop ∗ v (cid:0) ( d ′ d ′′ ϕ j ) ∧ n (cid:1) ∈ A n,n ( V v ). By Lemma 2.3.6 we know that such an ( n, n )-formhas support in the canonical skeleton V v ∩ N Σ = U and hence the same is true for theassociated Radon measure. Passing to the weak limit of currents, we prove the claim. (cid:3) Lemma 6.3.4. Let U be an open subset of N Σ and assume that the fan Σ is smooth. Let µ be a positive Radon measure on U such that µ ( U ∩ ( N Σ \ N R )) = 0 . Let µ ∞ denotethe invariant positive Radon measure on V := trop − ∞ ( U ) defined by µ . Then µ ∞ does notcharge pluripolar subsets of X anΣ , ∞ .Proof. By construction, µ ∞ does not charge V \ T an ∞ . Hence it is enough to check that µ ∞ does not charge any pluripolar subset D of V ∩ T an ∞ . Let D be such a pluripolar subset.Then Josefson’s Theorem (see [Jos78] or [Kli91, Theorem 4.7.4]) yields a psh function u : T an ∞ → R −∞ not identically −∞ such that D ⊂ { u = −∞} . Let U be a bounded convexopen subset of N R and V = trop − ∞ ( U ). Since a countable union of such subsets U covers N R , in order to show µ ∞ ( D ) = 0, is enough to show that µ ∞ ( { u = −∞} ∩ V ) = 0. Assumethat µ ∞ ( { u = −∞} ∩ V ) > 0. It follows from [Dem12, Corollary I.5.14] that the function ρ : U −→ R , r Z trop − ∞ ( { r } ) u ( t ) dP r ( t )is convex on the convex open subset U of N R where P r is the unique S ∞ -invariant probabil-ity measure on trop − ∞ ( { r } ). As U is bounded, we have R V udµ ∞ = R U ρdµ ∈ R . However,this integral is −∞ by our assumption µ ∞ ( { u = −∞} ∩ V ) > 0, which gives the desiredcontradiction. (cid:3) We switch to the global case. In the following, we assume that Σ is a complete fan in N R . We will see in this toric setting that the global approaches to psh functions descibedin 6.2.7 and 6.2.9 have natural combinatorial analogues. We consider a concave piecewise linear function Ψ on Σ with integral slopes and withdual polytope ∆ = { x ∈ M R | Ψ( u ) ≤ h x, u i ∀ u ∈ N R } of dimension n . This means that theinduced toric line bundle L is big and semiample on the proper toric variety X Σ . Moreover,we have deg L ( X ) = n ! · vol(∆) using the normalized volume such that vol( N R /N ) = 1.Then the canonical Green function g := − Ψ is convex and we set θ := c (Ψ , g ). The non-pluripolar Monge–Amp`ere operator of a θ -psh function ϕ : N Σ → R −∞ is defined by(6.9) µ ϕ := lim k →∞ { ϕ> − k } ( θ + d ′ d ′′ max( ϕ, − k )) ∧ n as a strong limit of measures. We omit here and in the following the case ϕ ≡ −∞ . Proposition 6.3.6. Under the hypotheses above, the following properties hold.(i) µ ϕ is a positive Radon measure of total mass µ ϕ ( N Σ ) ≤ n ! · vol(∆) .(ii) If ϕ is bounded, then µ ϕ is equal to the Monge–Amp`ere measure ( d ′ d ′′ ϕ + θ ) ∧ n from6.2.4 and we have µ ϕ ( N Σ ) = n ! · vol(∆) .(iii) µ ϕ is the image measure of the Monge–Amp`ere measure ( θ | N R + dd c ( ϕ | N R )) ∧ n on N R with respect to the inclusion j : N R → N Σ . Proof. The first claim in (ii) is obvious from the definition in (6.9). Then the second claimin (ii) follows directly from Proposition 6.3.1 if Σ is smooth and L is ample using thecorresponding claim in the complex case. In general, the second claim follows from thisspecial case by passing to a projective toric desingularization and writing the pull-back of L as a limit of ample Q -line bundles. This proves (ii) .For unbounded ϕ , we deduce from (ii) that µ ϕ ( N Σ ) ≤ vol(∆) and hence the Borelmeasure µ ϕ on N Σ is locally finite. This proves (i) .It is clear that µ ϕ and ( θ | N R + dd c ( ϕ | N R )) ∧ n agree on the bounded subsets { ϕ > − k } of N R for every k ∈ N . Since these subsets form a covering of N R , the measures agree on N R and we get (iii) . (cid:3) Definition 6.3.7. We define E ( N Σ , θ ) as the set of θ -psh functions ϕ : N Σ → R −∞ with µ ϕ ( N Σ ) = n ! · vol(∆) and E ( N Σ , θ ) := { ϕ ∈ E ( N Σ , θ ) | ϕ ∈ L ( µ ϕ ) } . For ϕ ∈ E ( N Σ , θ ),we define the Monge–Amp`ere measure ( d ′ d ′′ ϕ + θ ) ∧ n := µ ϕ . Remark 6.3.8. To compare with the global complex approach in 6.2.7, we assume thatΣ is a smooth fan and that Ψ is a strictly concave function on the complete fan Σ withintegral slopes. This means that the corresponding toric line bundle L is ample. Let θ ∞ be the first Chern current of a fixed canonical metric of the toric line bundle L an ∞ . Thereis also a smooth Fubini–Study metric on L an ∞ such that the associated first Chern form isa K¨ahler form ω on the complex manifold X anΣ , ∞ . These two reference metrics lead to anisomorphism from the space of θ ∞ -psh functions to the space of ω -psh functions. Let u bethe ω -psh function corresponding to a θ -psh function f , then we define the non-pluripolarMonge–Amp`ere operator of f by µ f := µ u and then we can subsequently adjust all thedefinitions in 6.2.7 replacing ω by θ .Let ϕ be a θ -psh function on N Σ . By Theorem 5.2.12, the corresponding S ∞ -invariantfunction ϕ ∞ := ϕ ◦ trop ∞ on X anΣ , ∞ is θ ∞ -psh. It follows immediately from the definitions ofthe non-pluripolar Monge–Amp`ere operators that µ ϕ = trop ∞ , ∗ ( µ ϕ ∞ ). By Theorem 5.2.12,the map ϕ → ϕ ∞ induces canonical isomorphisms between E ( N Σ , θ ) (resp. E ( N Σ , θ )) andthe space of S ∞ -invariant functions in E ( X anΣ , ∞ , θ ∞ ) (resp. E ( X anΣ , ∞ , θ ∞ )).We now give two proofs of the existence and uniqueness of solutions of the tropical toricMonge–Amp`ere equation. The first one uses convex analysis through Theorem 6.2.13 andthe second one uses the comparison with the complex situation. Lemma 6.3.9. Let ∆ ⊂ ∆ be bounded convex sets with the same Lebesgue measure. Thenthe interior ∆ ◦ of ∆ in N R is contained in ∆ .Proof. Assume that ∆ ◦ ∆ . Then there is a point p ∈ ∆ ◦ \ ∆ and a ball B ( p, ε )centered at p which is contained in ∆ ◦ . By the separation theorem for convex sets, thereis a non-zero affine function H such that H ( p ) ≥ H ( x ) ≤ x ∈ ∆ . The set B ( p, ε ) ∩ { H > } has positive Lebesgue measure and is contained in ∆ ◦ \ ∆ . Thereforethe Lebesgue measure of ∆ is strictly bigger than the Lebesgue measure of ∆ . (cid:3) Theorem 6.3.10. Let Σ , Ψ , ∆ and θ be as in 6.3.5. Let µ be a positive Radon measureon N Σ such that µ ( N R ) = µ ( N Σ ) = n !vol(∆) . Then there exists a θ -psh function ϕ ∈ E ( N Σ , θ ) such that ( d ′ d ′′ ϕ + θ ) ∧ n = µ . Moreover, if ϕ and ϕ are two such functions, then ϕ − ϕ is constant.Proof using convex analysis. Since µ ( N R ) = µ ( N Σ ) = n !vol(∆), the measure µ is the imagemeasure of a Radon measure µ in N R such that µ ( N R ) = n !vol(∆). By Theorem 6.2.13there exist a convex function f : N R → R such that MA( f ) = µ and ∆ ◦ ⊂ ∆( f ) ⊂ ∆. We ROPICAL TORIC PLURIPOTENTIAL THEORY 41 define the function ϕ : N Σ → R −∞ by ϕ ( x ) = lim sup y → xy ∈ N R ( f ( y ) + Ψ( y )) . Note that the values of this function belong to R −∞ because, by Lemma 6.2.14 the function f + Ψ is bounded from above. By Proposition 5.2.6, the function ϕ is θ -psh. By Proposition6.3.6, the measure µ ϕ is the image measure of ( θ | N R + dd c ( ϕ | N R )) ∧ n = MA( f ) and hence µ ϕ = µ . We conclude that µ ϕ ( N Σ ) = µ ( N Σ ) = n !vol(∆) proving ϕ ∈ E ( N Σ , θ ).For uniqueness, assume that ϕ and ϕ are θ -psh functions in E ( N Σ , θ ) which solve theMonge–Amp`ere equation. For i = 1 , 2, write f i = ϕ i | N R − Ψ. Since ϕ i is θ -psh, we geta convex function f i on N R . Since ϕ i is continuous on the compact set N Σ and does notattain the value + ∞ , it is bounded above. By Lemma 6.2.14, we have ∆( f i ) ⊂ ∆. Since ϕ i ∈ E ( N Σ , θ ), the Lebesgue measure of ∆( f i ), that agrees with the total mass of theMonge–Amp`ere measure of f i , agrees with the Lebesgue measure of ∆. By Lemma 6.3.9,we have ∆ ◦ ⊂ ∆( ϕ i ). It follows from Theorem 6.2.13 that f − f is constant. (cid:3) Proof using the complex Monge–Amp`ere equation. We assume first that Σ is a smooth fanand that Ψ is strictly concave in Σ. Then X anΣ , ∞ is a complex manifold and Ψ induces atoric meromorphic section s of the holomorphic ample toric line bundle L an ∞ on X anΣ , ∞ againby 5.1.7. The canonical Green function g = − Ψ induces the canonical metric k k ∞ , can on the toric line bundle L an ∞ and we set θ ∞ := c ( L an ∞ , k k ∞ , can ). Let µ ∞ be as in 6.3.2.Introducing a K¨ahler form ω as in 6.3.8 and applying Theorem 6.2.8, we deduce that thereis ϕ ∞ ∈ E ( X anΣ , ∞ , θ ∞ ), unique up to adding a constant, solving the complex Monge–Amp`ereequation ( dd c ϕ ∞ + θ ∞ ) ∧ n = µ ∞ . Uniqueness shows that ϕ ∞ is S ∞ -invariant. By Theorem 5.2.12, there is a θ -psh function ϕ with ϕ ∞ = ϕ ◦ trop ∞ . By Remark 6.3.8, we have ϕ ∈ E ( N Σ , θ ) and (6.8) yields ( dd c ϕ + θ ) ∧ n = µ and ϕ is uniquely determined by these conditions up to adding a constant.If Ψ is not strictly concave, but only concave, and the polytope ∆ is full dimensional,then the line L is no longer ample but it is still big and nef. Then we use the generalizationof Theorem 6.2.8 in [BEGZ10, Theorem A] and proceed in the same way as before.If Σ is not smooth, we choose a projective toric desingularisation X Σ ′ → X Σ correspond-ing to a smooth fan Σ ′ subdividing Σ. Then the pull back L ′ of L to X Σ ′ is still big and nef.Let θ ′∞ be the first Chern current of the canonical metric of ( L ′ ) an ∞ and let θ ′ := c (Ψ , − Ψ)the canonical first Chern current on N Σ ′ . Applying again [BEGZ10, Theorem A], we de-duce that there is a θ ′∞ -psh function ϕ ′∞ on X anΣ ′ , ∞ , unique up to adding a constant, solvingthe complex Monge–Amp`ere equation ( dd c ϕ ′∞ + θ ′∞ ) n = µ ′∞ for the invariant measure µ ′∞ on X anΣ ′ , ∞ induced by µ . Here, we note that µ does not charge the boundary and hencemay be seen as a Radon measure on N Σ ′ . As above, we get a θ ′ -psh function ϕ ′ on N Σ ′ solving the real Monge–Amp`ere equation ( dd c ϕ ′ + θ ′ ) ∧ n = µ and which is unique up toadding a constant. It follows from Remark 5.2.9 that ϕ ′ descends to a θ -psh function ϕ andsince the Monge–Amp`ere measures and µ do not charge the boundary, we see that ϕ solves( dd c ϕ + θ ) ∧ n = µ , and is unique up to adding a constant. (cid:3) To compare with the global approach in 6.2.9 for a non-archimedean field K withvaluation v , we do not have to assume that the fan Σ is smooth. We fix as before acontinuous Green function g for the polarization function Ψ of the toric line bundle ( L, s )and put θ = c (Ψ , g ). For a θ -psh function ϕ : N Σ → R −∞ , Theorem 5.3.11 gives acorresponding toric continuous singular psh metric k k v on L an v . We claim that k k v is alsosemipositive in the sense of the global approach of Boucksom–Jonsson explained in 6.2.9.Indeed, we apply Proposition 5.4.3 to the convex Green function g := ϕ + g . We obtaina decreasing sequence ( g n ) n ∈ N of rational piecewise affine convex Green functions with pointwise limit g . Let k k v,n be the metric on L an v corresponding to the Green function g n .Then k k v,n is by construction a Fubini–Study metric, k k v is the increasing limit of themetrics k k v,n , and hence the metric k k v is semipositive in the sense of 6.2.9.For non-pluripolar Monge–Amp`ere operators, we have µ ϕ = trop v, ∗ ( µ k k v ). The map ϕ 7→ k k v induces canonical isomorphisms between E ( N Σ , θ ) (resp. E ( N Σ , θ )) and thespace of toric metrics in E ( L an v ) (resp. E ( L an v )). Since µ k k v does not charge the boundary X anΣ ,v \ T an v , we have the useful formula(6.10) µ k k v = j v (cid:0) c ( L an v | T an v , k k v ) ∧ n (cid:1) where we use the image measure with respect to the inclusion j v : T an v → X anΣ ,v . ApplyingProposition 6.3.3 to the dense torus T and using that the real and the non-archimedean non-pluripolar Monge–Amp`ere operators do not charge the boundaries, we see that µ k k v = µ ϕ,v ,i.e. µ k k v is the push-forward measure of µ ϕ with respect to the inclusion N Σ → X anΣ ,v ofthe canonical skeleton.We illustrate the theory in the case of a projective line. Example 6.3.12. Let X Σ = P , then the associated tropical toric variety N Σ is [ −∞ , ∞ ].We pick the piecewise linear function Ψ( u ) = min( u, 0) on R . The associated toric divisoron P is [ ∞ ]. It induces the very ample toric line bundle L = O P (1). The dual polytopeis ∆ = [0 , 1] and we have deg L ( X ) = vol(∆) = 1. Then the canonical Green form is theconvex function g := max( − u, 0) on R and θ := c (Ψ , g ) is the Dirac current δ { } on[ −∞ , ∞ ]. In the following, we fix a real parameter 0 < α < ϕ ( u ) := ϕ α ( u ) := u ≤ , − u if 0 ≤ u ≤ , − u α α if 1 < u, −∞ if u = ∞ .We also allow α = 0 by using ϕ ( u ) := − log( u ) for 1 < u . The unbounded function ϕ is θ -psh and solves the real Monge–Amp`ere equation d ′ d ′′ ϕ + θ = µ for the positive Radonmeasure(6.11) µ := µ α := (1 − α )1 { u ≥ } d ′ u ∧ d ′′ uu − α on [ −∞ , ∞ ]. Note here that µ ([ −∞ , ∞ ]) = 1 and hence ϕ ∈ E ( N Σ , θ ). Basic analysis showsthat we have ϕ ∈ E ( N Σ , θ ) (i.e. ϕ integrable with respect to µ ) if and only if α < / z on P C with trop ∞ ( z ) = − log | z | . Then S ∞ is the unit circle. The canonical metric on L an ∞ induced by Ψ has firstChern current θ ∞ induced by the Haar probability measure on S ∞ . Note that θ ∞ is thepositive (1 , ∞ , ∗ ( θ ∞ ) = θ illustrating [BGJK20, Theorem 7.1.5]. The S ∞ -invariant positive Radon measure µ ∞ from 6.3.2 is here given by(6.12) µ ∞ = α − πi · {| z |≤ /e } · dz ∧ d ¯ z ( − log | z | ) − α z ¯ z . Note that µ ∞ is the unique S ∞ -invariant Radon measure with trop ∞ , ∗ ( µ ∞ ) = µ . Hence(6.12) follows from (6.11), (2.4) and the projection formula in [BGJK20, Corollary 5.1.8].For α = 0, we recover the Poincar´e metric of the punctured disk. Applying Remark 6.3.8or a direct computation show that ϕ ∞ := ϕ ◦ trop ∞ is an unbounded θ -psh function in E ( P , an C , θ ∞ ) solving the Monge–Amp`ere equation dd c ϕ ∞ + θ ∞ = µ ∞ . Moreover, we have ϕ ∞ ∈ E ( P , an C , θ ∞ ) if and only if α < / K with valuation v . Similary as in thecomplex setting, S v is the unit circle in P , an K . Let k k v, can be the canonical metric on L an v ROPICAL TORIC PLURIPOTENTIAL THEORY 43 induced by Ψ, then ϕ induces the continuous singular psh metric k k v := e − ϕ k k v, can on L an v which is in E ( L an v ) and solves the Monge–Amp`ere equation c ( L an v , k k v ) n = µ v by 6.3.11.Moreover, we have k k v ∈ E ( L an v ) if and only if α < / µ v = µ α,v has finite energy if and only if α < / 2. Indeed, if α < / ϕ v = ϕ α,v ∈ E ( L an v ) and hence µ α,v is of finite energy by [BJ18, Proposition 7.2]. It isclear that µ α,v has not finite energy for α > / θ v -psh function ϕ β,v for any β with 1 / > β ≥ − α . For α = 1 / 2, we see that µ α,v has infinite energy byintegrating against another test function g from E ( L an v ) which is defined by g ( u ) := 1 − u / (1 + log( u )) / u ≥ 1, by 1 − u for u ≤ u = −∞ at the boundary points.The example above shows that there is a finite positive Radon measures µ v on thecanonical skeleton N Σ of X anΣ ,v which does not charge the boundary N Σ \ N R and which isnot of finite energy. This explains that the following theorem goes beyond [BJ18]. Theorem 6.3.13. Let L be an ample toric line bundle on a proper toric variety X Σ over anon-archimedean field K with valuation v . Let µ v be a positive Radon measure on X anΣ ,v sup-ported in the canonical skeleton N Σ with µ v ( N Σ \ N R ) = 0 and with µ v ( X anΣ ,v ) = deg L ( X Σ ) .Then there exists a unique up to scaling toric continuous singular psh metric k k v on L an v solving the non-archimedean Monge–Amp`ere equation (6.13) c ( L an v , k k v ) ∧ n = µ v . Proof. Choosing a toric meromorphic section s of L , we have seen in 5.1.7 that ( L, s ) inducesa piecewise linear function Ψ on Σ with integral slopes. Since L is ample, the function Ψ isconcave on N R . Choosing the canonical Green function g := − Ψ, we set θ := c (Ψ , g ). Let µ be the measure on N Σ determined by µ v . It satisfies the hypothesis of Theorem 6.3.10.Hence, there exists a θ -psh function ϕ ∈ E ( N Σ , θ ) such that ( d ′ d ′′ ϕ + θ ) ∧ n = µ .Applying Proposition 6.3.3 to the dense torus T and using that the real and the non-archimedean Monge–Amp`ere measures do not charge the boundaries, we see that the pshmetric k k v on L an v corresponding to ϕ solves (6.13) and is the unique up to scaling toriccontinuous singular metric solving this non-archimedean Monge–Amp`ere equation. (cid:3) Remark 6.3.14. If the measure µ v is of finite energy, then Theorem 6.3.13 follows fromTheorem 6.2.10 of Boucksom–Jonsson which goes beyond the toric case and also gives thenuniqueness of solutions in the space E ( X anΣ ,v ). As we have seen in Example 6.3.12, there arepositive Radon measures µ v on the canonical skeleton N Σ in X anΣ ,v with µ v ( N Σ \ N R ) = 0which do not charge the boundary and which are not of finite energy. Hence Theorem6.3.13 is also a strengthening of Theorem 6.2.10 and holds without any restrictions on thenon-archimedean field K . Remark 6.3.15. The above proof shows that Theorem 6.3.13 generalizes to L big and nef.In this toric case, we can use (6.10) as a definition of the non-pluripolar Monge–Amp`ereoperator.7. Monge–Amp`ere equations on totally degenerate abelian varieties We apply our toric correspondence results to totally degenerate abelian varieties over anon-trivially valued non-archimedean field ( K, | | ). The goal is to generalize Liu’s solution ofthe non-archimedean Calabi–Yau problem [Liu11]. We denote by v := − log | | the additivevaluation of K and by Γ := v ( K × ) ⊂ R the additive valuation group. Polarized tropical abelian varieties. Recall from [FRSS18, Definition 2.4] the def-inition of a polarized tropical abelian variety. Definition 7.1.1. A polarized tropical abelian variety of dimension n with polarization ofdegree d is given by a quadruple (Λ , M, [ · , · ] , λ ) where (i) M and Λ are finitely generated free abelian groups of rank n , (ii) the pairing [ · , · ] : Λ × M → R is bilinear, (iii) the bilinear form Λ R × M R → R induced by [ · , · ] is non-degenerate, (iv) λ : Λ → M is a homomorphism with λ ) = d , (v) the induced pairing [ · , λ ( · )] : Λ × Λ → R is symmetric and positive-definite.A polarized tropical variety (Λ , M, [ · , · ] , λ ) is called Γ -rational if [Λ , M ] ⊂ Γ. A morphismof polarized tropical abelian varieties ( f, g ) : (Λ , M, [ · , · ] , λ ) → (Λ ′ , M ′ , [ · , · ] ′ , λ ′ ) is given byhomomorphisms f : Λ → Λ ′ and g : M ′ → M such that [ f ( · ) , · ] ′ = [ · , g ( · )] and λ = g ◦ λ ′ ◦ f .Let (Λ , M, [ · , · ] , λ ) be a polarized tropical abelian variety. We put N := Hom Z ( M, Z ).The natural pairing h· , ·i : N R × M R → R leads to a unique monomorphism ι : Λ → N R suchthat [ · , · ] = h ι ( · ) , ·i . Note that (iii) is equivalent to require that ι (Λ) is a lattice in N R . Fromnow on, we will identify Λ with this sublattice of N R using ι . Then we have [ · , · ] = h· , ·i onΛ × M . We write deg(Λ , M, [ · , · ] , λ ) := d := ( λ )) for the degree of (Λ , M, [ · , · ] , λ ).The associated tropical abelian variety is the real torus N R / Λ with integral structuregiven by N and we will always denote the quotient homomorphism by π : N R → N R / Λ.Tropical abelian varieties were first considered in [MZ08]. Furthermore the tropical abelianvarieties N R / Λ are tropical spaces in the sense of [JSS19] and hence Lagerberg forms aredefined on N R / Λ (see Remark 2.2.2). Remark 7.1.2. Let b ( · , · ) : N R × N R → R be the unique symmetric bilinear form whichsatisfies b ( · , · ) = [ · , λ ( · )] on Λ × Λ. The non-degeneracy in (iii) yields that λ extends to anisomorphism λ R : Λ R = N R → M R and that(7.1) b ( x, y ) = h x, λ R ( y ) i for all x, y ∈ N R . Indeed (7.1) is clear on N × Λ, shows that b ( N, Λ) ⊂ Z and extends bylinearity to N R × N R . Using that λ R is an isomorphism, we also deduce from (7.1) that thebilinear form b is non-degenerate and hence (v) shows that b is positive definite. Any basis u , . . . , u n of M R determines coordinates on N R . Let u ′ , . . . , u ′ n denotethe dual basis of N R . The positive translation invariant Lagerberg (1 , θ := θ (Λ ,M, [ · , · ] ,λ ) := n X k,l =1 b ( u ′ k , u ′ l ) d ′ u k ∧ d ′′ u l ∈ A , ( N R )does not depend on the choice of the basis. There is a unique ω ∈ A , ( N R / Λ) with π ∗ ( ω ) = θ . We call θ (resp. ω ) the canonical (1 , -form of the polarized tropical abelianvariety (Λ , M, [ · , · ] , λ ) on N R (resp. N R / Λ ) . This fits to the setting of § N Σ = N R , Ψ := 0, the Green function g ( u ) := b ( u, u ) for Ψ noting that θ = c (Ψ , g ). Choosingthe basis u , . . . , u n such that the dual basis u ′ , . . . , u ′ n is a Z -basis of Λ, one computes(7.3) Z N R / Λ ω n = Z F Λ θ n = n ! · λ ) = n ! · p deg(Λ , M, [ · , · ] , λ ) , It is not enough to assume that the canonical maps M → Hom(Λ , R ) and Λ → Hom( M, R ) are injectiveas then the images of M and N are not necessarily sublattices. A counterexample is Λ = Z + Z π ⊂ R and M = Z with pairing [ a, m ] = a ( m + νm ) where we fix ν ∈ R \ Q . Using λ ( m + m π ) = ( m , m ) ∈ M ,all other conditions are also satisfied. ROPICAL TORIC PLURIPOTENTIAL THEORY 45 where F Λ is a fundamental domain for the action of Λ on N R .7.2. Tropicalization of polarized totally degenerate abelian varieties. We workover a non-archimedean field K with valuation v and non-trivial value group Γ := v ( K × ). A totally degenerate split abelian variety over K is an abelian variety A over K suchthat there exists a morphism of analytic groups π : T an → A an for some split torus T over K , which induces an isomorphism A an = T an / Λ for a lattice Λ in T an . Here a lattice in T an means a closed discrete subgroup Λ of T ( K ) such that trop v maps Λ isomorphically onto acomplete lattice of N R , where N is the cocharacter lattice of T . If S denotes the maximalaffinoid torus in T an , then T is the split torus with character lattice M := Hom ( A , G m , )given by homomorphisms of analytic groups, where A is the maximal affinoid torus π ( S )in A an and G m , is the maximal affinoid torus in G anm .We will always identify Λ with the lattice trop v (Λ) in N R along trop v . Note that trop v induces a map trop v : A an → N R / Λ called the canonical tropicalization map of A . Moreover,there is a canonical section(7.4) ι v : N R / Λ −→ A an of trop v which identifies N R / Λ with a closed subset of A an , called the canonical skeleton ,such that trop v is a strong deformation retraction (see [Ber90, 6.5]). Let A be a totally degenerate split abelian variety over K with A an = T an / Λ as in7.2.1, let M be the character lattice of the torus T and let N := Hom Z ( M, Z ). Then[ · , · ] : Λ × M −→ R , ( γ, m ) v ( m ( γ ))is a bilinear pairing. We denote by T ∨ the torus with character lattice Λ. Restricting thecharacters of M to Λ identifies M with a lattice in ( T ∨ ) an . Then the dual abelian variety A ∨ of A has the uniformization ( A ∨ ) an = ( T ∨ ) an /M . A polarization φ : A → A ∨ induceshomomorphisms T → T ∨ and λ : Λ → M . It is verified in [FRSS18, Proposition 4.7] that(7.5) trop v ( A, φ ) := (Λ , M, [ · , · ] , λ )is a polarized tropical abelian variety. Recall that a morphism f : ( A, φ ) → ( A ′ , φ ′ ) ofpolarized abelian varieties over K is given by a homomorphism f : A → A ′ of abelianvarieties over K such that φ = f ∨ ◦ φ ′ ◦ f . Then ( A, φ ) trop v ( A, φ ) induces a functor(7.6) trop v : (cid:18) polarized totally degeneratesplit abelian varieties over K (cid:19) −→ (cid:18) Γ-rational polarizedtropical abelian varieties (cid:19) . The rank of M equals the dimension of A and the polarized tropical variety trop v ( A, φ )has the same degree as the polarized abelian variety ( A, φ ) [BL91, Theorem 6.15]. Observethat it follows from results of Bosch and L¨utkebohmert that the functor (7.6) is essentiallysurjective [BL91, § 2, Theorem 6.13]. Definition 7.2.3. Let ( A, φ ) be a polarized totally degenerate split abelian variety over K with tropicalization (7.5) . Let θ ∈ A , ( N R ) and ω ∈ A , ( N R / Λ) denote the canonical(1 , v ( A, φ ) from (7.2). We call(7.7) θ v := trop ∗ v ( θ ) ∈ A , ( T an ) and ω v := trop ∗ v ( ω ) ∈ A , ( A an )the canonical (1 , -forms of ( A , φ ). Remark 7.2.4. In the situation of 7.2.2, there is always a non-zero m ∈ N such that thepolarization mϕ is induced by an ample line bundle L on A . In fact, we can always choose m = 2 [MFK94, Proposition 6.10]. If K is algebraically closed, then we can even choose m = 1. Let π : T an → A an denote the uniformization morphism. The line bundle L an on A an carries a canonical metric k k can [GK17, Example 8.15] and we have(7.8) c ( π ∗ L an , π ∗ k k can ) = m θ v and c ( L an , k k can ) = m ω v . This follows from the description of the canonical metric in [GK17, Example 8.15]. It isshown in [BL91, Lemma 2.2] that the line bundle π ∗ L an on T an is trivial.7.3. Tropicalization of decomposed polarized complex abelian varieties. We workcomplex analytically and consider a polarized complex abelian variety ( A, H ) of dimension n . The complex manifold A an admits a uniformization V A /U A where U A is the lattice givenby the image of H ( A an , Z ) in the tangent space at the origin V A := T A an . The polarizationis given by a Riemann form H , i.e. a positive definite sesquilinear form H : V A × V A → C (antilinear in the second argument) whose imaginary part E := Im( H ) takes integral valueson the lattice U A . Recall that we can recover H from E by the formula(7.9) H ( v, w ) = E ( iv, w ) + iE ( v, w ) ( v, w ∈ V A ) . Remark 7.3.1. A decomposition for ( A, H ) is a direct sum decomposition U A = U ⊕ U for subgroups U , U of U A which are isotropic for E [BL04, Chapter 3 § Z -basis a , . . . , a n , b , . . . , b n of the lattice U A such that E is given in this basis by a matrix (cid:18) D − D (cid:19) for a diagonal matrix D = diag( d , . . . , d n ) with d j ∈ N > satisfying d j | d j +1 for j =1 , . . . , n − 1. The matrix D is uniquely determined by U A and E . We call D the type and d := det D the degree of the polarized complex abelian variety ( A, H ). Any symplecticbasis as above defines a decomposition for ( A, H ) if we put U = h a , . . . , a n i and U = h b , . . . , b n i and all decompositions come from such a symplectic basis [BL04, § Definition 7.3.2. A decomposed polarized complex abelian variety ( A, H, U , U ) is a po-larized complex abelian variety ( A, H ) together with a fixed decomposition U A = U ⊕ U .A morphism of decomposed polarized complex abelian varieties is a morphism of polarizedabelian varieties which preserves the decomposition.Decomposed polarized complex abelian varieties of dimension n and type D admit amoduli space which can be realized as a quotient of the Siegel upper half plane H n [BL04,Proposition 8.3.3].Let ( A, H, U , U ) be a decomposed polarized complex abelian variety of dimension n anddegree d . Put N := U and let T := Spec C [ M ] be the multiplicative torus with characterlattice M := Hom( N, Z ). As H is positive definite and hence E is non-degenerate, we seethat the C -span of N is V A , i.e. V A = N C := N ⊗ Z C . We use the homomorphisms e : C −→ C × , e ( z ) := exp(2 πiz ) and t : C × −→ R , z 7→ − log | z | with t ◦ e ( z ) = 2 π Im( z ) to get the maps(7.10) 2 π Im N : N C = N ⊗ Z C e N −→ N ⊗ Z C × = T an trop ∞ −→ N ⊗ Z R = N R where e N := id N ⊗ e , trop ∞ = id N ⊗ t and Im N := id N ⊗ Im. Note that e N : V A → T an maps U isomorphically onto a discrete closed subgroup Λ := e N ( U ) of T an . Moreover, Λmaps under trop ∞ isomorphically onto the complete latticetrop ∞ (Λ) = 2 π Im N ( U ) ⊂ N R . By abuse of notation similarly as in 7.2.1, we will identify Λ with the lattice trop ∞ (Λ).Note that A an = V A /U A ≃ ( V A /U ) / ( U A /U ) ≃ T an / Λ leading to the following definition. Definition 7.3.3. The complex Tate uniformization of the decomposed polarized complexabelian variety ( A, H, U , U ) is defined by A an ≃ T an / Λ. The map trop ∞ : A an → N R / Λinduced by trop ∞ : T an → N R is called the canonical tropicalization map of ( A, H, U , U ). ROPICAL TORIC PLURIPOTENTIAL THEORY 47 The complex Tate uniformization A an ≃ T an / Λ is not intrinsically associated to thepolarized abelian variety ( A, H ), it depends on the decomposition U A = U ⊕ U .We also have a non-degenerate pairing[ · , · ] : Λ × M −→ R , [ w, m ] := m (trop ∞ ( w )) ( w ∈ Λ , m ∈ M ) , and a homomorphism λ : Λ −→ M, λ ( w )( u ) := E ( v, u ) ( w = e N ( v ) ∈ Λ = e N ( U ) , v ∈ U , u ∈ N ) . Given w = e N ( v ) ∈ Λ as above, we use that N = U is isotropic for E , (7.10) and (7.9) for[ w, λ ( w )] = E ( v, trop ∞ ( w )) = 2 πE ( i Im N ( v ) , Im N ( v )) = 2 πH (Im N ( v ) , Im N ( v )) . We conclude that(7.11) trop ∞ ( A, H, U , U ) := (Λ , M, [ · , · ] , λ )is a polarized tropical abelian variety of dimension n and degree d . The latter is seen bychoosing a symplectic basis as in Remark 7.3.1. The assigment (7.11) induces a functor(7.12) trop ∞ : (cid:18) decomposed polarizedcomplex abelian varieties (cid:19) −→ (cid:18) polarized tropicalabelian varieties (cid:19) . which preserves the dimension and the degree. Proposition 7.3.4. The functor (7.12) is essentially surjective.Proof. Let (Λ , M, [ · , · ] , λ ) be a polarized tropical abelian variety. As in Subsection 7.1, weconsider Λ as a lattice in N R . We get a decomposed lattice U := U ⊕ U := N ⊕ πi Λ ⊂ N R ⊕ iN R = N C and consider the alternating map E : U × U −→ Z given by E (cid:18) x + i π a, x ′ + i π a ′ (cid:19) := b ( a, x ′ ) − b ( a ′ , x ) ( x, x ′ ∈ N, a, a ′ ∈ Λ) . We have seen after (7.1) that E has indeed values in Z . We denote the unique extension of E to a real bilinear form on N C also by E . It is given by(7.13) E ( x + iy, x ′ + iy ′ ) := 2 π ( b ( x ′ , y ) − b ( x, y ′ )) ( x, y, x ′ , y ′ ∈ N R )and hence we have E ( iz, iw ) = E ( z, w ) for z, w ∈ N C . By [BL04, Lemma 2.1.7], there is aunique hermitian form H on N C such that E = Im( H ). By (7.9) and (7.13), we get(7.14) H ( x + iy, x + iy ) = E ( i ( x + iy ) , x + iy )) = 2 π (cid:0) b ( x, x ) + b ( y, y ) (cid:1) ( x, y ∈ N R ) , which shows that H is positive definite. Hence H is a Riemann form and N C /U is theanalytification of a complex abelian variety A . Now ( A, H, U , U ) is a decomposed polarizedcomplex abelian variety which maps under trop ∞ to (Λ , M, [ · , · ] , λ ). (cid:3) Let ( A, H ) be a polarized complex abelian variety with canonical translation invariant(1 , e ω ( A,H ) := i n X k,l =1 H (cid:18) ∂∂z k , ∂∂z l (cid:19) dz k ∧ d ¯ z l ∈ A , ( V A ) . and induced K¨ahler form ω ( A,H ) ∈ A , ( A an ). Both forms do not depend on the choice ofholomorphic coordinates z , . . . , z n on V A . We call θ ( A,H ) := p ∗ ( ω ( A,H ) ) the canonical (1 , -form on the Tate uniformization p : T an → A an . We compare these forms to their tropicalanalogues from (7.2). We have introduced in 2.2.6 a natural map trop ∗∞ : A p,q ( T an ) → A p,q ( N R ) where N = Hom( U , Z ) is the cocharacter lattice of the torus T an . There is alsoan induced map trop ∗∞ : A p,q ( A an ) → A p,q ( N R / Λ). Proposition 