A comparison of positivity in complex and tropical toric geometry

Given a smooth complex toric variety we will compare real Lagerberg forms and currents on its tropicalization with invariant complex forms and currents on the toric variety. Our main result is a correspondence theorem which identifies the cone of invariant closed positive currents on the complex toric variety with closed positive currents on the tropicalization. In a subsequent paper, this correspondence will be used to develop a Bedford-Taylor theory of plurisubharmonic functions on the tropicalization.

A smooth complex toric variety X Σ is a smooth algebraic variety over C with an open immersion of an algebraic torus T and an algebraic extension of the action of T on itself to X Σ . Such a variety is encoded by the combinatorial structure of a fan Σ in the vector space N R := N ⊗ Z R where N is the cocharacter lattice of T. We denote by T an := T(C) and X an Σ := X Σ (C) the respective complex analytic manifolds of complex points. Inside T an , there is a maximal compact torus S. The quotient X an Σ /S is denoted by N Σ . The topological space N Σ has a canonical stratification indexed by the cones of Σ, analogous to the stratification of X Σ into orbits. The incidence relations between cones of Σ translate to incidence relations between strata with the inclusions reversed. If τ is a face of the cone σ ∈ Σ, denoted as τ ≺ σ, then the corresponding strata satisfy N (σ) ⊂ N (τ ). The stratum corresponding to the cone {0} is equal to N R . It is called the dense stratum. Each stratum N (σ) of N Σ has a canonical structure of a finite dimensional real vector space equipped with a Z-structure. For τ ≺ σ ∈ Σ, there is a linear projection π σ,τ : N (τ ) −→ N (σ).
The space N Σ is a classical object in the theory of toric varieties where it appears also under the name manifold with corners (see e.g. [AMRT10,Oda88]]). In tropical geometry, it is called the Kajiwara-Payne tropicalization of X Σ . By construction, it comes with a natural map trop : X an Σ → N Σ which is a proper map of topological spaces. On the spaces X an Σ and N Σ , there are sheaves of bigraded algebras of smooth differential forms. Both are denoted A ·,· or, when we want to stress the underlying space, by A ·,· X an Σ and A ·,· N Σ respectively. The well-known sheaf of complex smooth differential forms A ·,· X an Σ is a sheaf of C-algebras and plays a central role in complex analysis and complex geometry. The sheaf A ·,· N Σ is a sheaf of R-algebras and was introduced by Smacka, Shaw and the third author [JSS19], based on work of Lagerberg [Lag12]. The smooth differential forms on X an Σ are called complex forms while the forms on N Σ are called Lagerberg forms. In both cases, the elements of A 0,0 are called smooth functions.
We explain briefly the definition of Lagerberg forms. More details are given in Section 3. If U is an open subset of the finite dimensional real vector space N (σ), Lagerberg [Lag12] has introduced the bigraded R-algebra where A · (U ) denotes the usual R-algebra of real smooth differential forms on U . For an open set U ⊂ N Σ , denote by U σ := N (σ) ∩ U its strata. Then a Lagerberg form on U is defined as a collection of forms (ω σ ) σ∈Σ , with ω σ ∈ A ·,· (U σ ), satisfying the following compatibility conditions. For every pair of cones τ ≺ σ and every point p ∈ U σ , there is a neighborhood V ⊂ U of p with V τ = π −1 τ,σ (V σ ) ∩ V and (1.1) ω τ | Vτ = π * τ,σ (ω σ | Vσ ) on V τ . The compatibility conditions (1.1) are, roughly speaking, saying that close to the boundary, Lagerberg forms are constant in the direction towards the boundary. Although this condition does not seem entirely natural from an archimedean point of view, it is very natural from both a tropical [JSS19] and a non-archimedean point of view [Jel19]. Moreover, it has very strong consequences. For instance, if ω is a form of bidegree (p, q), then the support of ω is disjoint to any stratum of dimension smaller than min(p, q). There are natural differential operators d , d of bidegree (1, 0) and (0, 1) turning A ·,· N Σ into a double complex analogous to the usual differential operators ∂ and∂ on A ·,· X an Σ . There is also a theory of integration for Lagerberg forms similarly to the complex case.
Therefore trop * sends symmetric forms to real forms. Furthermore, this morphism respects positivity and integration of top dimensional forms.
If U is contained in the dense stratum N R , then (1.2) is an isomorphism. In general, this is no longer true. The reason for this is the compatibility conditions (1.1). These results will be shown in Section 4. The normalization factors in equation (1.3) are almost forced by the compatibility with integration and the compatibility between the Lagerberg involution and complex conjugation (1.4). If we do not insist on compatibility with integration or with the bigrading, other identifications between Lagerberg forms and invariant complex forms are possible. For instance, the map (1.2) differs from the interpretation of Lagerberg forms as S-invariant forms given in [CD12, Remarque (1.2.12)].
The main interest of this paper will be currents. To define currents on an open subset U ⊂ N Σ , we first introduce a topology on the space of Lagerberg forms with compact support A ·,· c (U ). The definition is similar to the complex case with additional input caused by the compatibility condition (1.1) towards the boundary (see Subsection 3.2 for details). A Lagerberg current of type (p, q) on U is a continuous linear map T : A n−p,n−q c (U ) → R. We denote the space of currents of type (p, q) by D p,q (U ). By duality, the involution J defines an involution on D ·,· , hence a notion of symmetric currents. We call T ∈ D p,p (U ) positive, if it is symmetric and T (α) ≥ 0 for all positive Lagerberg forms α ∈ A n−p,n−p c (U ). Let again V = trop −1 (U ). Since the map trop is proper, the dual of the map trop * from (1.2) induces a C-linear map (1.5) trop * : D p,q (V ) −→ D p,q (U ) ⊗ R C defined by trop * (T )(α) = T trop * (α) for all T ∈ D p,q (V ) and α ∈ A n−p,n−q to follow a similar argument. In fact, it is more convenient to go for the dual notion of co-coefficients as follows: For simplicity, we consider the case X an Σ = C n with coordinates z = (z 1 , . . . , z n ), a situation that can always be achieved locally. Then N Σ = (R ∪ {∞}) n has coordinates (u 1 , . . . , u n ) and the tropicalization map is given by trop(z) = (− log |z 1 |, . . . , − log |z n |). The cones of Σ are the faces of R n ≥0 . Any cone in Σ has the form σ L := {u ∈ R n ≥0 | u i = 0 ∀i / ∈ L} for some L ⊂ {1, . . . , n} and the corresponding stratum of N Σ is given by A key ingredient in the proof of the Correspondence Theorem is the following Decomposition Theorem along the above stratification. with uniquely determined currents T σ such that U \ (E I∪J ∪ N (σ)) is a null set with respect to the Radon measure T IJ σ for any σ = σ L ∈ Σ. The decomposition (1.6) does not depend on the choice of the coordinates u 1 , . . . , u n hence gives a canonical decomposition for any positive current T ∈ D p,p (U ) on any open subset U of N Σ as we show in Theorem 6.1.6. A similar statement is well known on the complex toric manifold X an Σ and we show in Subsection 6.1 that both canonical decompositions are closely related via trop * .
For a positive Lagerberg current T ∈ D p,p (U ), we will prove that T = trop * (S) for a positive current S on V = trop −1 (U ) if and only if T has C-finite local mass. The latter is a local condition on U given in Definition 6.2.1. We then show that a closed positive Lagerberg current has C-finite local mass, completing the proof of surjectivity. For injectivity of trop * , an additional argument is required. All this is done in Subsection 7.1.
As an application, we will prove in Theorem 7.2.4 a tropical analogue of the Skoda-El Mir Theorem for Lagerberg currents in the toric setting: Let U be an open subset of N Σ and let E be a union of strata closures in N Σ . We consider a closed positive current T ∈ D p,p (U \E) which has C-finite local mass on U . Then we can extend T by zero to a closed positive Lagerberg current on U . For details, we refer to Subsection 7.2. A consequence of the Tropical Skoda-El Mir theorem is that in the canonical decomposition (1.6), if T is closed, then all the currents T L are closed.
The motivation for the present work is the following. It is known that there is no way to continuously extend the wedge product on A ·,· to D ·,· . Bedford-Taylor theory provides a way to define products of certain closed positive currents on X an Σ . In a subsequent paper, using the Correspondence Theorem and Bedford-Taylor theory for complex manifolds, we develop a Bedford-Taylor theory on N Σ . For instance, Bedford-Taylor theory on N Σ will have the following application: Let K be a field endowed with a non-archimedean complete absolute value | | v . From the fan Σ, we may construct a toric variety X Σ,K . We may also consider the analytification X an Σ,K of the toric variety X Σ,K as a Berkovich space and the corresponding tropicalization map trop K : X an Σ,K −→ N Σ which should be viewed as a non-archimedean analogue of the tropicalization map trop of the complex toric manifold X an Σ considered before. For any open subset U of N Σ , these tropicalization maps lead to a natural bijective correspondence between invariant continuous plurisubharmonic functions on trop −1 (U ) and invariant continuous plurisubharmonic functions on trop −1 K (U ) in the sense of Chambert-Loir and Ducros [CD12,§5.5]. Moreover, we show that the complex Bedford-Taylor theory corresponds to the Bedford-Taylor theory of Chambert-Loir and Ducros [CD12,§5.6].
We explain in more detail the content of the paper. In Section 2, we introduce complex and Lagerberg multilinear forms. We interpretate the Lagerberg forms as the complex forms which are invariant under a natural involution F . Then we discuss several positivity notions of these forms. In the complex case, weakly positive, positive and strongly positive multilinear forms are well-known. They form strictly convex cones of maximal dimension. Moreover, the cone of strongly positive forms is dual to the one of weakly positive forms, while the cone of positive forms is self dual. Similar notions are defined in [Lag12] for Lagerberg multilinear forms on a real vector space. The notions of weakly and strongly positive Lagerberg forms are however somewhat pathological. We show in Example 2.3.6 that the cone of strongly positive forms is not of full dimension and that the cone of weakly positive forms is not strictly convex. This also implies that the notions of weakly and strongly positive in the complex and Lagerberg case do not correspond exactly. For this reason, we will mainly restrict ourselves to positive forms.
In Section 3, we will first introduce the partial compactification N Σ associated to a fan Σ and we will describe its topology. Then we introduce Lagerberg forms on N Σ . Similarly as in complex analysis, we endow the space of compactly supported Lagerberg forms with a locally convex topology and we define the dual notion of Lagerberg currents. The upshot of this section is that Lagerberg forms and currents on N Σ satisfy similar properties as their complex analogues.
In Section 4, we study positivity of Lagerberg forms and compare them to invariant complex differential forms. In particular, we prove Theorem A. We first deal with the dense torus T in Subsection 4.1 before we consider arbitrary smooth toric varieties in Subsection 4.2.
Section 5 is devoted to positivity of Lagerberg currents and the relation with positivity of complex currents. We define the map trop * and discuss the compatibility of the different notions of positivity with respect to this map. We also define the co-coefficients of a Lagerberg current and show that, analogously to the complex case, the co-coefficients of a positive Lagerberg current are Radon measures. We show in Example 5.2.8 that there are weakly positive Lagerberg currents whose co-coefficients are not Radon measures. This is caused by the fact that the cone of weakly positive multilinear Lagerberg forms is not strictly convex.
In Section 6, we prove the decomposition theorem (see Theorem C). Moreover we introduce the concept of C-finite local mass. Intuitively, a current has C-finite local mass if it has local finite mass as an invariant complex current (see Definition 6.2.1 for details). We give in Example 6.2.3 a positive current T that does not have C-finite local mass. Nevertheless, it is of the form T = trop * (S) for a complex current S, but this current cannot be chosen to be positive.
Finally, Section 7 is devoted to the proof of Theorem B and of the tropical Skoda-El Mir theorem. In Appendix A, there is a reminder on Borel and Radon measures.
The set N of natural numbers includes zero. We write R ∞ = R ∪ {∞} and R ≥t := {u ∈ R | u ≥ t} for any t ∈ R. In the notation A ⊂ B, we allow that A = B.
In this paper, N usually denotes a free abelian group of rank n with dual M . We denote by N R and M R their scalar extensions to R. By a fan Σ in N R , we mean a fan consisting of strictly convex rational polyhedral cones in N R . We denote the associated (complex) toric variety by X Σ and the associated partial compactification of N R by N Σ (see §3.1).
A topological space is called locally compact if it has a basis consisting of relatively compact subsets; but it is not necessarily Hausdorff. A topological vector space is assumed to be Hausdorff. Our conventions on Radon measures are summarized in Appendix A.

Positivity on real and complex vector spaces
Let V be a real vector space of dimension n. Write V C = V ⊗ R C for the associated complex vector space, V * = Hom R (V, C) = Hom C (V C , C) for the complex dual and V = Hom R (V, R) for the real dual. Let V C denote the real vector space underlying V C with the complex structure determined by λ(v ⊗ µ) = v ⊗ (λµ). We denote the complex dual of V C by V * . Observe that V * agrees with the space of antilinear maps from V C to C. Let V be a copy of V . We consider the exterior algebras The elements of Λ p,q V * are called complex (p, q)-forms while the elements of Λ p,q V are called Lagerberg (p, q)-forms.
2.1. The complex situation. We recall some definitions from complex geometry. The identity induces antilinear maps σ : V C → V C and σ : Λ p V C → Λ p V C which we denote by w →w. The inverse of σ is also denoted by σ : V C → V C . The antilinear maps σ induce antilinear maps σ : V * → V * and σ : V * → V * , that extend uniquely to an antilinear involution of the R-algebra Λ ·,· V * . This involution sends Λ p,q V * to Λ q,p V * . We continue to write σ(ω) =ω for complex forms ω. A complex form ω is called real ifω = ω.
We choose a real basis e 1 , . . . , e n of V . We obtain dual complex bases du 1 , . . . , du n of V * and dū 1 , . . . , dū n of V * determined by du i (e j ⊗ 1) = δ ij = dū i (e j ⊗ 1). Then we have Definition 2.1.1. The canonical orientation of the vector space V C is the orientation determined by the real form ω n = du 1 ∧ idū 1 ∧ · · · ∧ du n ∧ idū n ∈ Λ n,n V * .
The form ω n depends on our choice of a basis, but the orientation does not.
An ω ∈ Λ p.p V * is called positive if it belongs to the convex cone spanned by A complex (p, p)-form ω ∈ Λ p,p V * is called weakly positive if, for every strongly positive form η of type (n − p, n − p), there is a real number γ ≥ 0 with We denote by Λ p,p +,s V * , Λ p,p + V * and Λ p,p +,w V * the cones of strongly positive, positive and weakly positive (p, p)-forms respectively.
Observe that our weakly positive forms are called positive in [Dem12, §III.1.A].
Definition 2.1.2. To η ∈ Λ p,p V * we associate a sesquilinear form |η| on Λ p V C by the rule Moreover, for q = n − p, there is a duality pairing We denote by Λ p,p V * R the subspace of real elements. Then the pairing (2.1) induces a pairing Note that the assignment η → |η| is canonical and gives an isomorphism between Λ p,p V * and the space of sesquilinear forms on Λ p V C . On the other hand, the duality pairing ·, · depends on the choice of basis but only up to a non-zero positive number. Proof. The antilinear involution σ on Λ p,p V * is given by Thus, the sesquilinear form |η| is hermitian. The converse is proved analogously. Assume now that η ∈ Λ p,p V * is a positive form and x ∈ Λ p V C . Then we can write with γ j ∈ R ≥0 and α j ∈ Λ p,0 V * . Then, since i p 2 (−1) proving that |η| is positive semidefinite. Conversely, assume that |η| is positive semidefinite. Then by the spectral theory of Hermitian forms, there are γ j ∈ R ≥0 and α j ∈ Λ p,0 V * = (Λ p V C ) * such that showing that η is a positive form. Lemma 2.1.5. Any weakly positive form in Λ p,p V * is real.
Proof. The fact that ω n and any positive or strongly positive form are real follows directly from the definition. Let now ω be a weakly positive form in Λ p,p V * . This means that for each strongly positive form η in Λ q,q V * , there is a non-negative real number γ such that Since η, γ and ω n are real, this implies that Hence (ω −ω) ∧ η = 0 for any strongly positive form η in Λ q,q V * . By Lemma 2.1.4, Λ q,q V * admits a basis of strongly positive elements. Therefore (ω −ω) ∧ η = 0 holds for any η ∈ Λ q,q V * . By duality, we get ω −ω = 0 and hence ω is real.
Corollary 2.1.6. For p ∈ N, there are inclusions of closed convex cones For q := n − p, the cones Λ p,p +,s V * and Λ q,q +,w V * are dual to each other and the cone Λ p,p + V * is the dual of Λ q,q + V * with respect to the real duality pairing (2.2). Proof. By definition, the spaces of strongly positive forms and of positive forms are convex cones contained in Λ p,p V * R . Lemma 2.1.5 implies that Λ p,p +,w V * is contained in Λ p,p V * R . Then Λ p,p +,w V * is a closed convex cone as the dual of the convex cone Λ p,p +,w V * . We next show that the convex cone of strongly positive forms is closed. Choose any hermitian metric in the complex vector space Λ p V * and let S ⊂ Λ p V * be the unit sphere. The set K ⊂ S of totally decomposable elements is closed as it is the preimage of the Grassmanian Gr(p, V * ) under the projection S → P(Λ p V * ). Since S is compact, so is K. Let K ⊂ Λ p,p V * be the image of K under the continuous map Then K is a compact set that does not contain 0. Thus the convex cone over K is closed [Roc70, Cor. 9.6.1]. Since the convex cone over K is Λ p,p +,s V * , we deduce that the space of strongly positive forms is a closed convex cone. Being Λ p,p +,s V * closed and convex, it agrees with its double dual. Therefore Λ p,p +,s V * is the dual of Λ p,p +,w V * . We next prove that Λ p,p + V * is self dual. Let γ ∈ Λ n V * . Then γ = zdu 1 ∧ · · · ∧ du n for some z ∈ C and hence i n 2 γ ∧γ = zzω n . For α ∈ Λ p V * and β ∈ Λ q V * , we conclude that is a positive multiple of ω n . Here we have used the identity i p 2 i q 2 (−1) pq = i n 2 . This proves that the cone Λ p,p + V * is contained the dual of Λ q,q + V * . To prove the converse, we introduce the isomorphism ϕ : Then, for any pair of forms η ∈ Λ p,p V * and α ∈ Λ q V * , the equality η ∧ i q 2 α ∧ᾱ = |η|(ϕ(α), ϕ(α))ω n is satisfied. Therefore, if η belongs to the dual cone of Λ q,q + V * , then the sesquilinear form |η| is positive semidefinite. By Proposition 2.1.3, the form η is positive proving the reverse inclusion. We deduce that the cone Λ p,p + V * is closed because it is a dual cone. Remark 2.1.7. For p = 0, 1, n−1, n, the notions of strong positivity, positivity and weakly positivity agree (see [Dem12,  2.2. The real situation. We now shift to positivity of Lagerberg forms following [Lag12]. We consider again a real basis e 1 , . . . , e n of V which induces dual bases d u 1 , . . . , d u n of V and d u 1 , . . . , d u n of V . We denote by JV another copy of our n-dimensional R-vector space V and let us denote by J : V → JV the identity map. There is a unique involution on V ⊕ JV that extends J and which we also denote by J. From now on, we make the identification V := Hom(JV, R) and then duality yields an involution on V ⊕ V which we also call J. There is a unique algebra homomorphism on Λ ·,· V that extends J. It is again an involution mapping Λ p,q V onto Λ q,p V . This map, also denoted by J, is called the Lagerberg involution.
We call ω strongly positive if it belongs to the convex cone spanned by We call ω positive if it belongs to the convex cone spanned by A symmetric Lagerberg (p, p)-form ω ∈ Λ p,p V is called weakly positive if for every strongly positive form η of type (n − p, n − p), there is a real number γ ≥ 0 with ω ∧ η = γτ n .
We will also denote as Λ p,p +,s V , Λ p,p + V and Λ p,p +,w V the spaces of Lagerberg (p, p)-forms that are strongly positive forms, positive and weakly positive respectively. From our definition, we deduce that the product of positive Lagerberg forms is positive.
Note that, again, the assignment η → |η| is canonical and gives an isomorphism between Λ p,p V and the space of bilinear forms on Λ p V . On the other hand, the duality pairing ·, · depends on the choice of a basis, but only up to a positive number.
Proposition 2.2.4. A form η ∈ Λ p,p V is symmetric if and only if the bilinear form |η| is symmetric. A form η is positive if and only |η| is a positive semidefinite symmetric form.
Proof. The proof is similar to the complex case and is given in [Lag12, Proposition 2.1] Denote now by Λ p,p sym V the subspace of symmetric elements and let q := n−p. The duality pairing of Definition 2.2.3 induces a real duality pairing, denoted by the same symbol, ·, · : Λ p,p sym V ⊗ Λ q,q sym V → R. We give the analogue of Corollary 2.1.6 which was stated before [Lag12, Lemma 2.2].
Corollary 2.2.5. For p ∈ N, there are inclusions of closed convex cones For q := n − p, the cones Λ p,p +,s V and Λ q,q +,w V are dual to each other and the cone Λ p,p + V is the dual of Λ q,q + V with respect to the above real duality pairing. Proof. The arguments are as in Corollary 2.1.6 replacing complex by real numbers and sesquilinearforms by symmetric bilinear forms.
Remark 2.2.6. As in the complex case, strong positivity agrees with positivity and weak positivity for p = 0, 1, n − 1, n. Similarly as in [Dem12, Example III 1.10], there are positive Lagerberg forms that are not strongly positive for any 2 ≤ p ≤ n − 2.
2.3. Comparison between the real and complex situations. We aim for an identification of the space of Lagerberg forms Λ ·,· V with a subspace of the space of complex forms Λ ·,· V * that preserves positivity as much as possible.
Definition 2.3.1. The vector space V * = Hom R (V, C) has an antilinear involution F coming from complex conjugation in C. We extend F to V * ⊕ V * in such a way that F and σ (the complex conjugation from 2.1) anticommute: There is a unique antilinear involution of the R-algebra Λ ·,· V * which extends F . We denote this extension also by F .
Note that F induces an antilinear involution of any Λ p,q V * . In coordinates, we have We see V ⊂ V * and V ⊂ V * as the subspaces of F -invariant elements. Note that V looks like the space of real elements of V * , while, due to the twisted definition (2.4) of F in V * , the elements of V look like the imaginary elements of V * .
We extend the inclusion V ⊕ V → V * ⊕ V * to an R-algebra homomorphism This inclusion sends d u j to du j and d u j to idū j .
Proposition 2.3.2. We have the following compatibilities for the inclusion (2.7). Proof. Since F is an antilinear involution, the space of F -invariant elements of Λ ·,· V * has real dimension 2 2n which agrees with the dimension of Λ ·,· V . Since, by construction, the Lagerberg forms are invariant under F , both spaces agree. The remaining statements are direct computations.
Lemma 2.3.3. The involution F maps strongly positive (resp. weakly positive, resp. positive) complex forms to strongly positive (resp. weakly positive, resp. positive) complex forms.
Proof. For any complex (p, 0)-form α, antilinearity of F and (2.6) give We conclude that F preserves positivity of complex forms. Multiplicativity of F and (2.8) show also that F preserves strong positivity of forms.
Since ω n is the image of τ n under the above identification, it is fixed under F . Using that F is an involution, we deduce from duality that F preserves weak positivity as well. Proof. It is easily seen that if ω ∈ Λ p,p V is a positive Lagerberg form, then it is also a positive complex form by using the embedding (2.7) and that Jω = i pω . Conversely, assume that ω is a Lagerberg (p, p)-form that is positive as a complex form.
We claim that is enough to show that if η = (−1) with γ s ∈ R ≥0 and α s ∈ Λ p,0 V * . Using the claim and the F -invariance of ω, we have that is a positive Lagerberg form.
The next example shows that the analogues of Proposition 2.3.5 for strongly and weakly positive forms do not hold. Hence the converse of Proposition 2.3.4 (i) and (ii) is wrong.
Example 2.3.6. We show that the space of strongly positive Lagerberg forms does not span the space of symmetric Lagerberg forms. In particular, the cone of strongly positive Lagerberg forms has empty interior and not every symmetric Lagerberg form can be written as a difference of strongly positive ones.
By Lemma 2.1.4, every complex form ω can be written as a complex linear combination of strongly positive complex forms ω i . If ω is real, then applying σ to (2.9) and using that the ω i are real, we see that we may take the λ i to be real. If ω is further invariant under F , then applying F to (2.9) we see that replacing ω i by 1/2(ω i + F (ω i )) we may assume the ω i to be F -invariant. Note that F (ω i ) is strongly positive by Lemma 2.3.3.
We have just shown Λ p,p sym V = (Λ p,p +,s V * ) F R . Since we showed in Example 2.3.6 that Λ p,p +,s V R Λ p,p sym V , we find that (Λ p,p +,s V * ) F ⊂ Λ p,p +,s V . By Proposition 2.3.2, this means exactly that not every Lagerberg form that is strongly positive as a complex form is strongly positive as a Lagerberg form.
The next example illustrates this phenomenon.
Example 2.3.8. Let V still be a real vector space of dimension four. Choose a basis e 1 , . . . , e 4 and corresponding bases of V * , V * , V and V as before. The complex form is strongly positive by definition. By Lemma 2.3.3, the form ω = 1 2 (η + F (η)) is strongly positive. Moreover, it is F -invariant and hence ω may be seen as a Lagerberg form by Proposition 2.3.2. We claim that ω is not strongly positive as a Lagerberg form.
A direct computation shows that This shows that ω is a positive Lagerberg form, that the associated symmetric bilinear form |ω| has rank 2 and that (ker |ω|) ⊥ is the 2-dimensional subspace We pick any decomposition with α ∈ Λ 2,0 V . Then α j ∈ (ker |ω|) ⊥ , since for any v ∈ ker |ω| ⊂ Λ 2 V , we have If ω were strongly positive, we would have a decomposition like (2.10) where α j ∈ Λ 2,0 V is a product of (1, 0)-forms. However, this is not possible because (ker |ω|) ⊥ does not contain any non-zero real decomposable element as the following argument shows. Assume that

This implies the equations
The point ρ determines a point ρ in the Grassmannian Gr(2, 4) with Plücker coordinates The previous equations imply that the Plücker coordinates of ρ satisfy the equations Moreover, the Plücker equations for Gr(2, 4) are reduced to the single equation We conclude that the Plücker coordinates of ρ satisfy the equation (2.11) y 2 + z 2 = 0 that has no real solutions except the trivial one. The fact that ω is strongly positive as a complex form is reflected by the fact that (2.11) has non-trivial complex solutions.

Lagerberg forms and Lagerberg currents on partial compactifications
For convex geometry we will use the notation and conventions set up in [GK17,Appendix]. 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.
3.1. Partial compactifications. A strictly convex rational polyhedral cone σ ∈ N R is a polyhedron defined by finitely many equations of the form ϕ( . ) ≥ 0 with ϕ ∈ M , that does not contain a positive dimensional linear subspace. A rational polyhedral fan Σ in N R is a polyhedral complex all of whose polyhedra are strictly convex rational cones. In this paper we make the convention that a fan is always a rational polyhedral fan.
A cone is called smooth if it is generated by a subset of a Z-basis of N . A fan Σ is called smooth if each cone of Σ is smooth. For σ ∈ Σ we define the monoid Definition 3.1.1. Let Σ ⊂ N R be a rational polyhedral fan. We consider the disjoint union and call N Σ equipped with the topology introduced in Remark 3.1.2 the partial compactification of N R associated to Σ.
Remark 3.1.2. The partial compactification N Σ carries the following topology which is Hausdorff. It is also locally compact and has a countable basis and hence it is metrizable. Let us briefly recall its definition.
First, we define the partial compactification of N (σ) for a single cone σ ∈ Σ by setting The set N σ is naturally identified with the monoid homomorphisms Hom Mon (S σ , R ∞ ). We equip it with the subspace topology of R Sσ ∞ . Using a finite set of generators ϕ 1 , . . . , ϕ k for the monoid S σ , we can realize Hom Mon (S σ , R ∞ ) as a closed subspace of R k ∞ with the induced topology (see [Pay09,Remark 3.1] and use [Pay09, Lemma 2.1]).
For a face ρ of σ, we note that N ρ is an open subset of N σ . This is used to define a topology on the partial compactification N Σ by gluing the partial compactifications N σ , σ ∈ Σ, along the open subsets induced by common faces.
We give a second description of the topology of N Σ . To this end, we fix an Euclidean metric in N R . For a cone ν of Σ, the Euclidean metric allows us to identify ν ⊥ with a subspace of N R and, through the projection π ν , with the space N (ν). Again, we consider first the case of a single cone σ ∈ Σ. For a point u ∈ N σ , there is a unique face ν of σ with u ∈ N (ν). Let u 0 ∈ ν ⊥ ⊂ N R be the corresponding point and let U be a neighborhood of u 0 in ν ⊥ . For each face τ ≺ ν, the cone ν induces a cone π τ (ν) contained in N (τ ). For each p ∈ ν, we write The topology of N σ is defined by the fact that {W (ν, U, p)} U,p is a basis of neighbourhoods of u in N σ for any u ∈ N σ . As before the topology of N Σ is defined by gluing along the open subsets N τ of N σ whenever τ ≺ σ.
The first definition of the topology is given by Kajiwara [Kaj08] and by Payne [Pay09], the second definition is from [AMRT10, I.1]. The topologies coincide as the above basis of neighbourhoods works also for the first definition by [Pay09,Remark 3.4].
To prove that N Σ is Hausdorff, we use that the quotient of a topological space (in this case the disjoint union of the N σ ) by an equivalence relation is Hausdorff if the canonical map to the quotient is open and the graph of the equivalence relation is closed [Bou71, Ch. I §8.3 Prop. 8]. As the map to the quotient is open by construction, it is enough to show that, for cones σ 1 and σ 2 with τ = σ 1 ∩ σ 2 , the map is a closed immersion. This follows easily from the first description of the topologies of N σ 1 and N σ 2 by choosing a finite set of generators of S σ 1 and S σ 2 and observing that the union of both sets is a set of generators of S τ .
Note that N Σ is locally compact because the N σ provide an open covering of N Σ and each of them is locally compact. Finally every N σ has a countable basis and hence also N Σ .
be the split complex torus with cocharacter lattice N . Let X Σ denote the toric variety over C with dense torus T determined by the fan Σ in N R . Let X an Σ denote the analytification of X Σ , i.e. the set of complex points X Σ (C) with its structure of an analytic space.
There is a well-known continuous map (see for example [AMRT10, I.1, p.2], [Kaj08, which is nowadays called tropicalization map as it is given by glueing on the affine open To illustrate the topology on the partial compactification N Σ , we give the following lemma where the notation coincides with the one in [AMRT10, I.1, p.5]. Lemma 3.1.4. Let Σ ⊂ N R be a fan, and let N Σ be its associated partial compactification. Given p ∈ N R and v ∈ |Σ|, the limit p + ∞v := lim µ→+∞ p + µv exists in N Σ . Moreover, p + ∞v ∈ N (σ) for the unique cone σ ∈ Σ such that v ∈ relint(σ).
Proof. We use the description of the topology of N Σ given by the basis of neighborhoods W (σ, U, q), so we fix an Euclidean metric in N R . Let p 0 ∈ σ ⊥ be the point corresponding to π σ (p), U a neighborhood of p 0 in σ ⊥ and q ∈ σ. It is enough to show that there is a µ 0 > 0 and for all µ ≥ µ 0 , the condition p + µv ∈ W (σ, U, q) holds. Since π σ (p) = π σ (p 0 ) and q ∈ σ, we deduce that p − p 0 − q ∈ σ R . Since v ∈ relint(σ), there is a µ 0 such that for all µ ≥ µ 0 , we have p − p 0 − q + µv ∈ σ and hence p + µv ∈ q + U + σ = W (σ, U, q).
Example 3.1.5. As an example, we consider the toric variety P 2 and its tropicalization shown in Figure 1. Note that for v = (0, 1), the point p + ∞v is the point in N (σ 3 ) lying vertically above p. 3.2. Lagerberg forms and Lagerberg currents. Recall from [Lag12] that for every open subset U of N R , there is a bigraded R-algebra of Lagerberg forms A ·,· (U ) with differentials d and d of bidegree (1, 0) and (0, 1). Lagerberg forms were introduced by Lagerberg in loc. cit. under the name superforms. They are defined as where A · (U ) denotes the R-algebra of real valued smooth differential forms on U .
We choose a basis of N which defines coordinates u 1 , . . . , u n on N R . Then we may write a Lagerberg form α as where I = {i 1 < · · · < i p } and J = {j 1 < · · · < j q } range over all subsets of {1, . . . , n}, where f IJ are smooth real functions on U and we use the multi-index notation There are natural differentials d : A p,q (U ) → A p+1,q (U ) and d : A p,q (U ) → A p,q+1 (U ), which are in coordinates given by The product of the bigraded R-algebra A ·,· (U ) is alternating and we denote it by ∧. The algebras A ·,· (U ) form a sheaf on N R that is denoted by A ·,· or by A ·,· N R . The algebra A ·,· (U ) carries a natural involution J that permutes bidegrees and is deter- Let us fix a fan Σ ⊂ N R . Following [JSS19, Definition 2.4], smooth forms on open subsets of N Σ are defined as follows.
Definition 3.2.1. Let U ⊂ N Σ be an open subset. For every σ ∈ Σ, we write U σ := U ∩ N (σ). A Lagerberg form of type (p, q) on U is given by a family ω = (ω σ ) σ∈Σ with ω σ ∈ A p,q (U σ ) satisfying the following local condition. For each p ∈ U σ , there exists a neighborhood V of p in U such that for all τ ≺ σ we have . We denote by A p,q (U ) the real vector space of Lagerberg forms of type (p, q) on U . There are unique differentials d : defines a sheaf A p,q of real vector spaces on the topological space N Σ . If we want to stress the fact that A p,q is a sheaf on N Σ , we will denote it by A p,q N Σ , The support supp(ω) of a Lagerberg form ω ∈ A p,q (U ) is the closed subset of points of U where ω has a non-zero germ in the stalk. The space of Lagerberg forms of type (p, q) on an open subset U of N Σ with compact support is denoted A p,q c (U ).
x in x of the sheaf of Lagerberg forms on the real vector space N (σ). This follows from Definition 3.2.1.
This is caused by the fact that forms of large degree vanish automatically at the boundary.
We next discuss a topology on the space A p,q c (U ). This topology is modeled on the topology of the space of test forms used in analysis, see for instance [Rud73,§6]. In fact, for an open subset U of N Σ , we will define topologies on certain subspaces of A p,q c (U ) and use a limit process to define a topology on A p,q c (U ). Moreover, we shall describe the convergent sequences in A p,q c (U ). In the following, we fix a basis u 1 , . . . , u n of M which defines coordinates (u 1 , . . . , u n ) : N R ∼ → R n and allows to write Lagerberg forms on U in terms of standard forms d u I ∧ d u J for subsets I, J ⊂ {1, . . . , n}.
holds for all i and all cones σ, τ ∈ Σ with τ ≺ σ and for each m ∈ N using the supremum norm K of continuous real functions on the compact set K. The family of norms (3.4), where m ∈ N varies, defines on A p,q K (U, (V i ) i ) the structure of a locally convex topological vector space which is complete with respect to a translation invariant metric and hence it is a Fréchet space. The induced topology is denoted by We put on A p,q c (U ) the topology τ , defined as the limit topology in the category of locally convex topological vector spaces. Note that this may be different from the direct limit in the category of topological vector spaces.
As mentioned previously, the topology of A p,q c (U ) is modeled on the classical topology on the space of test functions in [Rud73,§6] and its formal properties are very similar. For instance, if U is not compact, then A p,q c (U ) is not metrizable. Nevertheless the topology on A p,q c (U ) has many nice properties and the fact that is not metrizable is only a minor issue. Remark 3.2.5. The spaces A p,q c (U ) have the same properties as the test function spaces in [Rud73,Chapter 6]. This is a consequence from the fact that E := A p,q c (U ) is an LF -space as introduced by Dieudonné and Schwartz [DS49]. This means that the vector space E is a countable union of strictly increasing Fréchet spaces E k such that the topology on E k agrees with the induced topology from E k+1 . Indeed, using that U has a countable basis, it is clear that the direct limit can be by described by using countable many (K k , (V k,i ) i ) such that the compact subset K k lies in the interior of K k+1 and such that a subfamily of that E has a canonical structure as a locally convex space which is the finest structure such that the topology on E n agrees with the induced topology and it follows that E is the direct limit of the E n in the category of locally convex spaces.
