Toroidal b-divisors and Monge-Amp\`ere measures

We generalize the intersection theory of nef toric b-divisors on smooth and complete toric varieties developed in \cite{botero} to the case of smooth and complete toroidal embeddings. As a key ingredient we show the existence of a limit measure, supported on the weakly embedded rational conical complex attached to the toroidal embedding, which arises as a limit of discrete measures defined via tropical intersection theory on the conical complex. We prove that the intersection theory of nef toroidal Cartier b-divisors can be extended continuously to nef toroidal Weil b-divisors and that their degree can be computed as an integral with respect to this limit measure. As an application, we show that a Hilbert--Samuel type formula holds for nef toroidal Weil b-divisors.

The theory of b-divisors (where b stands for "birational") was introduced by Shokurov [Sho03] in the context of Mori's minimal program. Since then, b-divisors have appeared in many contexts, for instance in the work by Fujino [Fuj12] on base point free theorems, in the work of Küronya and Maclean [KM13] on the Zariski decomposition of divisors, and in the proof of the differentiability of the volumes of divisors by Boucksom, Favre and Jonsson [BFJ09]. Moreover, b-divisors have been associated to dynamical systems in [BFJ08a] and to psh-functions in [BFJ08b]. In the last paper, b-divisors whose support is a single point are studied. In particular, a top intersection product, that can be −∞, among (relatively) nef b-divisors is defined and it is proved that such top intersection products can be computed by means of a Monge-Ampère type measure in a valuation space. In the paper [BKK16], a b-divisor is associated to the invariant metric on the line bundle of Jacobi forms, and it is shown that such a b-divisor is integrable, in the sense that its top self intersection product is well defined and finite. Moreover, it is proved that, considering this b-divisor, one recovers a Chern-Weil formula, that says that the top self intersection product of the b-divisor is computed as the integral on an open subset of a power of the first Chern form of the metrized line bundle of Jacobi forms, and a Hilbert-Samuel formula that states that the asymptotic growth of the dimension of the space of Jacobi forms is governed by the top self intersection product of the associated b-divisor. In addition it is shown that in this case the associated b-divisor is toroidal and that its top self intersection product can be computed using toric methods.
It is expected that the results of [BKK16] can be extended to the invariant metrics on automorphic line bundles on mixed Shimura varieties, i.e. that the associated b-divisors are toroidal and their degrees computable using toric techniques. In this spirit, in [Bot17], the first author studied the theory of toric b-divisors on toric varieties and showed that much of the theory of ordinary divisors on toric varieties can be extended to the setting of b-divisors.
The aim of the present paper is to study toroidal b-divisors. In particular, we generalize two results of [BFJ08b] from the local case to the global toroidal case. Namely, that there is a well defined top intersection product between (global i.e. not only supported on a single point) nef b-divisors and that this top intersection product is given by a Monge-Ampère type measure on a rational polyhedral complex. We moreover prove a Hilbert-Samuel formula for nef and big toroidal b-divisors and a Brunn-Minkowski type inequality.
We have chosen to restrict ourselves to toroidal b-divisors because they appear naturally in the applications to mixed Shimura varieties and they are technically simpler than arbitrary b-divisors. Nevertheless, some of the results of this paper might be extended to arbitrary b-divisors by replacing the rational polyhedral complex by a suitable Berkovich space.
Toroidal b-divisor come in two flavors: Cartier and Weil. We explain this briefly. Let U ֒→ X be a fixed smooth and complete toroidal embedding of dimension n with associated (weakly embedded smooth) conical rational polyhedral complex Π X (Proposition 3.16 and Definition 3.23). Let R sm (Π X ) be the directed system of all smooth subdivisions of Π X . This is a directed set under the relation An element Π ′ in R sm (Π X ) corresponds to a smooth and complete toroidal embedding U ֒→ X Π ′ together with a toroidal, proper birational morphism π Π ′ : X Π ′ → X which is a proper allowable modification of X (Theorem 3.31).