7.3.5. Let ( A, H, U , U ) be a decomposed polarized complex abelian varietywith A an = V A /U A ≃ T an / Λ as above. We have (7.15) θ ( A,H ) = trop ∗∞ ( θ trop ∞ ( A,H,U ,U ) ) and ω ( A,H ) = trop ∗∞ ( ω trop ∞ ( A,H,U ,U ) ) . Proof. Write trop ∞ ( A, H, U , U ) = (Λ , M, [ · , · ] , λ ). Let us pick a Z -basis u , . . . , u n of M .The corresponding holomorphic coordinates z , . . . , z n on V A and w , . . . , w n on T an aregiven by trop ∗ ( u j ) = w j = exp(2 πiz j ). We have dw j = 2 πiw j dz j and we taketrop ∗ ( d ′ u j ) = − dw j √ πw j , trop ∗ ( d ′′ u j ) = − id ¯ w j √ π ¯ w j . from [BGJK20, (4.5)]. Using also that H = 2 πb on the isotropic real span of U (see (7.14)),we deduce easily the claim from the definition of the canonical forms. (cid:3) Remark 7.3.6. Observe that T an is not simply connected. In fact, it is a domain ofholomorphy and hence a complex Stein manifold. Then we can use Cartan’s theorem B tocompute the holomorphic Picard group of T an via the exponential sequence as H ( T an , Z ),which, by the K¨unneth formula, is easily seen to be non-zero.Using that H ( A an , Z ) ≃ Λ Hom( U A , Z ) (see [BL04, Lemma 2.1.3]), the first Chern classassociates to any line bundle L on A an integer valued alternating form on the lattice U A . We say that L induces the polarization H if the first Chern class of L correspondsto E = Im H . Clearly, such line bundles are ample and they are unique up to Pic ( A ).Moreover, any polarization H on A is induced by an ample line bundle [BL04, Theorem2.2.3 and Proposition 4.5.2]. Lemma 7.3.7. Let ( A, H, U , U ) be a decomposed polarized abelian variety. Let L be anample line bundle on A inducing the polarization and p : T an → A an the Tate uniformizationcorresponding to the decomposition. Then p ∗ L an is a trivial line bundle on T an .Proof. Any line bundle L an on A an has trivial pull-back to V A and hence we can write L an = ( V × C ) /U A for an action of the group U A given by U A × ( V A × C ) −→ V A × C , ( u ; v, z ) ( v + u, α u ( v ) z )for a cocycle u α u of the group U A in H ( V A , O × V A ). If L ∈ Pic ( A ), then c ( L ) is zeroas well as the corresponding alternating form E . Then there is a unique character χ of U A such that L an is given by the cocycle α u ≡ χ ( u ) [BL04, Proposition 2.2.2].Now let L be an (ample) line bundle inducing the polarization H . To show that p ∗ ( L an ) istrivial on T an , we have to prove that L an can be given by a cocycle u → α u with α u ≡ u ∈ U as then the trivialization descends to V A /U ≃ T an . It is shown in [BL04, Lemma3.2.2 and thereafter] that there is a line bunde L on A which induces the polarization H and a cocycle u e u of L which is trivial on U (note that U = Λ and U = Λ in[BL04]). This e u is the factor of automorphy of classical theta functions for L .Since L ⊗ L − ∈ Pic ( A ), there is a character χ such that u → χ ( u ) e u is a cocycle for L .There is a unique ℓ ∈ Hom( U A , Z ) such that χ ( u ) = exp(2 πiℓ ( u )) for all u ∈ U A . If wechange the trivialization to a frame h ∈ H ( V A , O × V A ), then the cocycle changes to α u ( v ) = h ( v + u ) − e u ( v ) h ( v ) ( u ∈ U A , v ∈ V A ) . Now let ℓ be the unique complex linear form on V A which agrees with ℓ on U . Then h := exp ◦ (2 πiℓ ) does the job as α u = exp( − πiℓ ( u )) χ ( u ) e u is trivial on U . (cid:3) Corresponding polarized abelian varieties and psh functions. We continue todenote by K a non-archimedean field with valuation v = − log | | and value group Γ. Wedefine a correspondence between polarized totally degenerate split abelian varieties over K and decomposed polarized complex abelian varieties. The goal of this subsection is to relatepsh functions on corresponding abelian varieties. ROPICAL TORIC PLURIPOTENTIAL THEORY 49 Definition 7.4.1. A polarized totally degenerate split abelian variety ( A v , φ ) over the com-plete non-archimedean field K corresponds to the decomposed complex polarized abelianvariety ( A ∞ , H, U , U ) if their tropicalizations trop v ( A v , φ ) and trop ∞ ( A ∞ , H, U , U ) areisomorphic as polarized tropical abelian varieties. Remark 7.4.2. For any polarized totally degenerate split abelian variety over K thereexists a corresponding decomposed complex polarized abelian variety by Proposition 7.3.4.Furthermore a decomposed complex polarized abelian variety, whose associated polarizedtropical abelian variety is Γ-rational, corresponds to a polarized totally degenerate splitabelian variety over K by 7.2.2. However, corresponding abelian varieties do not determineeach other up to isomorphism.In the following consider a polarized totally degenerate split abelian variety ( A v , φ ) over K which corresponds to the decomposed complex polarized abelian variety ( A ∞ , H, U , U ).We will read the isomorphism of associated polarized tropical abelian varieties as an iden-tification. Hence we have,(7.16) trop v ( A v , φ ) = (Λ , M, [ · , · ] , λ ) = trop ∞ ( A ∞ , H, U , U ) . From 7.2.2 and Definition 7.3.3, we obtain the Tate uniformizations π v : T an v −→ A an v , π ∞ : T an ∞ −→ A an ∞ , with kernels identified with Λ along the tropicalizations. Here T an v and T an ∞ is the non-archimedean resp. the archimedean analytification of the split algebraic torus with cochar-acter lattice N := Hom Z ( M, Z ). From our data, we get complex (1 , θ ∞ := θ ( A,H ) on T an ∞ and ω ∞ := ω ( A,H ) on A an ∞ , smooth (1 , θ v := θ ( A,φ v ) on T an v and ω v := ω ( A,φ v ) on A an v , and Lagerberg (1 , θ := θ (Λ ,M, [ · , · ] ,λ ) on N R and ω := ω (Λ ,M, [ · , · ] ,λ ) on N R / Λ. Wehave seen in Proposition 7.3.5 and Definition 7.2.3 that trop ∗∞ ( θ ) = θ ∞ and trop ∗ v ( θ ) = θ ∞ . Let U be an open subset of N R . The cone of θ -psh functions on U is denoted byPSH( U, θ ). For w ∈ {∞ , v } , we have U w := trop − w ( U ) ⊂ T an w and W := trop − w ( U ) ⊂ T an w are invariant under the action of the compact torus S w . We denote by PSH( U w , θ w ) S w thecone of θ w -psh functions which are invariant under the action of S w . Recall from Theorems5.2.12 and 5.3.11 that we have a canonical isomorphism(7.17) PSH( U, θ ) −→ PSH( U w , θ w ) S w , ϕ ϕ ∞ := ϕ ◦ trop w of cones. By Proposition 5.2.5, all these functions ϕ and ϕ w are continuous. If they are notidentically −∞ , then the functions are finite. We assume that U is a Λ-invariant subset of N R . Equivalently, this means that thereis an open subset W of N R / Λ with U = π − ( W ) for the quotient map π : N R → N R / Λ.Then U w is Λ-invariant and we have U w = π − w ( W w ) for the open subset W w := trop − w ( W )of A an w for w ∈ { v, ∞} .If we apply Definition 5.2.4 literally, then we get a natural generalization of the class of θ -psh functions for any tropical space and any closed symmetric (1 , θ . We applythis for the open subset W of the tropical space N R / Λ and the canonical form ω , thenthe ω -psh functions on W are given by the Λ-invariant θ -psh functions on U and for thecorresponding cones we get PSH( W, ω ) ≃ PSH( U, θ ) Λ . Note that S w -acts on A an w in a unique way such that the Tate uniformization π w : T an w → A an w becomes S w -equivariant. We denote by PSH( W w , ω w ) S w the cone of S w -invariant ω w -pshfunctions on W w . Proposition 7.4.5. For any subset W of N R / Λ and w ∈ {∞ , v } , we have an isomorphism (7.18) PSH( W, ω ) ∼ −→ PSH( W w , ω w ) S w , ϕ ϕ ◦ trop w . Proof. This holds as the isomorphism (7.17) for U w := π − w ( U ) is Λ-equivariant. (cid:3) Monge–Amp`ere equations on corresponding abelian varieties. Let K be anon-trivially valued non-archimedean field with valuation v . The goal of this subsection isto show existence of an invariant solution of the Monge–Amp`ere equation on a polarizedtotally degenerate split abelian variety over K with respect to a positive Radon-measuresupported on the canonical skeleton. We will generalize our result to arbitrary abelianvarieties in Section 8 where we also address uniqueness. Let ( A v , φ ) be a polarized totally degenerate split abelian variety over K whichcorresponds to the decomposed complex polarized abelian variety ( A ∞ , H, U , U ), i.e.trop v ( A v , φ ) = (Λ , M, [ · , · ] , λ ) = trop ∞ ( A ∞ , H, U , U )as in (7.16). For w ∈ {∞ , v } , we have the Tate uniformization π w : T an w → A an w with kernelidentified with Λ along trop w , where T is the split algebraic torus with character lattice M and cocharacter lattice N . Observe that the maximal compact torus S w in T an w acts on A an w . Let ω ∈ A , ( N R / Λ) be the canonical form of the polarized tropical abelian variety(Λ , M, [ · , · ] , λ ) introduced in (7.2). If ω w denotes the canonical form of the polarization of A w , then it follows from (7.7) and (7.15) that ω w = trop ∗ w ( ω ) and hence ω w is S w -invariant. In the setup of 7.5.1, we want to fix corresponding invariant measures. We startwith a positive Radon measure µ on the tropical abelian variety N R / Λ.By push-forward along the proper map ι v in (7.4) to the canonical skeleton of the totallydegenerate abelian variety A v , we get a positive Radon measure µ v := ι v ( µ ) on A an v . Wehave trop v ( µ v ) = µ for the proper map trop v .By [BGJK20, Corollary 5.1.17], the positive S ∞ -invariant Radon measures on T an ∞ corre-spond bijectively to the positive Radon measures on N R , i.e. there is a unique S ∞ -invariantpositive Radon measure µ ∞ on A an ∞ with trop ∞ ( µ ∞ ) = µ .Let n denote the common dimension of A ∞ , A v and N R . In the following we assume that(7.19) µ ( N R / Λ) = n ! p deg(Λ , M, [ · , · ] , λ ) = Z N R / Λ ω ∧ n where we used (7.3) on the right. Using that the tropicalization functor (7.12) keeps thedegree and that trop ∗∞ leaves integrals invariant, we note that (7.19) is equivalent to(7.20) µ ∞ ( A an ∞ ) = n ! p deg H ( A an ∞ ) = Z A an ∞ ω ∧ n ∞ . Using that tropicalization (7.6) preserves the degree, we deduce that (7.19) is equivalent to(7.21) µ v ( A an v ) = n ! q deg φ ( A v ) = Z A an v ω ∧ nv . These equivalent conditions are necesssary to solve the following Monge–Amp`ere equations. Proposition 7.5.3. In the setup of 7.5.1 and with the measures µ, µ ∞ , µ v from 7.5.2, themaps ϕ ϕ w := ϕ ◦ trop w for w ∈ {∞ , v } induce bijections between(i) the set of functions ϕ ∈ PSH( N R / Λ , ω ) \ {−∞} such that ( ω + d ′ d ′′ ϕ ) ∧ n = µ ,(ii) the set of