All properties of test function spaces from [Rud73, Chapter 6] were shown in [DS49] more generally for LF -spaces and so they apply to A p,q c (U ). In fact, an LF -space is not only sequentially complete, but a complete Hausdorf space [DS49, Corollary of Theorem 6]. For our paper, we need mainly the following results about sequences. Even if the space A p,q c (U ) is not metrizable, for many purposes, sequences are enough.
Proposition 3.2.7. Let T : A n−p,n−q c (U ) → R be a linear functional. Then the following conditions are equivalent.
(i) The map T is continuous.
As in the classical case of distributions, we now define currents as the topological dual of the space of smooth forms with compact support.
and a product and ω ∈ A n−q,n−p c (U ) the Lagerberg current J(T ) ∈ D q,p (U ) is given by Integration of Lagerberg forms gives examples of currents. We start by recalling the integration theory of Lagerberg forms. If U ⊂ N Σ is an open subset and η ∈ A n,n (U ) is a Lagerberg form with compact support, using the chosen basis of M , we write Denote by dλ the Lebesgue measure on N R induced by the lattice N . The integral of η is defined as Since the support of any compactly supported Lagerberg form of type (n, n) is a compact subset of N R , the integral is finite. Since two isomorphisms N ∼ = Z n differ by a matrix of determinant ±1, the integral does not depend on the choice of coordinates.
Example 3.2.10. Let U ⊂ N Σ be an open subset. We will use the map This map is a morphism of A ·,· (U )-modules compatible with the actions of d , d and J.
Example 3.2.11. Let U be an open subset of N Σ . For every real Radon measure µ on U (see Appendix A), there exists a unique Lagerberg current T ∈ D n,n (U ) such that Indeed, for a compact subset K of U and a finite covering (V i ) i∈I of K, the canonical maps are morphisms of locally convex vector spaces. By the universal property of the direct limit, the composition of these maps induces a continuous map A 0,0 . This shows our claim. Proposition 3.2.12. Let (U i ) i∈I be an open cover of an open subset U of N Σ . Then there exists a partion of unity subordinate to the given cover (U i ) i∈I , i.e. a countable, locally finite open cover (V j ) j∈J of U together with a map s : J → I such that V j ⊂ U s(j) for all j ∈ J and a collection of non-negative functions f j ∈ A 0,0 c (V j ) such that j∈J f j ≡ 1.
Proof. By general arguments, see [War83, Theorem 1.11], it is sufficient to show that, given a point x and a neighborhood V of x in U , there exists a function f ∈ A 0,0 c (V ) that is constant equal to 1 on a neighborhood of x. This statement is clearly local, so we may assume that our fan Σ is generated by a single cone σ and N Σ = N σ . We have seen in Remark 3.1.2 that a choice of k generators of the cone σ leads to a realization of N σ as a closed (polyhedral) subset of R k ∞ . Now the existence of a partition of unity for an open subset of R k ∞ from [JSS19, Lemma 2.7] readily shows the claim.
Let U be an open subset of N Σ . Recall from Appendix A that the space C 0 c (U, R) of real valued continuous functions on U with compact support has a canonical structure of a locally convex topological vector space.
Proof. It is enough to show that, for any continuous function f ∈ C 0 c (U ), there is a sequence of functions g k ∈ A 0,0 c (U ), k ≥ 0, that converges to f in the topology of C 0 c (U ). Let K be the support of f . Using that N Σ is locally compact, we can easily find open subsets U 1 , U 2 and compact subsets K 1 , K 2 with By Proposition 3.2.12, the Stone-Weierstrass Theorem [Rud64, Theorem 7.32] implies that the R-algebra . Again by Lemma 3.2.12, there is a smooth function 0 ≤ ρ ≤ 1, whose support is contained in U 1 and with ρ| K ≡ 1. Then the sequence of smooth functions given by g k = ρh k converges to f in C 0 c (U ).

Positivity for complex invariant forms and Lagerberg forms
In this section we study positive forms on a smooth complex toric variety that are invariant under the action of the compact torus and compare them to positive Lagerberg forms. We keep the setting from Section 3. 4.1. Invariant forms in the case of the torus. We start with the case of the complex algebraic torus T = Spec C[M ] of dimension n with character lattice M and cocharacter lattice N . We fix a splitting N ∼ = Z n that induces holomorphic coordinates z 1 , . . . , z n on T as well as linear coordinates u 1 , . . . , u n on N R . As before we denote the associated complex manifold M ⊗ Z C × ∼ = (C × ) n by T an . We will also consider the real compact torus We will denote by A either the sheaf of complex differential forms on T an or the sheaf of (real) Lagerberg forms on N R . The context will always allow us to distinguish between them. If not we will denote the former as A T an and the latter as Remark 4.1.1. Let V be an S-invariant subset of T an . The subalgebra A(V ) S is a direct factor of A(V ) because averaging with respect to the Haar probability measure µ S of S defines a canonical projection where a : T an → T an denotes translation by a ∈ T an . Definition 4.1.2. Let F be the antilinear involution of the sheaf of C-algebras A =A T an determined by F (f ) =f for f ∈ A 0,0 and by . . , n. The S-invariant one-forms dz j /z j and idz j /z j (j = 1, . . . , n) generate a subsheaf of R-algebras B of A such that A ·,· = A 0,0 ⊗ R B ·,· . Observe that this definition of B does not depend on the choice of the splitting N ∼ = Z n which induces the coordinates z 1 , . . . , z n . The antilinear involution F above is the R-linear endomorphism on A T an given as the tensor product of complex conjugation on A 0,0 with the identity on B. We conclude that the involution F is indeed well defined. Proof. For f ∈ A 0,0 (V ) and j = 1, . . . , n, we have a * f = a * f , a * (dz j /z j ) = dz j /z j , and a * (idz j /z j ) = idz j /z j .
We deduce that a * F (ω) = F (a * ω) for any ω ∈ A(V ) and the claim follows. Proof. If ω ∈ A(V ) S , then we can write where the functions f I,K are smooth complex valued functions of n real variables. We denote by ∂ j f I,K the partial derivative with respect to the j-th variable. Clearly Then F (∂ω) is equal to Thus the commutativity between ∂ and F follows from equation (4.1).
Therefore the commutativity of F and i∂ also follows from (4.1).
The next example shows that S-invariance of ω is necessary in Lemma 4.1.4. The next goal is to give an identification between the algebra of S-invariant forms that are also F -invariant, and the algebra of real Lagerberg forms on N R . This identification will respect the natural differential operators. Recall that J denotes the Lagerberg involution introduced in Remark 3.2.2. for all ϕ ∈ A 0,0 (U ) and such that Proof. We first recall some formulas from complex analysis in one variable. If z = re iθ and u = − log |z| = − log(r), since r 2 = zz, we have For j = 1, . . . , n write r j = |z j | = (z jzj ) 1/2 . Then trop * (u j ) = − log(r j ).
Therefore, equations (4.2) and (4.3) imply Since trop * is an algebra homomorphism, we deduce that, if ω ∈ A(U ) is written as then the corresponding form on the torus is given by This proves the uniqueness of the map trop * . Conversely, using equation  The next remark shows that Lagerberg forms and the involution F are pointwise described by the linear algebra in Section 2.
Remark 4.1.10. Choose a point x ∈ T an and let y = − log |x| = trop(x) denote its image under the tropicalization map. Write V = T y N R for the tangent space to N R at y. Let V , V , V * and V * be defined as Section 2. There is an isomorphism This isomorphism is compatible with the map trop * in the sense that trop * (ω)(x) = trop * x (ω(y)). Let τ n ∈ Λ n,n V be the Lagerberg orientation as in Definition 2.2.1. Then (4.9) trop * x (τ n ) = i n dz 1 ∧ dz 1 ∧ · · · ∧ dz n ∧ dz n (4π) n |x 1 | 2 . . . |x n | 2 .
Note that the denominator of this form is strictly positive in the torus T an therefore, any positivity notions on T an defined using the orientation trop * x (τ n ) or the orientation i n dz 1 ∧ dz 1 ∧ · · · ∧ dz n ∧ dz n agree.
Let ω ∈ A p,q (W ) for some open subset W of T an and x ∈ W . We get from (2.5) that (4.10) where F on the righthand side is the involution from Definition 2.3.1. As we have seen in Example 2.3.6, it is reasonable to restrict our attention to positive Lagerberg forms. Nevertheless we add the other positivity notions for further reference. Remark 4.1.13. Remark 4.1.10 shows that the correspondence (4.11) π −1/2 ∂ ←→ d and π −1/2 i∂ ←→ d from Proposition 4.1.7 was already used implicitly in Section 2 in order to preserve positivity and the bigrading between the complex and the Lagerberg forms.  Let us explain our choices which lead to the achieved correspondence (4.11): Our first condition is that trop * should be a differential homomorphism of R-algebras with respect to the differential operators d and d of A(U ) and natural differential operators ∂ and ∂ on A(V ). Naturality means here that ∂ , ∂ should be in the two-dimensional C-vector space spanned by ∂ and∂. The second condition is that trop * respects the bigrading which implies ∂ = a∂ and ∂ = bi∂ for some a, b ∈ C. Third, the range of trop * should be contained in the F -invariant forms in V which yields a, b ∈ R. Observe that these three conditions imply already that trop * preserves positivity of forms (see Lemma 4.1.12). Our fourth condition is (4.4) which forces a = b. The fifth condition is that trop * is compatible with integration (see Lemma 4.2.5 below). This gives ab = 1/π. Hence we have seen that the five conditions given above fix our choices in Proposition 4.1.7 up to a sign.

4.2.
Invariant forms in the case of a toric variety. The next goal is to partially extend Proposition 4.1.7 to toric varieties. Let Σ be a smooth fan in N R . Let X Σ be the corresponding smooth complex toric variety and let N Σ be the corresponding partial compactification of N R as in Section 3. We denote by X an Σ the complex manifold associated to X Σ . Recall from Remark 3.1.3 that the tropicalization map is a proper continuous map trop : X an Σ → N Σ that identifies N Σ with X an Σ /S. Remark 4.2.1. Let V ⊂ X an Σ be an S-invariant open subset. Then, in general, the involution F does not induce an involution of A ·,· (V ). In fact, if ω ∈ A ·,· (V ), then F (ω| V ∩T an ) may not extend to a smooth form on V , as the following example shows.
We set X Σ = A 1 C and V = X an Σ = C. Then dz ∈ A 1,0 (V ), but is not a smooth form in 0. Nevertheless, the next result implies that F can be extended to smooth S-invariant forms.
Lemma 4.2.2. Let V ⊂ X an Σ be an S-invariant open subset. Let ω ∈ A ·,· (V ) S be a smooth S-invariant form on V . Then F (ω| V ∩T an ) extends uniquely to a smooth form S-invariant form on V and hence F induces an antilinear involution on A ·,· (V ) S also denoted by F .
Proof. Since the statement is local we may assume that V is contained in the affine toric variety X σ for some σ ∈ Σ. We choose a system of toric coordinates (z 1 , . . . , z n ) and write ω = f I,K (z 1 , . . . , z n )dz I ∧ dz K .
Moreover, the S-invariance of V yields that V is invariant under complex conjugation of the coordinates. Since ω is S-invariant, each summand f I,K dz I ∧ dz K is also S-invariant and hence we can write for a unique smooth function g I,K ∈ C ∞ (V ). On V ∩ T an , this yields which can be uniquely extended to a smooth form on V because ω is smooth. By Lemma For any open S-invariant subset V ⊂ X an Σ , we will denote by A(V ) S,F the R-subalgebra of forms that are at the same time invariant under S and under F . As before we obtain a sheaf of bigraded R-algebras (trop * A X an Σ ) S,F on N Σ . Proposition 4.2.3. There is a unique morphism of sheaves of bigraded R-algebras For an open subset U of N Σ and ω ∈ A p,q (U ), we have (4.14) trop * (J(ω)) = i p+q trop * (ω).
Proof. This follows easily from the definitions and Proposition 4.1.7.