The toroidal Riemann-Zariski space of the toroidal embedding (U, X) is defined formally as the inverse limit with maps given by the proper toroidal birational morphisms π Π ′ ,Π ′′ : X Π ′′ → X Π ′ induced whenever Π ′′ Π ′ . Then, toroidal b-divisors can be viewed as toroidal divisors on X U . For As an application, following [Bot17, Section 5], we define the space of global sections of a toroidal b-divisor. The volume of a nef Weil b-divisor is defined in analogy to the volume of nef divisors by the asymptotic growth of the space of global sections. Moreover, to a nef Weil toroidal b-divisor D D D we can associate an Okounkov body ∆ D D D (Definition 5.6). Then we obtain the following extension of the Hilbert Samuel theorem (Theorem 5.15). As a corollary, we obtain a Brunn-Minkowski type inequality (Corollary 5.17). One of the key ingredients to prove the above theorems is the combinatorial machinery developed in Sections 1 and 2. The existence of the limit measure µ D D D associated to a nef toroidal b-divisor D D D follows directly from Theorem 2.20 and the existence of the mixed limit measure µ D D D 2 ,...,D D D n is a direct consequence of Corollary 2.24. These results are based on the convex analysis on polyhedral spaced developed by M. Sombra and the authors in [BBS].
Another key ingredient is a result in [Gro15] relating tropical and algebraic intersection numbers on complete toroidal embeddings (Theorem 4.6).
It is important to note that for the applications to algebraic geometry it is convenient to work with conical complexes provided with an integral structure. Nevertheless, when studying convex analysis on polyhedral complexes, the integral structure plays no role, only the affine structure does. Moreover, to write down explicit estimates it is handy to choose a Euclidean structure. Therefore, to study Monge-Ampère measures associated to nef toroidal b-divisors it is convenient to shift the focus from rational conical complexes to Euclidean ones.
As has been noted in [BKK16] and [Bot17], a nef toroidal b-divisor encodes the singularities of the invariant metric on an automorphic line bundle over a mixed Shimura variety of non-compact type along any toroidal compactification. This article together with the above mentioned ones lays the ground of a geometric intersection theory with singular metrics, satisfying Chern-Weil theory and a Hilbert-Samuel formula, to be applied to mixed Shimura varieties of non-compact type.
The article is organized as follows. In Section 1 we recall the tropical intersection theory on Euclidean conical polyhedral complexes as in [BBS]. This is an Euclidean version of the tropical intersection theory on weakly embedded rational conical polyhedral complexes developed in [Gro15].
In Section 2 we show the combinatorial version of our main result stated in Theorem A. For this, we define b-divisors on Euclidean conical complexes and introduce a positivity notion for them. We show that the top intersection product of such positive b-divisors exists, is finite and is given by the total mass of a week limit of discrete Monge-Ampère measures. This is done by introducing the notion of the size of a tropical cycle. This allows us to prove a Chern-Levine-Nirenberg type inequality (Lemma 2.14) from which we conclude the weak convergence of the discrete measures (Theorem 2.20).
In Section 3 we give the definition of a quasi-embedded rational conical polyhedral complex. In short a rational conical polyhedral complex is a conical polyhedral complex with a lattice structure. We recall the definition of toroidal embeddings and describe the rational conical polyhedral complex associated to a toroidal embedding (see [KKMS73] or [AMRT10] for further details). Following [Gro15], we also give a natural weak embedding of this complex. Moreover, we show that by adding boundary components one can modify the toroidal structure of a toroidal embedding in such a way that the rational conical polyhedral complex becomes quasi-embedded. Then we describe the proper toroidal birational modifications of a toroidal embedding which, on the combinatorial side, correspond to subdivisions of the corresponding rational conical polyhedral complex.
In Section 4 we state and prove our main results. We show that nef toroidal b-divisors have well defined top intersection products (Definitions 4.14 and 4.19 and Theorem 4.25). For this, we first relate the geometric intersection product of toroidal divisors with the rational tropical intersection product on quasi-embedded rational conical complexes (Theorem 4.6) (see [Gro15]). Then we use the convergence results of Section 2 in order to extend the top intersection product to nef toroidal b-divisors. However, note that the Monge-Ampére measures of Section 2 are defined in an Euclidean setting (no integral structure). Therefore we will use the comparison in section 3.2 to relate the rational tropical intersection product with the Euclidean one.
Finally, in Section 5, as an application, we give a Hilbert-Samuel type formula for nef and big toroidal b-divisors. This relates the degree of a nef toroidal b-divisor both with the volume of the b-divisor and with the volume of the associated convex Okounkov body (Definitions 5.3 and 5.6 and Theorem 5.15). As a corollary, we obtain a Brunn-Minkowski type inequality (Corollary 5.17). Acknowledgments We would like to thank Jürg Kramer, Robin de Jong, Walter Gubler, David Holmes and Klaus Kuenneman for many stimulating discussions. We want specially to thank Martín Sombra for helping us in proving the continuity properties of concave functions on polyhedral spaces [BBS].