functions ϕ ∞ ∈ PSH( A an ∞ , ω ∞ ) S ∞ \{−∞} such that ( ω ∞ + dd c ϕ ∞ ) ∧ n = µ ∞ ,(iii) the set of functions ϕ v ∈ PSH( A an v , ω v ) S v \ {−∞} such that ( ω v + d ′ d ′′ ϕ v ) ∧ n = µ v . It follows from 7.4.3 and 7.4.4 that the functions ϕ, ϕ ∞ , ϕ v are finite and continuous,hence the Monge–Amp`ere measures in (i) – (iii) are defined as in the first part of § Proof. We have seen in Proposition 7.4.5 that we have an isomorphismPSH( N R / Λ , ω ) ∼ −→ PSH( A an w , ω w ) S w , ϕ ϕ w := ϕ ◦ trop w ROPICAL TORIC PLURIPOTENTIAL THEORY 51 of cones for w ∈ {∞ , v } . It remains to see that ϕ is a solution of the tropical Monge–Amp`ereequation in (i) if and only if ϕ w := ϕ ◦ trop w is a solution of the Monge–Amp`ere equationin (ii) or (iii) , respectively. This can be checked locally on an open subset W of N R / Λand on the corresponding open subsets W w := trop − w ( W ) of A an w . Then we may assumethat the quotient homomorphism π : N R → N R / Λ maps an open subset U isomorphicallyonto W and hence the open subset U w := trop − w ( U ) of the Tate uniformization T an w ismapped isomorphically onto W w by π w . We have noticed in 7.1.3 that the canonical form θ := π ∗ ( ω ) ∈ A , ( N R ) fits into the setup of § θ -psh function f on U is a solution of ( θ + d ′ d ′′ f ) ∧ n = ˜ µ if and only if f w := f ◦ trop w is a solution of ( θ w + d ′ d ′′ f w ) ∧ n = ˜ µ w for the canonicalform θ w := trop ∗ w ( θ ) ∈ A , ( T an w ). Applying this to the lift f of ϕ | W to U and using that π ∗ w ( ω w ) = θ w , we deduce that ( ω + d ′ d ′′ ϕ ) ∧ n = µ if and only if ( ω w + d ′ d ′′ ϕ w ) ∧ n = µ w . (cid:3) Proposition 7.5.4. Let ( A ∞ , H, U , U ) be a decomposed polarized complex abelian varietyof dimension n with canonical form ω ∞ ∈ A , ( A an ∞ ) . Let µ ∞ be an S ∞ -invariant positiveRadon measure on A an ∞ with respect to the S ∞ -action on A an ∞ from 7.4.4 and with (7.22) µ ∞ ( A an ∞ ) = n ! p deg H ( A an ∞ ) . Then the complex Monge–Amp`ere equation ( ω ∞ + dd c ϕ ∞ ) ∧ n = µ ∞ is solved by a continuous finite ω ∞ -psh function ϕ ∞ on A an ∞ , unique up to adding a constant.Any such solution is S ∞ -invariant.Proof. Existence and uniqueness of a solution is known from complex K¨ahler geometry (seeTheorem 6.2.8). Here, we use that ω ∞ is a K¨ahler form and that µ ∞ does not chargepluripolar sets. To see the latter, we apply Lemma 6.3.4 by lifting the positive Radonmeasures µ ∞ and µ := trop ∞ ( µ ∞ ) locally to the Tate uniformization T an ∞ with cocharacterlattice N and to N R → N R / Λ, respectively. The canonical form ω ∞ and the measure µ ∞ isinvariant under S ∞ . It follows now from uniqueness of the solution up to adding constantthat the solutions are S ∞ -invariant. By 7.4.3, the function ϕ ∞ is finite and continuous. (cid:3) Corollary 7.5.5. Let (Λ , M, [ · , · ] , λ ) be a polarized tropical abelian variety with canonicalform ω on the associated tropical abelian variety N R / Λ . We assume the condition (7.23) µ ( N R / Λ) = n ! p deg(Λ , M, [ · , · ] , λ ) . for a positive Radon measure µ on N R / Λ . Then the tropical Monge–Amp`ere equation ( ω + d ′ d ′′ ϕ ) ∧ n = µ is solved by a continuous ω -psh function ϕ : N R / Λ → R , unique up to constants.Proof. By Proposition 7.3.4, there is a polarized decomposed complex abelian variety( A ∞ , H, U , U ) with trop ∞ ( A ∞ , H, U , U ) = (Λ , M, [ · , · ] , λ ). Let µ ∞ be the unique posi-tive S ∞ -invariant Radon measure on A an ∞ with trop ∞ ( µ ∞ ) = µ . We have seen in 7.5.2 that(7.23) is equivalent to (7.22). By Proposition 7.5.4, the complex Monge–Amp`ere equation( ω ∞ + dd c ϕ ∞ ) ∧ n = µ ∞ has a solution ϕ ∞ ∈ PSH( A an ∞ , ω ∞ ), unique up to adding a constant. Now the claim followsfrom Proposition 7.5.3. (cid:3) Corollary 7.5.6. Let ( A v , φ ) be a polarized totally degenerate split abelian variety over K with canonical form ω v ∈ A , ( A an v ) . Let µ v be a positive Radon measure supported on thecanonical skeleton of A an v with (7.24) µ v ( A an v ) = n ! q deg φ ( A v ) . Then there is a ϕ v ∈ PSH( A an v , ω v ) S v , unique up to adding constants, with (7.25) ( ω v + d ′ d ′′ ϕ v ) ∧ n = µ v . All such solutions ϕ v are continuous and finite.Proof. This follows from Proposition 7.5.3 and Corollary 7.5.5 (cid:3) Monge–Amp`ere equations on arbitrary non-archimedean abelian varieties In this section, K denotes a non-archimedean field ( K, | | ) with valuation v := − log | | and value group Γ := v ( K × ) ⊂ R . We will extend the solvability of the Monge–Amp`ereequation for totally degenerate split abelian varieties from Section 7 to arbitrary abelianvarietes over K . Instead of using θ -psh functions, we will use in this section the equivalentlanguage of psh metrics. Furthermore we will show that the solution is semipositive in thesense of Zhang which is then unique up to scaling by a result of Yuan and Zhang.In § § K is algebraically closed and non-trivially valued and usethe same strategy as in the totally degenerate case to solve the non-archimedean Monge–Amp`ere equation. Semipositivity will follow from a piecewise linear approximation of thetropical solution and then by relying on the results in [Gub10] about Mumford models. In § K .8.1. Raynaud uniformization and canonical tropicalization. We assume that thenon-archimedean field K is non-trivially valued and algebraically closed. We can associateto an abelian variety A over K in a functorial way an exact sequence of algebraic groups0 −→ T −→ E q −→ B −→ Raynaud extension of A and a discrete subgroup Λ of E ( K ) such that A an = E an / Λ, i.e. E an is the uniformization of A an . Here B is an abelian variety of good reductionover K and T is a torus over K (see [BL91] and [Ber90, 6.5]). The quotient homomorphism p : E an → A an is only an analytic morphism, but the Raynaud extension is algebraic. Remark 8.1.1. We denote the character lattice of the torus T by M and set N :=Hom Z ( M, Z ). For u ∈ M , the pushout of the Raynaud extension with respect to thecharacter χ u : T → G m gives rise to a translation invariant rigidified G m -torsor over B andhence to a rigidified translation invariant line bundle E u on B (see [BL91] and [FRSS18,3.2]). Note that the line bundle q ∗ ( E u ) is trivial over E and the above pushout constructiongives a canonical frame e u : E → q ∗ ( E u ). Following [FRSS18, 3.2]), we define the canonicaltropicalization map of E as the unique maptrop v : E an −→ N R satisfying h trop v ( x ) , u i = − log q ∗ k e u ( x ) k E u ( x ∈ E an , u ∈ M ) , where k k E u is the canonical metric of the rigidified line bundle E u . The canonical trop-icalization map agrees with the classical tropicalization map on T an and maps Λ onto acomplete lattice in N R (see [BL91, Theorem 1.2]). By abuse of notation, we will identify Λwith the lattice trop v (Λ) in N R . Then trop v induces a continuous maptrop v : A an −→ N R / Λwhich is called the canonical tropicalization of the abelian variety A . Remark 8.1.2. The canonical tropicalization map trop v : E an → N R admits a canonicalsection ι v : N R → E an which identifies N R with a closed subset of E an called the canonicalskeleton of E , such that trop v is a strong deformation retraction. Hence E an is contractible(see [Ber90, 6.5]). Passing to the quotient, we see that there is a canonical section ι v of ROPICAL TORIC PLURIPOTENTIAL THEORY 53 trop v which identifies N R / Λ with a closed subset of A an called the canonical skeleton of A such that trop v is a strong deformation retraction onto N R / Λ.In the following, we consider a polarized abelian variety ( A, φ ) over K of dimension g .Recall that its degree deg( A, φ ) is defined as the degree of the isogeny φ : A → A ∨ . Remark 8.1.3. Using the above notations, we have a natural bilinear pairingΛ × M −→ Z , ( a, u ) [ a, u ] := h trop v ( a ) , u i . Using the lift of φ to Raynaud uniformizations, it is shown in [BL91], [FRSS18] that φ in-duces a homomorphism λ : Λ → M and that (Λ , M, [ · , · ] , λ ) is a Γ-rational polarized tropicalabelian variety as defined in § v ( A, φ ) := (Λ , M, [ · , · ] , λ ) the tropicalizationof the polarized abelian variety ( A, φ ). This induces a functor(8.1) trop v : (cid:18) polarized abelianvarieties over K (cid:19) −→ (cid:18) Γ-rational polarizedtropical abelian varieties (cid:19) extending the tropicalization functor (7.6). Observe that the rank n of M might be smallerthan g = dim( A ) and equality occurs precisely in the totally degenerate case.As K is algebraically closed, there is an ample line bundle L on A such that φ = φ L is the isogeny φ L : A → A ∨ induced by L . By [BL91, Proposition 6.5 and Theorem 6.13],there is an ample line bundle H on B with q ∗ ( H an ) ∼ = p ∗ ( L an ) on E an . Let χ ( A, L ) and χ ( B, H ) be the Euler characteristics. Proposition 8.1.4. Under the above assumptions, we have deg( A, φ ) = χ ( A, L ) and χ ( A, L ) = deg L ( A ) g ! = p deg(trop v ( A, φ )) · χ ( B, H ) = p deg(trop v ( A, φ )) · deg H ( B )( g − n )! . Proof. This follows from the Riemann–Roch theorem and the vanishing theorem on abelianvarieties [Mum70, III.16] and the dimension formula in [BL91, Theorem 6.13]. (cid:3) Mumford models. We recall the construction of Mumford models from [Gub10, § § K is algebraicallyclosed and the valuation v is non-trivial, the value group Γ := v ( K × ) is dense in R . Again,we consider an abelian variety A over K with Raynaud extension0 −→ T −→ E q −→ B −→ . Let g be the dimension of A and n the rank of the torus T . The abelian variety B over K is the generic fiber of an abelian scheme B over K ◦ . For any scheme X over K ◦ , wedenote c X the formal completion along the special fiber and by X s its special fibre over theresidue field e K = K ◦ /K ◦◦ .We first fix the terminology from convex geometry expanding the conventions from § A polytopal decomposition Π of N R is a locally finite set Π of polytopes in N R such that the set Π contains with a polytope also all its faces, polytopes in Π intersectonly in common faces, and the support condition S ∆ ∈ Π ∆ = N R holds. The polytopaldecomposition Π is called integral Γ -affine if every polytope ∆ ∈ Π is integral Γ-affine (oftenalso called Γ -rational ). The star of Π in ω ∈ N R is the fan Σ := { σ ∆ | ∆ ∈ Π with ω ∈ ∆ } where σ ∆ is the cone in N R generated by ∆ − ω .A function f : N R → R is called piecewise