Remark 4.2.4. If U is an open subset of N R , then we have seen in Proposition 4.1.7 that the map (4.12) is an isomorphism. For an open subset U of N Σ , the map is obviously still injective, but in general no longer surjective. The latter can be seen already in the one dimensional case. Let N = Z and Σ the fan with a single maximal cone σ = R ≥0 . Then X Σ = A 1 C and N Σ = R ∪ {∞}. Consider the smooth function ϕ(z) = zz on X an Σ = C and the function f (u) = e −2u on N R = R. Then, on C × ⊂ C we have ϕ = trop * f , but f can not be extended to a smooth function on N Σ , because, by definition, a smooth function on N Σ has to be constant in a neighborhood of ∞. This gives an example of a non-surjective  Proof. Since η is a top degree Lagerberg form with compact support in U , it has compact support contained in U ∩ N R . Choosing integral linear coordinates in N R , we have η = f (u 1 , . . . , u n )d u 1 ∧ d u 1 ∧ · · · ∧ d u n ∧ d u n for a smooth function f on R n . By Proposition 4.1.7, we have trop * η = f (− log |z 1 |, . . . , − log |z n |) i n dz 1 ∧ dz 1 ∧ · · · ∧ dz n ∧ dz n (4π) n z 1 . . . z nz1 . . .z n .
Writing trop * η in polar coordinates z j = r j e iθ j and using that 4.2.6. We next discuss the notions of positivity for toric varieties. In the case of the complex smooth toric variety X an Σ the different notions of positivity are the usual ones: A complex differential form is strongly positive (resp. positive, weakly positive) if it is so pointwise.
In the case of N Σ a little bit more has to be said because the different fibers of the sheaves of forms A over a point p ∈ N Σ \ N R have a different nature.
Recall from Subsection 3.1 that given a point v ∈ N Σ , there is an unique orbit N (σ) with v ∈ N (σ) corresponding to a cone σ of the fan. Then there is an identification of fibers Therefore, the different notions of positivity for Lagerberg forms make sense for this fiber by using Definition 2.2.1 for V = N (σ). Then for U ⊂ N Σ , a Lagerberg form is strongly positive (resp. positive, weakly positive) if it is so fiber by fiber.  One can deduce from Example 2.3.8 that the converses of (i) and (iii) in the previous lemma are not always true.
Lemma 4.2.9. Let U be an open subset of N Σ and V = trop −1 (U ) ⊂ X an Σ . Every strongly positive complex differential form on V (resp. strongly positive Lagerberg form on U ) is positive and every positive complex differential form on V (resp. positive Lagerberg form on U ) is weakly positive. Moreover, if p = 0, 1, n − 1, n, then all three positivity notions agree on A p,p (U ) (resp. on A p,p (V )).
Proof. Since all notions of positivity of forms are checked fiber by fiber, the result follows from Corollary 2.1.6 and Remark 2.1.7 in the complex case and Corollary 2.2.5 and Remark 2.2.6 in the Lagerberg case.

Positivity for complex invariant currents and Lagerberg currents
Throughout this section, Σ will be a smooth fan in N R , X Σ will denote the associated toric variety, N Σ will denote the partial compactification of N R and X an Σ = X Σ (C) will denote the complex manifold associated to X Σ , with tropicalization map trop : X an Σ → N Σ .

Invariant currents. Let U be an open subset of N Σ and write V = trop −1 (U ). Let
A p.q c (U ) be the space of Lagerberg forms on U with compact support and similarly A p,q c (V ) is the space of complex forms on V with compact support. Since trop : V → U is proper, the map trop * in (4.12) induces a map . We now compare the topologies on the space of Lagerberg forms and the space of complex forms through the map trop * . The definition of the C ∞ -topology on both spaces is slightly different. The topology of A p,q c (U ) has been described in Definition 3.2.4. The topology of A p,q c (V ) is defined similarly. One first defines a topology on A p,q K (V ) for each compact K using norms similar to those in (3.4) taking into account all complex derivatives and then define the topology of A p,q c (V ) as the direct limit in the category of locally convex topological vector spaces. So the main difference is the use of finite coverings in Definition 3.2.4. From the definition of the topologies, it is easy to check that the map trop * : A p,q c (U ) → A p,q c (V ) is injective and continuous. Nevertheless, as the following examples show, trop * is in general not a homeomorphism onto its image endowed with the subspace topology. Moreover, the image of trop * is not closed in A p,q c (V ).
The sequence (f n ) n≥3 does not converge to zero in A 0,0 c (U ) because for any compact subset K of U there is no finite covering {V i } i of K with f n ∈ A 0,0 K (U, {V i }) for all n (see Proposition 3.2.6). Indeed, assume that such a compact K and covering {V i } exists. Since the point n ∈ R ∞ is in the support of f n , and these points converge to ∞, then ∞ ∈ K. Therefore one V i would contain ∞ and hence by definition of A 0,0 K (U, {V i }) all the f n | V i would have to be constant, which is not the case.
The support of ϕ n is contained in the closed annulus {z ∈ C | e n−2 ≤ |z| −1 ≤ e n+2 }. Since ρ is smooth with compact support, all the derivatives of ρ are bounded. From this and the condition above on the support of ϕ n , we deduce that for every pair of integers a, b, there is a constant C a,b such that It follows that the sequence (ϕ n ) n≥3 converges to the function 0 in A 0,0 c (V ). We conclude that the topology of A 0,0 c (U ) is not induced by the topology of A 0,0 c (V ).
Now we shift our attention to currents.
It defines an antilinear involution on the space of S-invariant complex currents D(V ) S , which we also denote by F , given by Proof. The first statement follows from Proposition 4.2.3. It remains to show for S ∈ D(V ) S,F and ω ∈ A c (U ) that S(trop * ω) ∈ R. From (5.2) and the fact that F is an involution, we deduce for η ∈ A c (V ) that S(η) = F S(F η). Using that S and trop * ω are F -invariant, we compute proving the lemma.
Remark 5.1.6. The map trop * fits in the following commutative diagram: U is an open subset of the dense orbit N R , then Corollary 4.1.8 yields that the map is an isomorphism. If U intersects the boundary, then trop * is not a closed immersion and hence trop * is not surjective (see Examples 5.1.11 and 5.1.12 below).
Lemma 5.1.7. Given ω ∈ A p,q (U ) and S ∈ D r,s (V ), we have the projection formula Proof. Given η ∈ A n−p−r,n−q−s c (U ), by Remark 3.2.9(ii) we have as trop * respects products and (5.3) is an immediate consequence.
Recall that for ω ∈ A p,q (U ), we have an associated Lagerberg current [ω] = U ω ∧ . in D p,q (U ). Similarly, we have a complex current [η] ∈ D p,q (V ) associated to a complex form η ∈ A p,q (V ). From Lemma 4.2.5 and 5.1.7, we immediately deduce the following result. . Every strongly positive current in D p,p (V ) (resp. in D p,p (U )) is positive, every positive current in D p,p (V ) (resp. in D p,p (U )) is weakly positive and every weakly positive current in D p,p (V ) (resp. in D p,p (U )) is real (resp. symmetric). Moreover, if p = 0, 1, n − 1, n, then all three positivity notions agree in D p,p (V ) (resp. in D p,p (U )).
Proof. The positivity claims follow from Lemma 4.2.9. By definition, a weakly positive current on U is symmetric. To see that a weakly positive current on V is real, we use Lemma 2.1.4. The latter implies that any real form on V is a real linear combination of strongly positive ones which easily yields the claim.
We will see in Theorem 7.1.5 that trop * induces a bijection between the space of closed positive complex currents on V = trop −1 (U ) that are invariant under F and S and the space of closed positive Lagerberg currents on U . In particular, every closed positive Lagerberg current is in the image of trop * . The following two counterexamples show that one can drop neither the closedness assumption nor the positivity assumption. The first example is a positive Lagerberg current and the second example is a closed Lagerberg current, neither of which are in the image of trop * .
, then f has compact support contained in U \ {∞}. Therefore the integral in (5.4) is finite. To show that T is a continuous functional, we use Proposition 3.2.7. Let ω n = g n d u ∧ d u, n ≥ 1 be a sequence of forms in A 1,1 c (U ) that converge to zero. Then there is a neighborhood V 0 of ∞ such that ω 0,n | V 0 = π * σ,0 (ω σ,n ) = 0 for all n, where σ = R ≥0 is the cone corresponding to the point ∞. Hence there is a compact K ⊂ U \ {∞} such that supp(g n ) ⊂ K for all K. Moreover g n K converges to zero. Therefore T (ω n ) also converges to zero. We conclude that T is a Lagerberg current. Clearly it is positive.
Assume that S is a current on V such that trop * S = T . Let f n and ϕ n be as in Example 5.1.1. Since the functions ϕ n converge to 0 in A 0,0 c (V ), we deduce that On the other hand, the assumption trop * S = T yields the desired contradiction as follows: n e x 2 −n 2 dx > 1.
Example 5.1.12. We keep the setting from Example 5.1.11. Then defines a linear functional T : A 0,0 c (U ) → R. Since any f ∈ A 0,0 c (U ) is constant in a neighborhood of ∞, the support of f is a compact subset of U \ {∞}. As in Example 5.1.11, we see that T defines a current in D 1,1 (U ). This current is closed because it has top degree. We claim that T is not of the form trop * (S) for any S ∈ D 1,1 (V ). We argue by contradiction and suppose that T = trop * (S). Let ρ and ψ be as in Example 5.1.2 and write ζ n = ψ(− log |z|)e −1/|z| ρ(− log |z| − n), ζ = ψ(− log |z|)e −1/|z| .
. Therefore lim n→∞ S(ζ n ) = S(ζ) ∈ C. This contradicts converging to −∞ for n → ∞, as a direct computation using integration by parts shows.
The positivity statements follow from Lemma 4.2.8.
On the dense orbit, we also have the converse implication. We will see in the next subsection that positive currents have measure coefficients. The key result to prove this is captured in the following proposition. We refer the reader to Appendix A for the convention we use about Radon measures. Proof. The following argument works for the complex and the Lagerberg case. Let K be a compact subset of U and let C 0 K (U, R) be the set of continuous real functions with support in K. As usual, this real vector space is endowed with the supremum norm sup . We also consider the subspace A 0,0 K (U, R) of smooth real functions with support in K and its subspace A 0,0 K (U, R ≥0 ) of non-negative functions. We claim that the restriction of T to A 0,0 K (U, R) is continuous with respect to the supnorm. Since we have smooth partitions of unity by Proposition 3.2.12, there is a nonnegative χ K ∈ A 0,0 c (U ) with χ K (x) = 1 for all x ∈ K. Now the positivity of T yields for all smooth functions f with compact support in K. This proves the claim.
Let f ∈ C 0 c (U, R). To define T (f ), we use Corollary 3.2.13 or its classical analogue to get a sequence (f n ) n∈N in A 0,0 c (U, R) which converges to f . It follows from the proof of that corollary as well as from Proposition 3.2.6 that we can find a compact subset K of U such that f n ∈ A 0,0 K (U, R) for all n ∈ N. The sequence (f n ) n∈N converges to f with respect to the sup-norm on K. By the continuity of T shown above, it is clear that T (f n ) converges to a real number. The limit T (f ) neither depends on the choice of the sequence nor on the choice of K. Note also that if f ≥ 0, then χ K · (f n + f n − f sup ) is a sequence in C 0 supp(χ K ) (U, R ≥0 ) converging to f with respect to the sup-norm. We conclude easily that the above defines a positive linear functional T on C 0 c (U, R) which extends the given functional. This is equivalent to have a positive Radon measure µ on U with T (f ) = U f dµ for every f ∈ C 0 c (U, R). Note that uniqueness is obvious as a positive linear functional on C 0 c (U, R) is continuous on C 0 K (U, R) (by the same argument as above) and so our definition of T (f ) is forced.
We can now prove that positive (n, n)-currents on open subsets of X an Σ or N Σ are the same as positive Radon measures.
Proposition 5.1.16. Let U be either an open subset of X an Σ or of N Σ . For every positive current T ∈ D n,n (U ), there is a unique positive Radon measure µ on U such that Conversely, every positive Radon measure µ on U induces a current T ∈ D n,n + (U ) by (5.6).
Proof. If T is a positive current in D n,n (U ), then it is a positive linear functional on A 0,0 c (U ). Hence the first claim follows from Proposition 5.1.15. Conversely, we have seen in Example 3.2.11 that every real Radon measure µ induces an element T ∈ D n,n (U ) with (5.6). Clearly, if µ is positive, then T is positive as well.