Most of the work of this paper has been conducted while the authors were visiting Humboldt University of Berlin, Regensburg University and the ICMAT. We would like to thank these institutions for their hospitality.

E
In this section we recall the tropical intersection theory on Euclidean conical polyhedral complexes as in [BBS]. This is an Euclidean version of the tropical intersection theory on weakly embedded rational conical polyhedral complexes developed in [Gro15].
1.1. Euclidean conical polyhedral complexes. We start with the definition of a quasi-embedded conical polyhedral complex endowed with a Euclidean structure. We also discuss morphisms and subdivisions of such complexes.
consisting of a topological space |Π| together with a finite covering by closed subsets σ α ⊆ |Π| and for each σ α , a finitely generated R-vector space M α of continuous, R-valued functions on σ α satisfying the following conditions. Let N α := Hom(M α , R) denote the dual vector space.
(1) For each α ∈ Λ, the evaluation map φ α : σ α → N α given by the assignment maps σ α homeomorphically to a strictly convex, full-dimensional, polyhedral cone in N α . (2) The preimage under φ α of each face of φ α (σ α ) is a cone σ α ′ for some index α ′ ∈ Λ, and we have that The intersection of two cones is a union of common faces. The R vector spaces M α give the complex a so called linear structure.
The following notations will be used.
(1) We will usually refer to a conical polyhedral complex just as a conical complex.
(2) By abuse of notation we will think of Π as the set of cones {σ α } α∈Λ . For every integer k 0 we write Π(k) for the set of cones of dimension k.
(3) The topological space |Π| = α∈Λ σ α is called the support of the conical polyhedral complex Π. (4) Given a cone σ ∈ Π, we will write M σ , N σ and φ σ for the corresponding R-vector space, dual vector space and evaluation map, respectively. We denote by , σ the pairing induced by the dual vector spaces M σ and N σ . We will usually omit the index "σ" from the pairing.
(5) We will identify a cone σ with its image in N σ . The linear structure of N σ induces a linear structure in σ. Therefore we can talk of linear maps between cones. (6) If τ is a face of σ we will write τ ≺ σ or σ ≻ τ. (7) We will denote by 0 σ the zero for the linear structure of N σ . Since σ is strictly convex, the set {0 σ } is a face of σ. By abuse of notation we will denote this face also as 0 σ . (8) The dimension of the polyhedral complex Π is sup σ∈Π {dim (M σ )}. We say that Π has pure dimension n if all the maximal cones have dimension n. (9) By the relative interior of a cone σ, denoted relint(σ) we mean the preimage under φ σ of the interior of the cone φ σ (σ) ⊆ N σ . (10) A conical complex is called simplicial if every cone φ σ (σ) is generated by an R-basis of N σ . Definition 1.4. A weakly embedded conical complex is a conical complex Π, together with a finite-dimensional R-vector space N Π , and a continuous map ι Π : |Π| → N Π which is linear on every cone σ of Π. We will usually denote a weakly embedded conical complex by the underlying conical complex Π.
Given a weakly embedded conical complex Π we write M Π = Hom N Π , R for the dual vector space of N Π . For every cone σ ∈ Π we write N Π σ for N Π ∩ Span (ι Π (σ)) and M Π σ = Hom N Π σ , R for its dual. The following notion is stronger than that of a weakly embedded conical complex. Definition 1.5. A weakly embedded conical complex is said to be quasi-embedded if the restriction of ι Π is injective in each cone. If Π is a quasi-embedded conical complex we will identify each vector space N σ with its image N Π σ in N Π . The following is a useful property of quasi-embedded conical complexes that is not true in general for weakly embedded ones. Lemma 1.6. Let Π be a quasi-embedded conical complex. Then the map ι Π : |Π| → N Π is proper.
Proof. Since |Π| is a finite union of closed cones σ, it is enough to show that ι Π | σ is proper.
Definition 1.7. A Euclidean conical complex is a quasi-embedded conical complex Π together with a Euclidean product on the real vector space N Π . This Euclidean structure induces compatible Euclidean structures in each vector space N σ . Definition 1.8. A morphism f : Θ → Π of conical complexes is a continuous map f : |Θ| → |Π| with the property that for every cone τ ∈ Θ there exists a cone σ ∈ Π such that f(τ) ⊆ σ, and that the restriction f| τ : τ → σ is linear.