affine with respect to a polytopal decomposition Π if for any polytope ∆ ∈ Π there are u ∆ ∈ M R = Hom( N, R ) and c ∆ ∈ R such that f = u ∆ + c ∆ on ∆. Such a function f is called piecewise Γ -affine if Π is integral Γ-affineand c ∆ ∈ Γ for all ∆ ∈ Π. We say that f has integral (resp. rational) slopes if u ∆ ∈ M (resp. u ∆ ∈ M Q ) for all ∆ ∈ Π. The recession function of f in ω is the unique function g : N R → R which is piecewise linear with respect to the star of Π in ω and satisfies g ( x − ω ) = u ∆ ( x ) for each ∆ ∈ Π and each x ∈ ∆.We call f : N R → R a locally piecewise affine function if f is piecewise affine with respectto a polytopal decomposition Π of N R . Here, the term “locally” emphasizes the fact thata polytopal decomposition is locally a polyhedral complex and hence a locally piecewiseaffine function is locally a piecewise affine function in the sense of our conventions in § Remark 8.2.2. Mumford’s construction associates to an integral Γ-affine polytopal de-composition Π of N R an admissible formal K ◦ -model ˆ E Π of E given by the formal analyticatlas { trop − v (∆) | ∆ ∈ Π } on trop − ( N R ) = ( ˆ E Π ) an (see [Gub10, § 4] for details).The irreducible components of the special fiber of ˆ E Π are in bijective correspondence tothe vertices of Π. In fact, the irreducible component Y ω corresponding to the vertex ω of Πis a fiber bundle over B s . The torus T ˜ K := Spec( ˜ K [ M ]) acts naturally on the fibers of Y ω over B s and each fiber is T ˜ K -equivariantly isomorphic to the T ˜ K -toric variety Y Σ associatedto the star Σ of Π in ω [Gub10, Proposition 4.8]. The fiber over zero is even canonicallyisomorphic to the T ˜ K -toric variety Y Σ . It follows in particular that Y ω is proper over B s .We call ˆ E Π the formal Mumford model of trop − v ( | Π | ) associated to Π. In fact, using toricschemes over K ◦ from [Gub13, § § E Π of E with formal completion ˆ E Π . Note that q extends uniquely tomorphisms ˆ E Π → ˆ B and E Π → B which we denote by q as well.Let f : N R → R be a piecewise Γ-affine function with respect to Π. If f has integralslopes, then f defines a formal model O ˆ E Π ( f ) of the trivial line bundle O trop − v ( N R ) living onˆ E Π which is determined by(8.2) f ◦ trop v = − log (cid:0) k k O ˆ E Π ( f ) (cid:1) , where k k O ˆ E Π ( f ) denotes the model metric on O antrop − v ( N R ) induced by O ˆ E Π ( f ).In the following, we consider a piecewise Γ-affine function f : N R → R with respect to apolytopal decomposition Π of N R and we assume that f has integral slopes. We fix a vertex ω of Π with associated irreducible component Y ω of the special fiber of ˆ E Π (see Remark8.2.2). The function f is called convex in ω if f is convex in a neighbourhood of ω orequivalently if the recession function g of f in ω is convex on N R . Lemma 8.2.3. Let ω be a vertex of Π . Then the function f is convex in ω if and only ifthe line bundle O ˆ E Π ( f ) | Y ω on the proper scheme Y ω is nef.Proof. It follows from Remark 8.1.1 that any u ∈ M induces a rigidified translation invariantline bundle E u on B which has a model E u on B algebraically equivalent to zero.For any piecewise Γ-affine function h with respect to Π such that h has integral slopes,the line bundle O ˆ E Π ( h ) has a unique meromorphic section s h extending the constant section1 from the generic fiber. This construction is linear in h . If h is equal to some u ∈ M (resp. equal to some c ∈ Γ), then O ˆ E Π ( h ) is numerically trivial as it is isomophic to thepull-back of E u (resp. a trivial line bundle).Suppose first that O ˆ E Π ( f ) | Y ω is nef. We have seen in Remark 8.2.2 that Y ω → B s is afiber bundle with fiber isomorphic to the proper toric variety Y Σ where the fan Σ is givenas the star of Π in ω . The restriction of O ˆ E Π ( f ) to the fiber of Y ω → B s over zero leads toa line bundle on Y Σ corresponding to the recession function g of f in ω . This restriction isalso nef and hence g is convex (see [CLS11, Theorem 6.3.12] or [Ful93, § − g is used there). ROPICAL TORIC PLURIPOTENTIAL THEORY 55 Conversely, let f be convex in ω . We have to show that O ˆ E Π ( f ) | Y ω is nef. Let P be theunion of all polytopes of Π with vertex ω . Then there is a unique formal open subset U ofˆ E Π with trop − ( P ) = U an . Note that Y ω is also an irreducible component of U s .For any ∆ ∈ Π with vertex ω , there is c ∈ Γ and u ∈ M such that f = u + c on ∆. Forthe formal metric k k induced by O ˆ E Π ( f ), the convexity of f | P yields that k s f − u − c k ≤ s f − u − c | U is a global section of O ˆ E Π ( f − u − c ) | U which is nowhere vanishing overthe formal open subset U ∆ of U with U an∆ = trop − v (∆). Now let C be any closed curvein Y ω . We pick any y ∈ C . By [GRW17, Proposition 2.17] there exists some x ∈ U an withreduction y . We choose ∆ ∈ Π with trop v ( x ) ∈ ∆. The above shows that the restriction ofthe global section s f − u − c | U to C does not vanish in y . Hence the intersection number c ( O ˆ E Π ( f )) .C = c ( O ˆ E Π ( f − u − c )) .C + c ( O ˆ E Π ( u )) .C + c ( O ˆ E Π ( c )) .C = div( s f − u − c ) .C is non-negative. This proves the claim. (cid:3) Let F be a line bundle on E an with F = q ∗ ( H ) for a rigidified line bundle H on B an .Then H has a unique rigidified model H on the formal abelian scheme ˆ B and q ∗ H ( f ) := q ∗ H ⊗ O ˆ E Π ( f )is a model of F living on the formal Mumford model ˆ E Π . Proposition 8.2.4. Consider F = q ∗ ( H ) as above. Let f : N R → R be a piecewise Γ -affinefunction with respect to the polytopal decomposition Π of N R . We assume that f has integralslopes and is convex in the given vertex ω of Π . We consider the dual polytope { ω } f := { u ∈ M R | u ( x − ω ) ≤ f ( x ) − f ( ω ) for all x in a neighborhood of ω } of the star Σ in ω where M = Hom Z ( N, Z ) . Then we have deg q ∗ H ( f ) ( Y ω ) = g !( g − n )! · deg H ( B ) · vol( { ω } f ) where vol is the Haar measure on M R such that the lattice M has covolume one.Proof. The proof is similar as in [Gub10, Proposition 5.18]. We use the canonical strataof a scheme as defined in [Ber99, § S of the special fiber of the Mumford model ˆ E Π and the open faces τ of Π given by S = red(trop − v ( τ )) where red is the reduction map of ˆ E Π [Gub10, Proposition 4.8].Similarly as in the proof of Lemma 8.2.3, the intersection product c ( O ˆ E Π ( f )) r · Y ω canbe performed for any r ≤ g by using proper intersections with suitable Cartier divisorsdiv( s f − u − c ) and hence can be represented by a positive linear combination of strata closuresin Y ω . The above correspondence shows that there are no strata S of Y ω with codim( S, Y ω ) >n . Then it follows from the projection formula with respect to q : Y ω → B s that(8.3) deg q ∗ H ( f ) ( Y ω ) = (cid:18) gn (cid:19) c ( H ) g − n .q ∗ (cid:16) c (cid:0) O ˆ E Π ( f ) (cid:1) n · Y ω (cid:17) . For any closed point b of B s , the fibre q − ( b ) is T ˜ K -equivariantly isomorphic to the toricvariety Y Σ over ˜ K associated to the star Σ of Π in ω . Along this isomorphism, the restrictionof O ˆ E Π ( f ) to q − ( b ) corresponds to the line bundle O Y Σ ( g ) for the recession function g of f in ω . Hence the degree of q − ( b ) with respect to O ˆ E Π ( f ) is equal to n ! · vol( { ω } f ) [Ful93, § c ( O ˆ E Π ( f )) n · Y ω is represented by a positivelinear combination of strata closures and that every stratum of ˆ E s maps onto B s , we maycompute the intersection product locally over B s , say over an open subset W of ˆ B which trivializes E and hence Y ω is over W s isomorphic to Y Σ × W s . Along this isomorphism, theline bundle O ˆ E Π ( f ) | q − ( W s ) is given by pull back of O Y Σ ( g ). This shows immediately that q ∗ (cid:0) c ( O ˆ E Π ( f )) n · Y ω (cid:1) = n ! · vol( { ω } f ) B s and hence the claim follows from (8.3) and deg H ( B s ) = deg H ( B ). (cid:3) Remark 8.2.5. A line bundle F on E an descends to A an if and only if F admits a Λ-linearization over the action of Λ on E an . In this case, we have F = p ∗ ( L an ) for the linebundle L an = F/ Λ on A an . By [BL91, Proposition 6.5], for any rigidified line bundle L on A there is a rigidified line bundle H on B an such that p ∗ ( L an ) ≃ q ∗ ( H ) as Λ-linearizedcubical sheaves. The line bundle H is unique up to a tensor product with a line bundle E u for some u ∈ M . Using the above isomorphism for identification, it is shown in [Gub10,4.3] that the Λ-linearization yields a canonical cocycle ( z λ : N R → R ) λ ∈ Λ and a canonicalsymmetric bilinear form b : Λ × Λ → R associated to L such that(8.4) z λ ( ω ) = z λ (0) + b ( ω, λ ) ∀ ω ∈ N R , λ ∈ Λ . and(8.5) z λ (0) ∈ Γ ∀ λ ∈ Λ . Moreover, the cocycle condition shows that λ z λ (0) is a quadratic function on Λ withassociated bilinear form b . The line bundle L is ample if and only if H is ample and b ispositive definite [BL91, Theorem 6.13]. In this case, b is the bilinear form of the polarizedtropical abelian variety trop v ( A, φ L ) considered in (8.1) and § H is thegeneric fiber of a unique rigidified ample line bundle H on ˆ B .A polytopal decomposition Π of N R is called Λ -periodic if for all ∆ ∈ Π and for all λ ∈ Λ \ { } the polytope ∆ + λ is a face of Π disjoint from ∆. For a Λ-periodic integralΓ-affine polytopal decomposition Π of N R , the following facts are shown in [Gub10, 4.3]: (i) The Mumford model ˆ E Π is a formal model of E an over K ◦ and A Π := ˆ E Π /M is aformal model of A an = E an /M over K ◦ . Furthermore the formal models ˆ E Π and A Π are locally isomorphic. (ii) Let f : N R → R be a piecewise Γ-affine function with respect to Π. Assume that f has integral slopes and let F := q ∗ H ( f ). If f satisfies the automorphy condition(8.6) f ( ω + λ ) = f ( ω ) + z λ ( ω ) ∀ ω ∈ N R , λ ∈ Λ , then there is a unique line bundle L on A Π with p ∗ ( L ) = F . Proposition 8.2.6. Every convex function f : N R → R satisfying (8.6) is a uniform limitof locally finite piecewise Γ -affine convex functions with rational slopes satisfying also (8.6) .Proof. Convexity, (8.6) and (8.4) yield that f ( ω ) grows like the positive definite quadraticform b ( ω, ω ). Fix ε > p ∈ N R . By the quadratic growth of f , the convex set C p = { x ∈ M R | f ( ω ) ≥ f ( p ) + x ( ω − p ) − ε/ ∀ ω ∈ N R } has nonempty interior and hence there is a point m p ∈ C ∩ M Q . Choose a value c p ∈ Γ with f ( p ) − m p ( p ) ≥ c p + ε/ ≥ f ( p ) − m p ( p ) − ε/ . This is possible because the value group Γ is dense in R . Then the function h p := c p + m p is Γ-affine with rational slope and satisfies h p ( p ) = c p + m p ( p ) ≤ f ( p ) − ε/ . Since m p ∈ C p , this implies that h p ≤ f on N R . Moreover, h p ( p ) = c p + m p ( p ) ≥ f ( p ) − ε/ . ROPICAL TORIC PLURIPOTENTIAL THEORY 57 By continuity of h p and f , there is an open neighborhood U p of p such that h p ( ω ) ≥ f ( ω ) − ε ∀ ω ∈ U p . We choose such a function h p for every point p ∈ N R . We may assume that h p + λ ( ω + λ ) = h p ( ω ) + z λ ( ω )for all λ ∈ Λ. Note that h p + λ is still Γ-affine with rational slope because the cocycle z λ isΓ-affine with integral slope. This follows from (8.4), (8.5) and Remark 7.1.2 using the factthat b is the bilinear form of a polarized tropical abelian variety.Let F Λ be the closure of a fundamental domain of the lattice Λ. Since F Λ is compact,there is a finite subset I ⊂ N R such that the open subsets ( U p ) p ∈ I cover F Λ . Then we define(8.7) h : N R −→ R , ω h ( ω ) := sup p ∈ I,λ ∈ Λ h p + λ ( ω ) . Obviously, we have f − ε ≤ h ≤ f . Since we have h p + λ ( ω + λ ) = h λ ( ω ) + z λ ( ω ), it is clearthat h satisfies the automorphy condition (8.6). It remains to show that h is a convex locallyfinite piecewise Γ-affine function. To see this, it is enough to show that the supremum in(8.7) is locally the maximum of finitely many of the functions h p . Recall that m p is thelinear part of the affine function h p . For ω ∈ N R and λ ∈ Λ, we have h p + λ ( ω ) ≤ h p ( ω ) ⇔ h p ( ω − λ ) + z λ ( ω − λ ) ≤ h p ( ω ) ⇔ z λ ( ω − λ ) ≤ m p ( λ ) . Using (8.4), we have z λ ( ω − λ ) = z λ (0) + b ( λ, ω − λ ) and hence the above inequalitiesare equivalent to z λ (0) − b ( λ, λ ) ≤ m p ( λ ) − b ( λ, ω ). Since the left hand side decreasesquadratically like ∼ − b ( λ, λ ) and the right hand side grows at most linearly for k λ k → ∞ ,we conclude that for any bounded set Ω, there is R ≥ h p + λ ( ω ) ≤ h p ( ω ) is satisfied for all ω ∈ Ω and all λ ∈ Λ with k λ k ≥ R . This proves thatlocally the supremum in (8.7) is the maximum of only finitely many h p . (cid:3) Remark 8.2.7. If f is a function as in Remark 8.2.5 (ii) satisfying (8.6) and if L is ample,then L in (ii) is ample and A Π is the formal completion of an algebraic model.Here is a sketch of proof. To show ampleness, by the formal GAGA-principle in [GD63,Th´eor`eme 5.4.5] and [FK18, Theorem I.10.1.2], it is enough to show that the restriction of L to any irreducible component Y of the special fiber of A Π is ample. There is a vertex ω of Π such that p maps Y ω isomorphically onto Y . By [Gub10, Proposition 4.12], strictconvexity of f yields that p ∗ ( L ) | Y ω is relatively ample with respect to the toric fiber bundle Y ω → B s . Recall that p ∗ ( L ) = q ∗ H ⊗ O ˆ E Π ( f ) and the line bundle H on ˆ B is ample as L isample. Using the global sections s f − ℓ − c | Y ω as in the proofs of Lemma 8.2.3 and Proposition8.2.4, the Nakai–Moishezon criterion shows that p ∗ ( L ) | Y ω is ample and hence the same istrue for L | Y .8.3. Monge–Amp`ere equations for abelian varieties. Our goal is to solve the invari-ant non-archimedean Monge–Amp`ere equation for arbitrary abelian varieties. In § K with non-trivial valuation v . In § g -dimensional polarized abelian variety ( A, φ ) over K . Let L be an ample linebundle on A which induces the polarization φ . Again, we denote by n the rank of the torus T in the Raynaud extension 0 −→ T −→ E q −→ B −→ . of A . Let θ ∈ A , ( N R ) and ω ∈ A , ( N R / Λ) be the canonical (1 , , M, [ · , · ] , λ ) := trop v ( A, φ ). We also rely on the canonical metric k k L of L . Proposition 8.3.1. For ϕ ∈ PSH( N R / Λ , ω ) , let ϕ v := ϕ ◦ trop v . Then k k L ( ϕ ) := e − ϕ v k k L is a continuous semipositive metric as introduced in Remark 5.3.6. Proof. The canonical metric k k L of L is determined by choosing a rigidification of L . Wefirst recall some facts from Remark 8.2.5. There is a rigidified ample line bundle H on B an such that p ∗ ( L an ) ≃ q ∗ ( H ) as Λ-linearized cubical sheaves. The ample line bundle L induces a cocycle ( z λ ) λ ∈ Λ and a scalar product b on N R . We have seen that Q ( λ ) := z λ (0)is a quadratic function in λ and that b is the associated bilinear form. Since b is also thebilinear form of trop v ( A, φ ) = (Λ , M, [ · , · ] , λ ), we have θ = d ′ d ′′ Q by 7.1.3. Using [GK17,Example 8.15], we have that(8.8) p ∗ k k L = e − Q q ∗ k k H . Let ˜ ϕ : N R → R be the lift of ϕ to N R . Then ˜ ϕ is a θ -psh function ϕ on N R which meansthat f := ˜ ϕ + Q is convex. For x ∈ N R and λ ∈ Λ, we deduce from (8.4) that f ( x + λ ) = ˜ ϕ ( x + λ ) + Q ( x + λ ) = ˜ ϕ ( x ) + Q ( x ) + Q ( λ ) + b ( x, λ ) = f ( x ) + z λ (0) . This means that f satisfies the automorphy condition (8.6). By Proposition 8.2.6, f isa uniform limit of locally finite piecewise Γ-affine functions f k on N R which have rationalslopes and which satisfy also (8.6). We have seen in the proof of Proposition 8.2.6 that z λ (0)is an affine function with integral slopes and hence there is a non-zero m k ∈ N such that m k f k is a piecewise Γ-affine function with integral slopes. Going the above steps backwards,we see that f k − Q is the lift of a unique ω -psh function ϕ k . By construction, k k L ( ϕ ) is theuniform limit of the metrics k k L ( ϕ k ) . As in § H be the model of H on the formalcompletion ˆ B of the abelian scheme B over K ◦ with generic fiber B . Then we have p ∗ k k ⊗ m k L ( ϕ k ) = e − Q − ϕ k,v ◦ p k k ⊗ m k q ∗ H = e − f k k k q ∗ H ⊗ mk = k k q ∗ H ⊗ mk ( m k f k ) . It follows from Remark 8.2.5 that k k ⊗ m k L ( ϕ k ) is the metric induced by a line bundle M k on aformal model A k = E k /M of A for a formal Mumford model E k of E induced by a integralΓ-affine Λ-periodic decomposition Π k of N R . We claim that these metrics are semipositive.To see this, we have to show that the restriction of M k to any irreducible component Y of A k is nef. Using A k = E k /M , we see that Y ≃ Y ω for an irreducible component Y ω of E k associated to a vertex ω of Π k . By Lemma 8.2.3, we have that O E k ( m k f k ) | Y ω is nef. Usingthe H is ample, we deduce that p ∗ M k = q ∗ H ( m k f k ) restricts to a nef line bundle on Y ω .This proves our intermediate claim and hence k k L ( ϕ k ) is a semipositive model metric. Bydefinition, we get that k k L ( ϕ ) := e − ϕ v k k L is semipositive. (cid:3) Remark 8.3.2. Let µ be a positive Radon measure on N R / Λ. Let ι v : N R / Λ → A an bethe canonical section of trop v used in Remark 8.1.2 to identify N R / Λ with the canonicalskeleton of A . Then we obtain a Radon measure on A an as the image measure(8.9) µ v := (cid:18) gg − n (cid:19) deg H ( B ) · ι v ( µ ) . where H is again the line bundle on B an with q ∗ ( H ) = p ∗ ( L an ). Theorem 8.3.3. We consider a positive Radon measure µ v on A an which is supported inthe canonical skeleton of A and satisfies µ v ( A ) = deg L ( A ) . Then there is a continuoussemipositive metric k k of L which satisfies the non-archimedean Monge–Amp`ere equation (8.10) c ( L, k k ) ∧ g = µ v . The continuous semipositive metric k k is unique up to scaling. For totally degenerate abelian varieties, this was shown by Liu [Liu11] in case of a measure µ v with smooth density proving also that then the solution is a smooth metric. ROPICAL TORIC PLURIPOTENTIAL THEORY 59 Proof. It was shown by Yuan and Zhang [YZ17, Corollary 1.2] that uniqueness of a contin-uous semipositive solution up to scaling holds on any projective variety over K .For existence, we note that the measure µ v is obtained from a unique Radon measure µ on N R / Λ as in Remark 8.3.2. By Proposition 8.1.4, we have(8.11) µ ( N R / Λ) = n ! p deg(trop v ( A, φ )) . By Proposition 7.5.5, there is ϕ ∈ PSH( N R / Λ) solving the tropical Monge–Amp`ere equation(8.12) ( ω + d ′ d ′′ ϕ ) ∧ n = µ. We claim that the semipositive metric k k L ( ϕ ) of L from Proposition 8.3.1 is a solution of(8.10). We use now the notation and the results from the proof of Proposition 8.3.1. Recallthat f = ˜ ϕ + Q is the uniform limit of the sequence f k = ˜ ϕ k + Q . Since k k L ( ϕ k ) is amodel metric for L , its Monge–Amp`ere measure is the discrete measure on A an supportedin the Shilov points associated to the irreducible components Y of the Mumford model A Π k and with multiplicity give by the degree of Y with respect to the model of the metric[GK17, Theorem 10.5]. By [Gub10, Proposition 4.8], Y corresponds to a vertex u of Π k ,determined up to translation by Λ, and the Shilov point of Y is just p ( u ) in the canonicalskeleton of N R . Proposition 8.2.4 shows that the non-archimedean Monge–Amp`ere measure c ( L, k k L ( ϕ k ) ) ∧ g is equal to(8.13) g !( g − n )! deg H ( B ) X u vol( { u } f k ) = (cid:18) gg − n (cid:19) deg H ( B )¯ ι v ( ω + d ′ d ′′ ϕ k ) ∧ n where we sum over a system of representatives of vertices u of Π k modulo Λ. For (8.13), weuse that n !vol( { u } f k ) is the multiplicity of the real Monge–Amp`ere measure of the piecewiseaffine f k in u [BPS14, Proposition 2.7.4] and then Remark 6.2.12. Using continuity ofthe tropical and the non-archimedean Monge–Amp`ere measure along uniformly convergentsequences, we deduce from (8.9), (8.12) and (8.13) that k k L ( ϕ k ) is a solution of (8.10). (cid:3) Descent. In this subsection, K is an arbitrary non-archimedean field with non-trivialvaluation v . We solve the invariant non-archimedean Monge–Amp`ere equation for anyabelian variety A over K . We use the solution from Theorem 8.3.3 for the base change of A to an algebraic closure K and apply a descent argument. We use continuous semipositivemetrics on an ample line bundle as introduced in Remark 5.3.6. Let L be an ample line bundle on a geometrically integral projective variety Y over K of dimension n . We consider an extension F/K of non-archimedean fields. The basechange of Y (resp. L ) is denoted by Y F (resp. L F ). Note that the base change morphisminduces a proper map π : Y an F → Y an . Let k k be a continuous metric on L an and let k k F := π ∗ k k be the base change metric on L an F .We recall results of Boucksom and Eriksson about the base change of semipositive metrics. Proposition 8.4.2. In the setting of 8.4.1, the following properties hold.