The following result is a prototype for our main correspondence result in Theorem 7.1.5. 5.2. Co-coefficients of currents. To extend Proposition 5.1.15 to (p, p)-currents, we place ourselves in the local setting. We assume in this subsection that N = Z n and that Σ is the fan given by the maximal cone R n ≥0 and its faces in N R = R n . Then N Σ = R n ∞ and X Σ is the smooth toric variety A n C . Recall that any smooth toric variety can be covered by toric affine varieties isomorphic to open subvarieties of this one. The splitting N = Z n induces coordinates (z 1 , . . . , z n ) in X an Σ and (u 1 , . . . , u n ) in N R . For I ⊂ {1, . . . , n}, we consider the following union of strata Definition 5.2.2. Let U ⊂ N Σ = R n ∞ be open and T ∈ D p,p (U ). Let q, I, J be as above. Then the co-coefficient T IJ is the Lagerberg distribution on U \ E I∪J defined by Remark 5.2.3. Note that we define T IJ ∈ D n,n (U \ E I∪J ) because, by definition of Lagerberg forms, we can only plug in functions that vanish in a neighborhood of E I∪J .
Remark 5.2.4. Conversely, given S IJ ∈ D n,n (V ) for all I, J ⊂ {1, . . . , n} with |I| = |J| = q = n − p, there is a unique complex current S ∈ D p,p (V ) with co-coefficients S IJ . Similarly, one can show that given T IJ ∈ D n,n (U \ E I∪J ) for all I, J ⊂ {1, . . . , n} with |I| = |J| = q = n − p, there is a unique T ∈ D p,p (U ) with co-coefficients T IJ .
Positive currents have measure co-coefficients, instead of just distribution co-coefficients. The complex case is given by [Dem12, Proposition III.1.14]: Proposition 5.2.5. Let V ⊂ X an Σ = C n and S ∈ D p,p (V ) be a weakly positive current. Then the co-coefficients S IJ are complex Radon measures on V that satisfy S IJ = S JI for all multi-indices I, J with |I| = |J| = q. Moreover, all S II are positive Radon measures and the total variation measures |S IJ | satisfy the estimates for any collection of real numbers λ k ≥ 0, k = 1, . . . , n, and λ I = k∈I λ k .
We have seen in Example 2.3.6 that in the tropical case it is reasonable to restrict our attention to positive Lagerberg forms and hence to positive Lagerberg currents, as opposed to weakly positive currents.
Proposition 5.2.6. Let U ⊂ N Σ = R n ∞ be an open subset and T ∈ D p,p (U ) a positive current. Then the co-coefficients T IJ are real Radon measures on U \ E I∪J that satisfy T IJ = T JI for all multi-indices I, J with |I| = |J| = q. Moreover, all T II are positive Radon measures and the total variation measures |T IJ | satisfy the estimates for any pair of real numbers λ I , λ J ≥ 0.
Proof. By definition, Lagerberg currents are real valued. The condition follows from the symmetry of T . For every f ∈ A 0,0 c (U \ E I∪J ), we have can be written as the difference of two positive Lagerberg forms Since the product of a positive Lagerberg current by a positive Lagerberg form is a positive Lagerberg current by Remark 2.2.2, we deduce from equations (5.10), (5.11) and from Proposition 5.1.15 that T IJ + T J,I is a real Radon measure on U \ E I∪J . From equation (5.9), it follows that T IJ is also a real Radon measure on U \ E I∪J . In the special case I = J, there is no need for (5.11) and we can directly deduce from the definition of the co-coefficient T II or from (5.10) that T II is a positive Radon measure on U \ E I .
Note that for every positive continuous function with compact support contained in U \ E I∪J and for every pair of non-negative real numbers λ I , λ J , we have the inequalities Expanding and using equation (5.9), we deduce ±2λ I λ J T IJ (f ) ≤ λ 2 I T I,I (f ) + λ 2 J T J,J (f ), from which equation (5.8) follows.
Observe that (5.8) is stronger than (5.7). This is because in Proposition 5.2.6 we are dealing with positive currents while Proposition 5.2.5 deals with weakly positive currents.
Example 5.2.7. We show that there can be no analogue of Proposition 5.2.6 for weakly positive Lagerberg currents. Example 2.3.6 gives a non-zero Lagerberg form ω of type (2, 2) on R 4 such that ±ω are both weakly positive. In fact ω was constructed such that ω ∧ η = 0 for each strongly positive Lagerberg form η of type (2, 2). By definition, the associated Lagerberg currents ±[ω] are weakly positive. We compute the co-coefficients of T = [ω]. For I = {3, 4}, by using the formula for ω in Example 2.3.6. We conclude that T JI = T IJ is the Lebesgue measure on R 4 and the same holds for I In Example 5.2.7, all co-coefficients were positive or negative Radon measures on R 4 .
Example 5.2.8. We construct from ω in Example 2.3.6 a weakly positive Lagerberg current T whose co-coefficients are not all Radon measures.
Since ω ∧ α = 0 for all strongly positive forms α, we have that ω ∧ T is weakly positive for every symmetric current T of type (0, 0). This already shows that it is very unlikely that every weakly positive current has measure coefficients. Indeed, one may check that for the Lagerberg currents and T = ω ∧ T , and the sets I = {3, 4} and J = {1, 2}, we have It is well-known that uniform convergence does not imply convergence of derivatives. Therefore the co-coefficient T IJ is not a Radon measure.

Tropicalization of positive currents
We consider a smooth fan Σ in N R with associated complex toric variety X Σ and corresponding tropicalization N Σ . In this section, we will describe precisely which positive Lagerberg currents are in the image of positive complex currents with respect to trop * . To this end, in the first subsection, we will introduce a canonical decomposition of positive currents in the complex and in the Lagerberg case and use it to give the desired characterization in the second subsection.
6.1. Decomposition of positive currents along the strata. In this subsection, we give a canonical decomposition of a positive current along the boundary strata. As usual, we handle the complex and the tropical case simultaneously. For simplicity, we will often give the arguments only in the case of Lagerberg currents as the complex case is completely similar and even easier as the exceptional sets E I for co-coefficients do not occur (see Subsection 5.2). At the end of the subsection, we will study functoriality of the canonical decomposition with respect to trop * . Definition 6.1.2. Since Σ is smooth, a given cone ρ in Σ is generated by part of a Z-basis b 1 , . . . , b n of N . Using the corresponding coordinates, we may view V ρ (resp. U ρ ) as an open subset of C n (resp. R n ∞ ). We call such an identification of V ρ (resp. U ρ ) with an open subset of C n (resp. R n ∞ ) a choice of toric coordinates on V ρ (resp. U ρ ). We usually denote the corresponding complex coordinates by z 1 , . . . , z n and the corresponding tropical coordinates by u 1 , . . . , u n .
Note that for a current T of bidegree (p, q) on V (resp. U ), the choice of toric coordinates on V ρ (resp. U ρ ) leads to well-defined co-coefficients (T | Vρ ) IJ (resp. (T | Uρ ) IJ ) for all I, J ⊂ {1, . . . , n} with |I| = |J| = q = n − p. Recall that, if T is positive, then in the complex case the co-coefficient (T | Vρ ) IJ is a Radon measure on the open set V ρ while in the Lagerberg case (T | Uρ ) IJ is a Radon measure on the open set U ρ \ E I∪J (see §5.2). This allows us to define the notion of a null set of a positive current.
Definition 6.1.3. Let U be an open subset of X an Σ (resp. N Σ ). We say that T ∈ D p,p (U ) has measure co-coefficients if for every ρ ∈ Σ and some choice of toric coordinates on U ρ , the co-coefficients T IJ are complex (resp. real) Radon measures on U ρ (resp. on U ρ \ E I∪J ).
Let T ∈ D p,p (U ) have measure co-coefficients and let A be a Borel subset of U . We say that A is a null set for T if, for all ρ ∈ Σ and some choice of toric coordinates on U ρ , the set A ∩ U ρ (resp. A ∩ U ρ \ E I∪J ) is a null set of the total variation measure (T | Vρ ) IJ (resp. (T | Uρ ) IJ ) for all I, J ⊂ {1, . . . , n} with |I| = n − p.
Using change of variables, it is easy to see that the conditions in Definition 6.1.3 do not depend on the choice of toric coordinates. For any α ∈ A p,q c (U ), the forms α k := ψ k α ∈ A p,q c (U ) have compact support in U \ A. If α is a positive form, then α k and α − α k are both positive.
For every T ∈ D n−p,n−q (U ) with measure co-coefficients, we have: (a) if A is a null set for T , then lim k→∞ T (α k ) = T (α); (b) if U \ A is a null set for T , then lim k→∞ T (α k ) = 0.
Proof. Let α ∈ A p,q c (U ) with support contained in the compact subset K of U . Since U is a metric space and since A ∩ K is compact, there is a strictly decreasing sequence (W k ) k∈N of relatively compact open subsets of U with k∈N W k = A ∩ K and W 0 ⊃ W 1 ⊃ W 1 ⊃ W 2 ⊃ · · · . Now a partition of unity (see Proposition 3.2.12 for the Lagerberg case) gives the existence of a function ϕ k ∈ A 0,0 c (W k ) with 0 ≤ ϕ k ≤ 1 and ϕ k ≡ 1 on W k+1 for every k ∈ N. We note that the ϕ k form a decreasing sequence of smooth functions with compact support which converges pointwise to the characteristic function of A ∩ K. For the sequence ϕ k := 1 − ψ k , we get the first claim. Evaluating T (α k ) in terms of the co-coefficients T IJ and using that the latter are Radon measures, we deduce (a) and (b) from the monotone convergence theorem. If α is positive, then every α k = ψ k α and every α − α k = ϕ k α is positive.
Remark 6.1.5. In the complex case of Lemma 6.1.4, if A is S-invariant, then we can choose the smooth functions ψ k to be S-invariant. Indeed, this follows easily from Lemma 6.1.4 by averaging the ψ k with respect to the Haar probability measure on S.
The decomposition theorem is the following result.
Theorem 6.1.6. Let U be an open subset of X an Σ (resp. of N Σ ) and let T be a positive current in D p,p (U ). Then there is a unique decomposition such that, for every σ ∈ Σ, the following two conditions are satisfied Moreover, the support of the current T σ is contained in V ∩ O(σ) an (resp. U ∩ N (σ)).
We call (6.1) the canonical decomposition of the positive current T .
Proof. We write the proof only in the Lagerberg case as the complex case is analogous. The statement about the support is a direct consequence of statement (ii) so we only need to prove the existence and uniqueness of the decomposition. Using a partition of unity provided by Proposition 3.2.12, the existence of such decomposition can be checked locally. Moreover, since currents form a sheaf, the unicity can also be checked locally. Therefore we choose a cone ρ ∈ Σ and replace U by U ρ . We also make a choice of toric coordinates, so we can assume from now on that U ⊂ R n ∞ . For each pair of subsets I, J with |I| = |J| = p we write with T IJ ± the positive and the negative part of the Radon measure T IJ . We denote by µ IJ ± the corresponding Borel measures. For any σ ∈ Σ, let us consider the immersion We define a new Borel measure µ IJ ±,σ on U \ E I∪J by first restricting µ IJ ± to the locally closed subset U ∩ N (σ) \ E I∪J and then using the image measure with respect to i IJ σ . Since the sets N (σ) form a stratification of N Σ , we get the decomposition of Borel measures. It follows that µ IJ ±,σ ≤ µ IJ ± . Note that in our setting, the Radon measures correspond to locally finite Borel measures (see Appendix A). We conclude that µ IJ ±,σ is a locally finite Borel measure and hence we get a real Radon measure T IJ σ on U \ E I∪J corresponding to µ IJ +,σ − µ IJ −,σ . By Remark 5.2.4, we obtain a unique current T σ ∈ D p,p (U ) with co-coefficients T IJ σ . Note that T σ has measure co-coefficients. By construction, the set U \ (E I∪J ∪ N (σ)) is a null set with respect to the Borel measure µ IJ +,σ + µ IJ −,σ associated to the total variation measure |T σ |. The decomposition T = σ T σ follows from (6.2). It remains to show uniqueness and property (i).
We prove uniqueness of T σ in the decomposition (6.1) by induction with respect to the partial ordering ≺ on Σ. We take σ ∈ Σ and we suppose that uniqueness of T τ is known for all τ = σ with τ ≺ σ. Recall from Remark 6.1.1 that U σ is an open subset of U and that A σ := U \ U σ is the closed subset of U given by the disjoint union of all N (τ ) ∩ U with τ ∈ Σ not a face of σ. Since T and all α − α k are positive, we deduce that T σ is positive.