A morphism of weakly embedded conical complexes consists of a morphism of conical complexes f : Θ → Π together with a morphism of finite-dimensional R-vector spaces f ′ : N Θ → N Π forming a commutative square with the weak embeddings.

|Θ| |Π|
A morphism of quasi-embedded or of Euclidean conical complexes is a morphism of weakly embedded conical complexes.
A subdivision is called simplicial if the conical complex Π ′ is simplicial. Remark 1.10. If Π is a weakly embedded conical complex and Π ′ is a subdivision, then Π ′ has an induced structure of weakly embedded conical complex. If Π is quasi-embedded or Euclidean then Π ′ is also quasi-embedded or Euclidean, respectively. Definition 1.11. Let Π be a conical complex. We define the directed set R(Π) to be the set of all subdivisions of Π with the partial order given by If Π ′′ Π ′ , we will denote by π Π ′ ,Π ′′ : Π ′′ → Π ′ the morphism of Euclidean polyhedral complexes that is the identity at the level of topological spaces. Since any two conical complexes have a common refinement, this gives R(Π) the structure of a directed set. We denote by R sp (Π) ⊆ R Π the subset of simplicial subdivisions with the induced partial order. This is a cofinal subset and thus has an induced structure of directed set.
1.2. The Euclidean tropical intersection product. Throughout this section Π will denote a Euclidean conical complex of pure dimension n with quasi-embedding given by ι Π : |Π| → N Π . The goal of this section is to define the Euclidean tropical intersection product between Euclidean Minkowski weights (Definition 1.13) and tropical cycles (Definition 1.15) on one side and R-Cartier divisors on Π (Definition 1.20) on the other. This is a Euclidean version of the tropical intersection product given in [Gro15]. For the interested reader, the articles [AR10], [FS97] and [Kat12] constitute a more thorough reference for tropical intersection theory on globally embedded polyhedral complexes with an integral structure.
We start with some definitions. These are the Euclidean versions of [Gro15, Section 3.1].
Definition 1.12. Let k 1 be an integer and let τ ∈ Π(k − 1) be a cone. For every cone σ ∈ Π(k) with τ ≺ σ we define the Euclidean normal vectorv σ/τ of σ relative to τ to be the unique unitary vector of N σ that is orthogonal to N τ and points in the direction of σ. By abuse of notationv σ/τ will also denote its image in N Π . If k = 1, we writev σ :=v σ/{0 σ } .
The set of weights is a real graded vector space denoted by W * (Π). The k-dimensional Euclidean Minkowski weights form an abelian group, which is denoted by M k (Π).
We can now define the pull-back of a Minkowski weight along a subdivision.
Definition 1.14. Let Π ′ be a subdivision of Π with its induced structure of Euclidean conical complex and denote by f : Π ′ → Π the corresponding morphism of Euclidean conical complexes. Let c ∈ M k (Π) be a Euclidean Minkowski weight. Then the pull-back of c by f is the Euclidean Minkowski weight where σ ′ ∈ Π ′ (k) and σ is the minimal cone of Π that contains σ ′ . This construction defines a group homomorphism More generally, Euclidean tropical cycles are defined as direct limits of Euclidean Minkowski weights over all subdivisions.
Definition 1.15. The group of Euclidean tropical k-cycles on Π is defined as the direct limit with maps given by the pull-back maps of Definition 1.14. If c is a k-dimensional Euclidean Minkowski weight on a subdivision Π ′ of Π, we denote by [c] its image in Z k (Π). We now define the objects where we want to compute top intersection numbers, namely balanced Euclidean conical complexes.
Definition 1.18. The Euclidean conical complex Π of pure dimension n is said to be balanced if there exists an n-dimensional Euclidean tropical cycle [Π] ∈ Z n (Π) represented by a Minkowski weight b ∈ M n (Π ′ ), for some subdivision Π ′ in R(Π), satisfying b(σ) > 0, ∀σ ∈ Π ′ (n). A Euclidean conical complex is called balanceable if it admits a structure of a balanced Euclidean conical complex.