(i) The continuous metric k k is semipositive if and only if k k F is semipositive.(ii) If k k is a continuous semipositive metric, then c ( L, k k ) ∧ n = π ∗ ( c ( L F , k k F ) ∧ n ) .Proof. Property (i) is [BE18, Theorem 7.32] and (ii) is shown in [BE18, § (cid:3) We keep the setting of 8.4.1. We assume that F/K is a finite Galois extension ofdegree d with Galois group G . Then base change induces a finite flat morphism Y F → Y and hence we have the norm of a line bundle L ′ on Y F as a line bundle N ( L ′ ) on Y (see[BE18, § A.8]). For a metric k k ′ of L ′ , we have the norm of the metric as a metric N ( k k ′ ) of N ( L ′ ). If k k ′ is a continuous semipositive metric, then N ( k k ′ ) is a continuous semipositivemetric [BE18, Proposition 8.22]. We apply this now to L ′ := L F for the ample line bundle L on Y . Then N ( L F ) = L ⊗ d and hence π ∗ ( N ( L F )) = L ⊗ dF . The finite Galois group G acts continuously on Y an and on L an F . For a continuous semipositive metric k k ′ of L F , the metric k k ′ σ := σ ∗ k k ′ is also acontinuous semipositive metric of L F and we deduce from the definitions that(8.14) π ∗ (cid:0) N ( L F ) , N ( k k ′ ) (cid:1) = O σ ∈ G ( L F , k k σ ) . Moreover, there is a metric k k on L with π ∗ k k = k k ′ if and only if k k ′ is G -invariant.Uniqueness of the complex Monge–Amp`ere equation was originally proven by Calabi. Innon-archimedean geometry, we have the following result of Yuan and Zhang. Theorem 8.4.4. Let L be an ample line bundle on a geometrically integral variety Y overthe non-trivially valued non-archimedean field K . If k k and k k ′ are continuous semipositvemetrics of L with c ( L, k k ) ∧ n = c ( L, k k ′ ) ∧ n , then there is r ∈ R > with k k ′ = r k k .Proof. In [YZ17, Corollary 1.2], this is proven for a non-trivially valued algebraically closednon-archimedean field. We will deduce from it the claim for any non-trivially valued non-archimedean field K . Let F be the completion of an algebraic closure K of K and let G be the Galois group of K/K . By [Ber90, Corollary 1.3.6], we have X an F /G ≃ X an as atopological space. Using that the profinite group G is compact, we note that c ( L F , k k F ) ∧ n is the unique G -invariant positive Radon measure on Y an F with image measure c ( L, k k ) ∧ n on Y an (see § c ( L F , k k F ) ∧ n = c ( L ′ F , k k ′ F ) ∧ n . By [YZ17, Corollary1.2], we get k k ′ F = r k k F for some r ∈ R > and hence k k ′ = r k k . (cid:3) We say that an abelian variety A is split over K if A has a Raynaud extension as in § T . Then we get a canonical tropicalization map trop v : A an → N R / Λas in Remark 8.1.1 where N is the cocharacter lattice of T . For any abelian variety A over K , there is a finite separable extension F of K such that A F is split over F .Now let L be a rigidified ample line bundle on the split abelian variety A and let0 −→ T −→ E q −→ B −→ A such that A an = E an / Λ for a lattice Λ of E ( K ). By [BL91,Proposition 6.5], there is a rigidified ample line bundle H on B an such that p ∗ ( L an ) ≃ q ∗ ( H ). We prove now Theorem E from the introduction which generalizes Theorem 8.3.3to any abelian variety over any non-trivially valued non-archimedean field K . Let S ( A ) bethe canonical skeleton of A from [Ber90, § L on anabelian variety A over K and a positive Radon measure µ v on A an which is supported in S ( A ) and satisfies µ v ( A ) = deg L ( A ). We have to show that(8.15) c ( L, k k ) ∧ g = µ v . has a solution given by a continuous semipositive metric k k of L unique up to scaling. Proof. Uniqueness up to scaling follows from Theorem 8.4.4 and hence it remains to showexistence. By 8.4.5, there is a finite Galois extension F/K such that A F is split over F . Let ϕ be solution of the associated tropical Monge–Amp`ere equation (8.12) as in the proof ofTheorem 8.3.3. Using ϕ v := ϕ ◦ trop v , we define the metric k k L ( ϕ ) of L F as in Proposition8.3.1. We have seen in the proof of Theorem 8.3.3 that k k ′ := k k L ( ϕ ) satisfies the non-archimedean Monge–Amp`ere equation 8.15 over the completion of the algebraic closure of F and hence also over F by using Proposition 8.4.2.Let G be the Galois group of F/K , let d := [ F : K ] and let N be the norm introducedin 8.4.3. By [Ber90, § S ( A ) = S ( A F ) /G and hence there is a unique G -invariant positive Radon measure µ ′ v on A an F which is supported in S ( A F ) and with image ROPICAL TORIC PLURIPOTENTIAL THEORY 61 measure µ v (see § µ v is invariant under the G -operation, it follows from theuniqueness theorem 8.4.4 that k k ′ = k k ′ σ for all σ ∈ G . Using (8.14), we conclude thatthe metric π ∗ N ( k k ′ ) of L ⊗ dF solves the non-archimedean Monge–Amp`ere equation on A an F for the measure d g · µ ′ v . Again by Proposition 8.4.2, it follows that the metric N ( k k ′ )of N ( L F ) = L ⊗ d satisfies the non-archimedean Monge–Amp`ere equation on A an for themeasure d g µ v . We conclude that k k := N ( k k ′ ) /d is the desired solution of (8.15). (cid:3) Appendix A. Proof of Theorem 6.2.13 We give a proof of uniqueness in Theorem 6.2.13 as the argument is omitted in Berman-Berndtsson [BB13] and Bakelman’s uniqueness proof for [Bak94, Theorem 17.1] relies on[Bak94, Lemma 10.2] which is false as the following example shows. We will adjust hisarguments. Example A.0.1. Let u, v the convex functions on G := R defined as follows u ( x ) = − x − , if x ≤ − ,x , if − ≤ x ≤ x − , if x ≥ , v ( x ) = ( − x − , if x ≤ , x − , if x ≥ . On the open interval Q = ( − , 1) we have v < u . On ∂Q = {− , } , we have v = u . Forthe multivalued subdifferentials, we have ∂u (1) = [2 , , ∂v (1) = { } . Hence ∂u (1) \ ∂v (1) = ∅ . Under these hypotheses, it was claimed in [Bak94, Lemma 10.2]that MA( v ) (cid:0) Q (cid:1) > MA( u ) (cid:0) Q (cid:1) , but we have MA( v ) (cid:0) Q (cid:1) = 4 = MA( u ) (cid:0) Q (cid:1) .We keep the same setting as in 6.2.11, i.e. N is a free abelian group of finite rank n withan identification N ≃ Z n and M := Hom( N, Z ). For a set W of M R , we denote by W ◦ the interior of W . In the following, we fix a convex body ∆ in M R and we consider a convexfunction f : N R → R with ∆ ◦ ⊂ ∆( f ) ⊂ ∆ for the stability set ∆( f ). For the multivaluedsubdifferential ∂f , it follows from [Roc70, Theorems 23.4 and 23.5] that(A.1) ∆ ◦ ⊂ ∂f ( N R ) ⊂ ∆( f ) ⊂ ∆ . We first prove a comparison principle . Proposition A.0.2. Let f and g be convex functions on N R ≃ R n and let ∆ be a convexbody in M R with ∆ ◦ ⊂ ∆( f ) ⊂ ∆ . For Ω := { x ∈ N R | f ( x ) < g ( x ) } , we have ∂g (Ω) ∩ ∆ ◦ ⊂ ∂f (Ω) . Proof. Let H ∈ ∂g (Ω) ∩ ∆ ◦ . Since H ∈ ∂g (Ω), there is an y ∈ Ω such that H ∈ ∂g ( y ).Since H ∈ ∆ ◦ ⊂ ∂ ( f )( N R ) using (A.1), there is an y ∈ N R such that H ∈ ∂f ( y ). Theconditions H ∈ ∂g ( y ) and y ∈ Ω yield(A.2) g ( y ) ≥ g ( y ) + H ( y − y ) > f ( y ) + H ( y − y ) . The condition H ∈ ∂f ( y ) yields(A.3) f ( y ) ≥ f ( y ) + H ( y − y ) . Equations (A.2) and (A.3) imply f ( y ) < g ( y ). Therefore y ∈ Ω proving claim. (cid:3) We will need also the following openness criterion. Proposition A.0.3. Let f be a convex function on an open convex set U of N R . Let x ∈ U and H ∈ ∂f ( x ) . If there is a compact neighbourhood B of x in U such that (A.4) f ( y ) > f ( x ) + H ( y − x ) for all y ∈ ∂B . Then ∂f ( B ) is a neighborhood of H . Proof. Since ∂B is compact, the condition (A.4) implies that there is a neighbourhood V of H such that f ( y ) > f ( x ) + H ( y − x ) for all H ∈ V and all y ∈ ∂B . We consider H ∈ V .We claim that for every point y ∈ U \ B , it is still true that f ( y ) > f ( x ) + H ( y − x ).To see that, consider the segment x y . There is a point y in the interior of x y such that y ∈ ∂B . Writing y = αx + (1 − α ) y with 0 < α < x g ( x ) := f ( x ) − f ( x ) − H ( x − x ) is convex, we have0 < f ( y ) − f ( x ) − H ( y − x ) ≤ αg ( x ) + (1 − α ) g ( y ) = (1 − α ) g ( y )proving f ( y ) > f ( x ) + H ( y − x ). We get ∅ 6 = { x ∈ U | f ( x ) ≤ f ( x ) + H ( x − x ) } ⊂ B which implies H ∈ ∂f ( B ) as claimed. (cid:3) Proof of uniqueness in Theorem 6.2.13. Let ∆ ⊂ M R be a convex body. For convex func-tions f, g on N R such that MA( f ) = MA( g ) and such that∆ ◦ ⊂ ∆( f ) ⊂ ∆ and ∆ ◦ ⊂ ∆( g ) ⊂ ∆ , we have to show that f − g is constant.Let f ∗ ( L ) := sup { x ∈ N R | L ( x ) − f ( x ) } be the conjugate function of f , also called the Legendre dual of f . It is a closed convex function on M R which takes the value ∞ preciselyoutside the stability set ∆( f ), and we have f ∗∗ = f (see [Roc70, § H ∈ ∆ ◦ such that ∂f ∗ ( H ) ∩ ∂g ∗ ( H ) = ∅ . We will seethat this contradicts the hypothesis that MA( f ) = MA( g ). For simplicity, we can assumethat H = 0. This amounts to translate ∆ and to add to f and g the same linear functionwhich does not change the Monge–Amp`ere measures. After adding a constant to f and to g , we can assume that min( f ) = min( g ) = 0. As before, let Ω := { x ∈ N R | f ( x ) < g ( x ) } .We are going to see that(A.5) 0 ∈ ∂f (Ω) ◦ , ∂g (Ω) . This and Proposition A.0.2 imply that MA( g )(Ω) = vol( ∂g (Ω)) < vol( ∂f (Ω)) = MA( f )(Ω),which contradicts the hypothesis MA( f ) = MA( g ).We show first that 0 ∈ ∂f (Ω) ◦ . Since H = 0 and min( f ) = 0, we have A := { x ∈ N R | f ( x ) = 0 } = ∂f ∗ (0) which, by hypothesis, is disjoint to { x ∈ N R | g ( x ) = 0 } = ∂g ∗ (0).Thus A ⊂ Ω. By [Roc70, Theorem 24.7], the set A is compact. Since Ω is open and A iscompact, there is a compact subset B of Ω with A ⊂ B ◦ . Then f | ∂B > 0. By PropositionA.0.3, ∂f ( B ) is a neighborhood of 0. Since ∂f ( B ) ⊂ ∂f (Ω), we get 0 ∈ ∂f (Ω) ◦ .Assume now that 0 ∈ ∂g (Ω). This means that there are sequences x k ∈ Ω and H k ∈ ∂g ( x k ) such that H k converges to zero and hence S := { H k | k ∈ N } ∪ { } is a compactsubset of M R . Since ∆ ◦ ⊂ ∂g ( N R ) by (A.1), we may assume that S is contained in theinterior ∆ ◦ of the domain ∆( g ) of g ∗ . Applying [Roc70, Theorem 24.7] to g ∗ , we deducethat ∂g ∗ ( S ) is compact. By [Roc70, Corollary 23.5.1], we have x k ∈ ∂g ∗ ( H k ) and hence wecan find a subsequence x i k that converges to a point y ∈ N R . By [Roc70, Theorem 24.4],we have 0 ∈ ∂g ( y ) and so g ( y ) = 0 < f ( y ). But x k ∈ Ω implies that f ( x k ) < g ( x k ) andhence f ( y ) ≤ g ( y ) by continuity leading to a contradiction. 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