To prove that T σ is positive, we consider again the closed subset A σ . Let α ∈ A n−p,n−p c (U ) be positive. By Lemma 6.1.4, there is a positive sequence (α k ) k∈N in A n−p,n−p c (U ) such that and lim k→∞ T τ (α k ) = 0 for all τ ∈ Σ which are not faces of σ. We conclude that lim k→∞ T σ (α k ) = T σ (α).
Using that T σ (α k ) ≥ 0 by the positivity of T σ and α k , we deduce that T σ is positive.
In the complex case, we will show next that the canonical current T σ from the canonical decomposition (6.1) is not always the push-forward of a current on V ∩ O(σ) an .
Example 6.1.7. Let us consider V = X an Σ = C 2 where Σ is the fan whose maximal cone is the positive quadrant in R 2 and let T ∈ D 1,1 (C 2 ) be given by T k,l α kl idz k ∧ dz l = α 1,1 (0, 1).
Using that the coefficients α kk of a positive (1, 1)-form are non-negative, we see that T is a positive current. We conclude that T = T σ for the cone σ generated by (1, 0). Note that the push-forward S of a current on the stratum closure O(σ) an = {z 1 = 0} can only have a non-zero co-coefficient S 2,2 . Since all the co-coefficients of T are zero except T 1,1 which is given by the Dirac measure δ (1,0) in the point (1, 0), we conclude that T σ is not the push-forward of a current on O(σ) an .
Note that the analogue of Example 6.1.7 in the Lagerberg case gives no counterexample as T is zero in this case since the point (∞, 0) belongs to E {1} . This is explained by the following result. We will see in the proof below that in the Lagerberg case, the necessary and sufficient condition in (a) is always satisfied which is the reason for (b) to be true.
Proof. We first deal with the complex case (a). Suppose that there is a current S on U such that ι * (S) = T σ for the closed immersion ι of U ∩ O(σ) an into U . Let ρ ∈ Σ and let us fix toric coordinates on U ρ such that we may view U ρ as an open subset of C n . Suppose that N (σ) ⊂ E I∪J . This means that there is k ∈ I ∪ J such that the coordinate z k vanishes on O(σ). Hence the restrictions of dz k , dz k to O(σ) are zero. This implies that T IJ σ = 0 using the definition of co-coefficients and ι * (S) = T σ .
Conversely, assume that for any ρ ∈ Σ and any choice of toric coordinates on U ρ , we have T IJ σ = 0 on U ρ whenever N (σ) ⊂ E I∪J . The coordinates z k of O(σ) are precisely those such that z k = 0 on O(σ). We conclude that the co-coefficients of a current S on T IJ σ is the push-forward of S IJ . This shows that T σ | Uρ is the push-forward of S ρ with respect to the closed immersion of U ρ ∩ O(σ) an into U ρ . It follows that the currents S ρ on the open covering U ρ of U glue to a current S on U with ι * (S) = T σ . Since T σ is positive and since we can extend compactly supported positive forms on U ∩ N (σ) to compactly supported positive forms U , it is clear that also S is positive. This proves (a).
In the case of a positive Lagerberg current on U , we note that Theorem 6.1.6 says that the support of T σ is contained in U ∩ N (σ). For any ρ ∈ Σ and any choice of toric coordinates on U ρ , the co-coefficient T IJ σ on U ρ \ E I∪J has support contained in supp(T σ ). If I, J satisfy N (σ) ⊂ E I∪J , then the support of the restriction of T σ to U ρ is contained in U ρ ∩ N (σ) ⊂ E I∪J and hence T IJ σ = 0 on U ρ \ E I∪J . We conclude that the crucial condition in (a) is always satisfied in the Lagerberg case and an easy adaption of the argument of (a) to the Lagerberg case gives a positive Lagerberg current S on U ∩ N (σ) whose image in U is T σ proving (b).
(ii) Assume that the current S is positive. Then T is positive as well and the total variation measures |S IJ | and |T IJ | of the Radon measures S IJ and T IJ satisfy Proof. Recall from Lemma 5.1.5 that T ∈ D p,p (U ). To prove (i), we use (4.5) to get To prove (ii), we recall from Proposition 5.1.13 that T is positive if S is positive. Now equation (6.4) follows from (6.3) by using |z IzJ | = i∈I e −u i j∈J e −u j .
The following result shows compatibility of the tropicalization map with the canonical decomposition of positive currents from Theorem 6.1.6. Proof. We first observe that with S each summand S σ is again S-and F -invariant by the uniqueness property of the canonical decomposition. To prove (6.5), we may argue locally on the base and so we may assume that U = U ρ and V = V ρ for some ρ ∈ Σ. We choose toric coordinates to view U as an open subset of R n ∞ and V as an open subset of C n . We write T := trop * (S) and denote by T = σ T σ its canonical decomposition. We pick σ ∈ Σ and we may assume S = S σ . Then we have to show that T = T σ . By our characterization of the canonical decomposition in Theorem 6.1.6, it is equivalent to show that U \ (N (σ) ∪ E I∪J ) are null sets with respect to the Radon measures |T IJ |. We note that trop −1 (N (σ)) = O(σ) and that V \ O(σ) is a null set with respect to the Radon measure S IJ using S = S σ and Theorem 6.1.6. Then (6.4) yields that U \ (N (σ) ∪ E I∪J ) is a null set with respect to |T IJ | proving (6.5).
6.2. Positive Lagerberg currents and local mass. Equation (6.4) gives a necessary condition for a positive Lagerberg current to be the image of an S-and F -invariant positive complex current. In fact, it naturally leads to the following definition which turns out to be also a sufficient condition.
Definition 6.2.1. Let U be an open subset of N Σ and let T ∈ D p,p (U ) be a positive Lagerberg current. We say that T has C-finite local mass if for all ρ ∈ Σ and some choice of toric coordinates u 1 , . . . , u n on U ρ as in Remark 6.1.2, the corresponding co-coefficients T IJ , which may be seen as real Radon measures on U ρ \ E I∪J with total variation measures |T IJ |, satisfy the condition that the Borel measures given as the image measures Proof. Suppose that S ∈ D p,p (V ) S,F is a positive current and let T = trop * (S). We will prove that T has C-finite local mass. Indeed, it follows from (6.4) that we have the identity of positive measures on U ρ \E I∪J for any ρ ∈ Σ and any choice of toric coordinates u 1 , . . . , u n on U ρ . Passing to image measures with respect to the open immersion j IJ ρ : U ρ \ E I∪J → U ρ and using that trop(|S IJ |) is a Borel measure on U ρ , we deduce that trop(|S IJ |) ≥ π −q 2 2q j IJ ρ i∈I e −u i j∈J e −u j |T IJ | ≥ 0.
By Proposition 5.2.5, the Borel measure |S IJ | is locally finite on V ρ and hence the Borel measure trop(|S IJ |) is locally finite on U ρ . We deduce that T has C-finite local mass. Conversely, let T be a positive Lagerberg current in D p,p (U ) with C-finite local mass. We consider the canonical decomposition T = σ∈Σ T σ from Theorem 6.1.6. First, we assume that T = T {0} for the minimal cone {0} ∈ Σ. By assumption, the co-coefficients are Radon measures T IJ on U ρ \ E I∪J such that the positive Borel measures are locally finite on U ρ for every ρ ∈ Σ. By Remark 5.1.6, there is a unique current R ∈ D p,p (V ∩ T an ) S,F with trop * (R) = T | N R . Since T is positive, it follows from Lemma 5.1.14 that R is a positive current. It follows from Proposition 5.2.5 that the co-coefficients R IJ of R are complex Radon measures on V ∩T an . Using (6.4), we get the following identity of Borel measures trop(|R IJ |) = π −q 2 2q i∈I e −u i j∈J e −u j |T IJ | on U ∩ N R . Using that T has C-finite local mass, we know that the image measure of the right hand side with respect to the open immersion j ρ : U ∩ N R → U ρ is a locally finite Borel measure on U ρ . We conclude that the image measure of |R IJ | with respect to the open immersion j ρ : V ∩ T an → V ρ is a locally finite Borel measure on V ρ by using j σ • trop = trop •j σ on V ∩ T an . Hence the Radon measure R IJ admits an image Radon measure j ρ (R IJ ) which is a complex Radon measure on V ρ . By Remark 5.2.4 and Example 3.2.11, there is a unique current in S ∈ D p,p (V ρ ) with co-coefficients S IJ induced by j ρ (R IJ ).
Since the restriction of the co-coefficients S IJ to the dense stratum V ∩T an is R IJ , it follows also from Remark 5.2.4 that S| V ∩T an = R. By construction, the boundary V ρ \ N R is a null set with respect to each Radon measure j ρ (R IJ ). By Lemma 6.1.4 and Remark 6.1.5, there is an increasing sequence ψ k ≥ 0 in A 0,0 c (V \ N R ) S which converges pointwise to the characteristic function of V \ N R such that for any α ∈ A q,q c (V ρ ) we have (6.6) lim It follows that the currents S, defined a priori only on V ρ for any ρ ∈ Σ, do not depend on the choice of toric coordinates on V ρ and hence agree on overlappings of the covering (V ρ ) ρ∈Σ . They define a current on V which we also denote by S. If α is positive, then ψ k α is also positive. By positivity of R and by (6.6), the current S is also positive. Since both ψ k and R are invariant with respect to S and F , we deduce from (6.6) that S ∈ D p,p (V ) S,F .
We have to check trop * (S) = T . Using j σ • trop = trop •j σ on V ∩ T an , we have the following identities of Radon measures on U ρ . By (6.3), we deduce the identity of Radon measures Now we skip the assumption T = T {0} from above and consider any positive current T with canonical decomposition T = σ∈Σ T σ from Theorem 6.1.6. By Proposition 6.1.8, for every σ ∈ Σ, there is a positive Lagerberg current P σ on U ∩ N (σ) such that T σ = (ι σ ) * (P σ ) for the closed immersion ι σ : U ∩ N (σ) → U . Now we apply the above case to the current P σ and to the open subset V ∩ O(σ) an of the toric variety O(σ). We conclude that there is a positive current Q σ on V ∩ O(σ) an such that trop * (Q σ ) = P σ . For the closed immersion an → V , we have the obvious relation trop •ι σ = ι σ • trop. Now we set which is a positive current on V . Then we get Then S := σ∈Σ S σ is a positive current on V satisfying trop * (S) = T .
Next we give an example of a positive Lagerberg current T such that T = trop * (S) for some complex current S, but for which no such S is positive.
Example 6.2.3. We consider the one-dimensional situation with N = Z, N Σ = R ∞ and Since the function 1 = e 2x e −2x does not have locally finite mass around ∞, this current is not of the form trop * (S) for a positive current S in V . Nevertheless T is in the image of trop * . Indeed, let S ∈ D 0,0 (U ) be the current given by Clearly this current is not weakly positive. Let now gd u ∧ d u be a Lagerberg form on U . Then g(u) = 0 for u 0 and hence This shows T = trop * S.

The correspondence theorem for closed positive currents
In this section, we consider a smooth fan Σ with associated toric variety X Σ and partial compactification N Σ . After considering positivity of currents in the previous sections, we add here the additional condition that the currents are closed. Recall that a complex (p, p)-current T (resp. a Lagerberg (p, p)-current S) is called closed if ∂T =∂T = 0 (resp. d S = d S = 0). In the first subsection, we show our main theorem. It states that the tropicalization map induces a bijective correspondence between S-and F -invariant closed positive currents on X Σ and closed positive Lagerberg currents on N Σ . In the second subsection, we derive from our main theorem a tropical version of the Skoda-El Mir theorem. . Then there is a constant C ∈ R ≥0 such that for all f ∈ C ∞ c (R M ), and for all v ∈ R M ≥0 such that (supp(f ) + P (v)) × supp(χ) ⊂ K, we have Proof. Let (e 1 , . . . , e n ) denote the standard basis of R n and write v = i∈M t i e i . Using a telescope argument, one easily reduces to the case where v = te i for some fixed i ∈ M . For a given t ∈ [0, v i ], we consider the auxiliary function From (supp(f ) + P (te i )) × supp(χ) ⊂ K, we get has finite mass in the region S := [R, ∞) M × [R, ∞] M c . We have For I, J ⊂ {1, . . . , n} with |I| = |J| = n − p, equation (5.8) yields Therefore the left hand side has finite mass in this region. Finally, using that e −u decreases faster than u −1−ε , we deduce that T has C-finite local mass. Proof. This can be proved locally on N Σ . Hence we may assume that Σ is a fan containing a single maximal dimensional cone with all its faces, so that N Σ = R n ∞ as in Definition 6.1.2. Then the cones of Σ are given by the faces We claim that all currents T σ , T σ and T σ are closed. It is clear that T agrees with T {0} on the dense stratum V ∩ T an and hence T {0} | V ∩T an is a closed positive current on V ∩ T an . By the Skoda-El Mir Theorem [Dem12, Theorem III.2.3] and using that the co-coefficients of T {0} are Radon measures on V , it follows that T {0} is closed on V . Using induction on codim(σ), we can prove similarly that T σ is closed for any σ ∈ Σ. The same arguments show that T σ is closed. Then T σ is closed as well.