Remark 1.19. If Π is a balanced Euclidean conical complex, then |Π| is a balanced polyhedral space in the sense of [BBS,Definition 3.27]. Thus we have at our disposal the theory of concave functions on polyhedral spaces developed in that paper.
We now define Cartier divisors on the Euclidean conical complex Π. The following definition is adapted from [Gro15, Definition 3.6].
Definition 1.20. An R-Cartier divisor on Π is a function φ : |Π| → R that is linear on each cone σ ∈ Π. We denote the group of R-Cartier divisors on Π by Div(Π) R . This is a finite dimensional real vector space. If φ is an R-Cartier divisor, then for every cone σ we may choose a linear function φ σ on N Π such that φ σ | σ = φ| σ . By abuse of notation φ σ will also denote the linear function restricted to N σ .
An R-Cartier divisor is always continuous as a function because the restrictions to the components of a finite closed covering are continuous. Moreover it is conical in the sense that, for all λ > 0, φ(λx) = λφ(x).
Remark 1.21. To simplify notation, we will usually omit the coefficient ring R from the notation, real coefficients being always implicit. Hence, when we say "a Cartier divisor on Π" we actually mean "an R-Cartier divisor on Π". For any k 1 we now construct a Euclidean tropical intersection product Definition 1.24. Let Π be a Euclidean conical complex, φ a Cartier divisor and c ∈ M k (Π) a k-dimensional Euclidean Minkowski weight on Π. Then the Euclidean tropical intersection The fact that φ · c is indeed a Euclidean Minkowski weight is proven in [BBS, Proposition 3.19].
The Euclidean tropical intersection product extends to a pairing between classes of Cartier divisors and Euclidean tropical cycles. Indeed, on the one hand, the Euclidean tropical intersection product is compatible with linear equivalence. Lemma 1.25. Let φ and φ ′ be linearly equivalent Cartier divisors and c a Euclidean Minkowski weight. Then Proof. If ψ is a linear function in N Π , then from the definition of the Euclidean tropical intersection product, we see that ψ · c = 0 from which the lemma follows.
On the other hand, the Euclidean tropical intersection product is compatible with the restriction to subdivisions.
The Euclidean tropical intersection product satisfies the following symmetry property.
Proposition 1.27. Let φ 1 and φ 2 be two Cartier divisors and c a Euclidean Minkowski weight. Then Proof. This is proved in [BBS,Proposition 3.15].
We can now define the intersection product between (classes of) Cartier divisors and Euclidean tropical cycles.
Definition 1.28. Let φ be a Cartier divisor on Π and let [c] ∈ Z k (Π) be a Euclidean tropical cycle represented by a Euclidean Minkowski weight c ∈ M k (Π ′ ). Then the bilinear pairing given by is well defined by Lemmas 1.26 and 1.25. We call this pairing the Euclidean tropical intersection product as well.
We define Euclidean tropical top intersection numbers of (classes of) Cartier divisors on Π.
Definition 1.29. Assume that Π is balanced. Let φ 1 , . . . , φ n be Cartier divisors on Π. The Euclidean tropical top intersection number φ 1 · · · φ n is defined inductively by This defined a multilinear map It is symmetric by Proposition 1.27. Remark 1.30. In [Gro15], the author works with weakly embedded conical complexes with an integral structure. As a consequence of working with a weakly embedded conical complex, only a tropical intersection product between tropical cycles and so called combinatorially principal Cartier divisors can be defined. In our setting, we assume that the complex is quasi-embedded. It follows that every Cartier divisor is combinatorially principal, hence arbitrary products between Cartier divisors and tropical cycles can be defined. As we will see later, the price to pay for this simplification in the algebro-geometric setting of Section 3.4 is that we will have to add more components at the boundary to be sure that the conical complex corresponding to a toroidal embedding is quasi-embedded. Moreover, in the study of Monge-Ampère measures it is more natural to replace the integral structure by an Euclidean structure.
1.3. b-divisors on Euclidean conical complexes. In this section we define Cartier and Weil b-divisors on a Euclidean conical complex Π. These are the main actors of his paper.
Definition 1.31. The group of R-Cartier b-divisors on Π is the direct limit where the connecting morphisms are the pull-backs defined in 1.23. In other words, an R-Cartier b-divisor is the choice of a subdivision Π ′ and an R-Cartier divisor φ on Π ′ with the equivalence relation generated by the relations whenever Π ′′ Π ′ and f : Π ′′ → Π ′ is the corresponding morphism. If an R-Cartier bdivisor is represented by (Π ′ , φ), we will denote it by the function φ, which, as a function, is independent of the choice of the subdivision, and call Π ′ a subdivision of definition of φ.
As before, we will usually omit the coefficient ring R from the notation and call R-Cartier b-divisors simply Cartier b-divisors.
The space of R-Weil divisors on Π is denoted by WDiv(Π) R . If Π ′ is a subdivision of Π, f : Π ′ → Π the corresponding morphism of rational conical complexes, and D ′ a Weil divisor on Π ′ , then the push-forward f * D ′ of D ′ by f is the R-Weil divisor given by In other words, if j : Π(1) ֒→ Π ′ (1) is the canonical inclusion of the sets of rays, then we have that f * D ′ = D ′ • j, so f * is just "to forget" the rays of Π ′ that are not in Π.
Also here, we will usually omit the coefficient ring R from the notation and refer to R-Weil divisors just as Weil divisors.
Remark 1.33. One should not identify Weil Cartier divisors and one dimensional weights. Both are maps from Π(1) to R but their role is very different. Definition 1.34. If φ is a Cartier divisor, then the associated Weil divisor D φ is defined as wherev ρ =v ρ/{0} is the generator of ρ of norm 1.
The following result follows as in the classical case of fans.

Lemma 1.35. If Π is a simplicial Euclidean conical complex then the map
is an isomorphism.
In view of Lemma 1.35 we can define the push-forward map of Cartier divisors on simplicial subdivisions.
Definition 1.36. Let Π ′′ Π ′ Π be subdivisions with Π ′ simplicial. Let f : Π ′′ → Π ′ be the corresponding morphism of conical complexes. Then the push-forward of Cartier divisors Div(Π ′′ ) → Div(Π ′ ) is defined as the composition We now define the space of Weil b-divisors as the inverse limit of Weil Divisors.
Definition 1.37. The group of R-Weil b-divisors on Π is the inverse limit where the connecting morphisms are the push-forwards defined in 1.32. In other words, an As is to be expected, we will usually omit the coefficient ring R from the notation and refer to R-Weil b-divisors just as Weil b-divisors.
Thanks to Lemma 1.35 and the fact that simplicial subdivisions are cofinal, we can define Weil b-divisors using Cartier divisors. In fact there is a canonical isomorphism Div(Π ′ ).
By abuse of notation, we will identify both limits. Given a subdivision Π ′ of Π the spaces of Cartier and Weil divisors on Π ′ are finitedimensional real vector spaces, hence they have a canonical topology. Definition 1.38. We endow the space Ca-b-Div(Π) with the direct limit topology (called the strong topology) and the space W-b-Div(Π) with the inverse limit topology (called the weak topology).
We can give the following characterization of the topological space W-b-Div(Π) R .
Let ρ x be the ray of |Π| containing x and choose any subdivision . By the compatibility with push forward condition of Weil b-divisors, this value is independent of the choice of Π ′ . Conversely It is easy to check that D D D Ψ is a Weil b-divisor, that both constructions are inverses of each other and that they are continuous. This concludes the proof.
The Euclidean tropical intersection product between Cartier divisors and Euclidean tropical cycles can easily be extended to Cartier b-divisors. If [c] is a Euclidean tropical cycle represented by a Minkowski weight c on a subdivision Π ′ , and φ is a Cartier b-divisor with subdivision of definition Π ′′ , then we can choose a common refinement Π ′′′ of Π ′ and Π ′′ and define φ · [c] = [φ · c] where the second product is computed in Π ′′′ . This product is independent from the choice of Π ′′′ by Lemma 1.26.
In fact we can extend the Euclidean tropical top intersection number of Definition 1.29 to the case where there is at most one Weil b-divisor involved.
is independent of the choice of the subdivision Π ′ .
Proof. In view of Lemma 1.26 we are reduced to prove the following projection formula. Let Π ′′ Π ′ be simplicial subdivisions of Π and f : Π ′′ → Π ′ the corresponding morphism. Moreover, let c 1 ∈ M 1 (Π ′ ) be a Euclidean Minkowski weight of dimension one on Π ′ and φ an Cartier divisor on Π ′′ . Then From the explicit description of the product in Definition 1.24 we deduce Equations (1.5) and (1.4) imply (1.3), which proves the lemma.