From Proposition 6.1.10, we get the equality of canonical decompositions To prove the proposition, it is enough to show for each σ = σ M ∈ Σ that T σ is zero , or equivalently that all co-coefficients T IJ σ are zero for all subsets I, J ⊂ {1, . . . , n} with |I| = |J| = n − p (see Remark 5.2.4). We prove this by induction on the cardinality of M ∩ (I ∪ J). For the initial step, we consider subsets M, I, J ⊂ {1, . . . , n} such that M ∩ (I ∪ J) = ∅. Let f be a real valued smooth function on U with compact support in U \ E I∪J = {u ∈ U | u i = ∞ ∀i ∈ I ∪ J}. Using trop * (T σ ) = 0 and (6.3), we obtain Using that T IJ σ is a Radon measure, we conclude by a standard approximation argument that (7.2) holds for all f ∈ C 0 c (U \ E I∪J ). For any g ∈ C 0 c (V \ trop −1 (E I∪J )), let g av be the natural projection onto the S-invariant functions given by averaging over the fibers of trop with respect to the probability Haar measures. By construction, there is a unique f ∈ C 0 c (U \ E I∪J ) with trop * (f ) = g av . The S-invariance of T σ implies that This means that the restriction of T IJ σ to the open subset V \ trop −1 (E I∪J ) is identically zero. Since V \ S M is a null set with respect to the Radon measure T IJ σ and since our assumption M ∩ (I ∪ J) = ∅ yields S M ⊂ V \ trop −1 (E I∪J ), we deduce that T IJ σ = 0. For the inductive step, the induction hypothesis is that T IJ ρ = 0 whenever ρ = σ L ∈ Σ with |L ∩ (I ∪ J)| < k for some k ≥ 1. Let M, I, J ⊂ {1, . . . , n} with |M ∩ (I ∪ J)| = k and m ∈ M ∩ (I ∪ J). We have to show that T IJ σ (f ) = 0 for σ = σ M and any f ∈ C ∞ c (V ). Using that V \ S M is a null set with respect to the Radon measure T IJ σ , we may assume that, in a neighborhood of S M , the function f depends only the variables z j , j ∈ M . By symmetry, we may also assume without loss of generality that m ∈ I. Write I = I \ {m}.
Let g ∈ C ∞ c (V ) be the function given by g = z m f . By the assumptions on f , there is a neighborhood of S M where ∂g ∂z m = f. and in this neighborhood, g depends only on the variables z j , j ∈ M , and z m . Consider the smooth form η = gdz I ∧ dz J on V which has compact support. Since T σ is closed and has support on S M , the assumptions on g lead to This function depends only on the variables (z j ) j ∈M and z m and, in a neighborhood of S M , satisfies ∂h j ∂z m = ∂g ∂z j .
For the compactly supported smooth form α j = h j dz j ∧ dz I ∧ dz J on V , we deduce as above that 0 = T σ (∂α j ) = T σ ± ∂g ∂z j dz j ∧ dz I ∧ dz J + j ∈M ∪J T σ ∂h j ∂z j dz j ∧ dz j ∧ dz I ∧ dz J .
This implies T σ ∂g ∂z j dz j ∧ dz I ∧ dz J = 0.
for all j ∈ M ∪ I which implies T IJ σ = 0 by (7.3) as before. This completes the induction and proves the result.
In general the map trop * : D p,p + (V )∩D p,p (V ) S,F → D p,p + (U ) is not injective as the following example shows.
Example 7.1.4. Consider P 1 C as a toric variety. Then the (0, 0) current that sends the form f (z)dz ∧ idz to the value f (0) is a non-zero positive invariant current, but its image by trop * is zero. Thus, in Proposition 7.1.3 the closedness condition is necessary.  Let ∆ be a polyhedron in N R of dimension p which is integral R-affine, i.e. a polyhedron given by finitely many inequalities ϕ ≥ c with ϕ ∈ Hom Z (N, Z) and c ∈ R. The argument map from polar coordinates induces an S-equivariant fibration of the S-invariant subset trop −1 (∆) ⊂ T an over a real torus of dimension n − p with fibers of complex dimension p. Integration of a complex (p, p)-form over the fibers and then integrating the resulting function on the real torus with respect to the probability Haar measure defines a complex current T ∆ ∈ D n−p,n−p (T an ). For details of the construction of T ∆ , we refer to [BH17, Definition 2.3].
Let δ ∆ denote the Lagerberg current of integration over ∆ defined in [Gub16,3.6]. The complex current T ∆ is by construction S-invariant. We get furthermore trop * (T ∆ ) = δ ∆ by a similar argument as in the proof of Lemma 4.2.5. Using the last equality and the fact that every element in A p,p (T an ) S is a C-linear combination of forms in trop * (A p,p (N R )), one verifies immediately that T ∆ is also F -invariant.
If C = (C , m) is a weighted integral R-affine polyhedral complex of pure dimension p with weights m in the sense of [Gub16, 3.1, 3.3], then one defines T C := ∆∈C dim ∆=p m ∆ T ∆ ∈ D n−p,n−p (T an ), δ C := ∆∈C dim ∆=p m ∆ δ ∆ ∈ D n−p,n−p (N R ).
If C is an effective tropical cycle, then T C is closed and positive by [BH17, Theorem 2.9]. The same is true for δ C by [Gub16,3.7]. We conclude that T C is the unique closed positive current in D n−p,n−p (T an ) S,F such that trop * (T C ) = δ C . 7.2. The analogue of the Skoda-El Mir Theorem for tropical toric varieties. We will prove a tropical analogue of the Skoda-El Mir Theorem. In this subsection, U is an open subset of N Σ and E is the intersection of U with a union of strata closures.
Let T ∈ D p,p (U \ E) be a positive Lagerberg current. We pick ρ ∈ Σ and choose toric coordinates on U ρ as in Definition 6.1.2. By Proposition 5.2.6, the co-coefficients T IJ are real Radon measures on U ρ \(E I∪J ∪E) with total variation measure |T IJ |. We will consider the open immersion i IJ E : U ρ \ (E I∪J ∪ E) −→ U ρ \ E I∪J . Definition 7.2.1. Let T ∈ D p,p (U \ E) be a positive Lagerberg current. We say that T is extendable by zero to U if for any ρ ∈ Σ, any toric coordinates on U ρ and all subsets I, J of {1, . . . , n} with |I| = |J| = n − p, the Radon measure T IJ admits an image Radon measure with respect to i IJ E (see Appendix A). In other words, there is a (unique) Radon measure i IJ E (T IJ ) on U ρ \ E I∪J which agrees with T IJ on U ρ \ (E I∪J ∪ E) such that U ρ ∩ E \ E I∪J is a null set with respect to i IJ E (T IJ ). Lemma 7.2.2. If T is extendable by zero to U , then there is a unique Lagerberg current T ∈ D p,p (U ) such that for any ρ ∈ Σ and all toric coordinates on U ρ , the co-coefficient ( T ) IJ is induced by the Radon measure i IJ E (T IJ ) for all I, J. Moreover, we have T | U \E = T and the Lagerberg current T is positive.
In the above situation, we say that T is the extension of T by zero to U .
Proof. For ρ ∈ Σ and a choice of toric coordinates, Remark 5.2.4 shows that there is a unique current T ∈ D p,p (U ρ ) with co-coefficients induced by the Radon measures i IJ E (T IJ ). Clearly, we have T | Uρ\E = T | Uρ\E and T does not depend on the choice of toric coordinates. Moreover, the extensions T constructed on the open covering (U ρ ) ρ∈Σ agree on overlapping. By glueing, we get a Lagerberg current on U also denoted by T . Using Lemma 6.1.4 for A := E ∩ U ρ , we deduce easily that T is again a positive Lagerberg current.
Definition 7.2.3. Let T ∈ D p,p (U \ E) be a positive Lagerberg current. Generalizing Definition 6.2.1, we say that T has C-finite local mass on U if for all ρ ∈ Σ the co-coefficients T IJ , which may be seen as real Radon measures on U ρ \ (E ∪ E I∪J ), satisfy the condition that the Borel measures on U ρ , given as the image measures j IJ E |T I,J | i∈I e −u i j∈J e −u j with respect to the open immersion j IJ E : U ρ \ (E ∪ E I∪J ) → U ρ , are locally finite on U ρ . Theorem 7.2.4. Let T ∈ D p,p (U \ E) be a closed positive Lagerberg current which has C-finite local mass on U . Then T is extendable by zero to U and the extension T of T by zero to U from Lemma 7.2.2 is a closed positive Lagerberg current on U .
Proof. We may assume that U = U ρ for some ρ ∈ Σ and we choose toric coordinates as in Definition 6.1.2. Since the function i∈I e −u i j∈J e −u j is locally finite on U \ E I∪J and since T has C-finite local mass on U , the Borel measure i IJ E (|T IJ |) is locally finite on U \ E I∪J and hence T is extendable by zero to U . By Lemma 7.2.2, the extension T of T by zero to U is a positive Lagerberg current on U .
Let V := trop −1 (U ) and let D := trop −1 (E). Note that D is the intersection of the open subset V of X an Σ with a union of strata closures and hence is a closed analytic subset of the complex toric manifold X an Σ . By our correspondence theorem (Theorem 7.1.5), there is a unique closed positive current S ∈ D p,p (V \ D) S,F cl,+ with trop * (S) = T . Using that T has C-finite local mass on U , it follows from (6.4) that S has finite local mass on V . The complex Skoda-El Mir Theorem (see [Dem12,Theorem III.2.3]) shows that the extension S of S by zero to V is a closed positive current on V . By construction, we have trop * ( S) = T and hence T is a closed Lagerberg current.
Corollary 7.2.5. Let T ∈ D p,p (U ) be positive and closed with canonical decomposition T = σ∈Σ T σ from Theorem 6.1.6. Then every Lagerberg current T σ is positive and closed.
Proof. By Theorem 6.1.6, every T σ is a positive Lagerberg current. We know from Proposition 7.1.2 that the closed positive Lagerberg current T has C-finite local mass. These two facts show that every T σ has C-finite local mass.
We prove that T σ is closed by induction on the dimension of σ. By construction T {0} is the extension by zero of T | N R ∩U . Hence T {0} is closed by Theorem 7.2.4. We assume now that T τ is closed for all τ with dim(τ ) < dim(σ). Hence T τ is closed. Since T σ | Uσ = T σ | Uσ we deduce that T σ | Uσ is closed. As T σ is the extension by zero of this last current, Theorem 7.2.4 implies that T σ is closed.

Appendix A. Reminder about Radon and regular Borel measures
For the convenience of the reader, we gather the used conventions about Radon measures and some basic facts. In this paper, we deal only with measures on locally compact Hausdorff spaces which have a countable basis, so let us consider such a space Y .
For K ∈ {R, C} and a compact subset Z of Y , we write C 0 Z (Y, K) for the space of Kvalued continuous functions on Y with support in Z equipped with the topology induced by the supremum norm. The space C 0 c (Y, K) of K-valued continuous functions on Y with compact support is the direct limit of the spaces C 0 Z (Y, K). We equip C 0 c (Y, K) with the direct limit topology in the category of locally convex topological vector spaces (see [Bou65, III §1 n o 1]).
We define the space of real Radon measures on Y as the topological dual of C 0 c (Y, R). Observe that our Radon measures are precisely the measures considered by Bourbaki [Bou65,