, Π ′ , c and φ Π ′ be as in Lemma 1.40. Then the Euclidean top intersection number of Z, φ 1 , . . . , φ k−1 and D D D is defined by One of the main motivations of this article is to extend Definition 1.41 to certain cases where all the divisors involved are Weil b-divisors and not just one.

M -A
Throughout this section Π will denote an n-dimensional balanced Euclidean conical complex with quasi-embedding given by ι Π : |Π| → N Π .
The goal of this section is to prove that given an admissible family C of nef Cartier bdivisors on Π (Definition 2.3), for any C-nef b-divisor φ on Π (Definition 2.7), its top intersection number exists, is finite, and is given by the total mass of a weak limit of discrete Monge-Ampère measures associated to the elements of the given admissible family (Corollary 2.22). This is done by introducing the notion of the size of a tropical cycle. This allows us to prove a Chern-Levine-Nirenberg type inequality (Lemma 2.14) from which we conclude the weak convergence of the discrete measures (Theorem 2.20).
2.1. C-nef b-divisors. Let | · | be the Euclidean norm on N Π and let The set S Π is compact since it is the inverse image of a compact space under a proper map by Lemma 1.6. Note that the Euclidean normal vectorsv σ/τ from Definition 1.12 are elements in S Π .
Since b-divisors can be described as conical functions in |Π|, we can view the spaces of b-divisors on Π as spaces of functions on S Π . Definition 2.1. We denote by C 0 (S Π ) the space of all real valued continuous functions on S Π with the topology of uniform convergence. We define the set Div S Π of Cartier divisors on S Π to be the set of continuous functions on S Π which are restrictions of Cartier b-divisors on Π, i.e. Div Note that that the sets Div S Π and Ca-b-Div(Π) are in bijection. Hence, the set Div S Π gives just a different way of looking at the set Ca-b-Div(Π). Similarly we define WDiv(S Π ) as the set of real valued functions on S Π with the pointwise convergence topology. By Lemma 1.39, the space WDiv(S Π ) is canonically homeomorphic to W-b-Div(Π).
We make the following remark.

Remark 2.2. By the Stone-Weierstrass Theorem, the subset Ca
We say that C is an admissible family of nef Cartier b-divisors on Π if the following properties are satisfied: (1) If φ 1 , . . . , φ r ∈ C, then the Euclidean tropical cycle φ 1 · · · φ r · [Π] is positive, i.e. if it belongs to Z + n−r (Π).
(2) If φ 1 and φ 2 are elements in C then for every choice of non-negative real numbers λ and µ, the linear combination λφ 1 + µφ 2 belongs to C.
If C is an admissible family of nef Cartier b-divisors on Π, an element φ ∈ C will be called C-nef. Example 2.6. We will see in the next section that if Π = Π X comes from the geometry of a smooth and complete toroidal embedding U ֒→ X satisfying certain mild hypothesis, then there is a canonical admissible family C of nef Cartier b-divisors on Π X induced by the collection of nef toroidal divisors on smooth toroidal modifications of X.
From now on we fix an admissible family C of nef Cartier b-divisors on Π.
Definition 2.7. We denote by Div S Π C ⊆ Div S Π the set of functions in Div S Π arising as restrictions of elements in C. Moreover, the space The following is a key result. Before stating it, recall that, for a subset A of a topological space T , the sequential closure |A| seq of A is the set of all points that are limits of sequences in A. Then |A| seq ⊆ A. The space T is called a Fréchet-Urysohn space if, for all A ⊆ T , the condition |A| seq = A holds. A Fréchet-Urysohn space is sequential, hence the topology of such spaces is determined by the convergent sequences.
. Moreover the topologies induced in this space by the one of WDiv S Π and the one of C 0 (S Π ) agree. That is, in WDiv S Π C the topology of pointwise convergence and that of uniform convergence are the same. In particular WDiv S Π C is metrizable. Proof. Let (f α ) α∈I be a net of C-nef Cartier b-divisors that converge to a Weil-b-divisor f. Choose a countable dense collection of points x 1 , x 2 , . . . of |Π|. Since the topology of the space of Weil-b-divisors is that of pointwise convergence, for any i > 0 there is an α i such that, for all α α i and all j i, the condition converges in a dense subset to f. By [BBS, Theorem 6.23], the sequence (f α i ) i>0 converges to a weakly concave (hence continuous) function g, that agrees with f on the points x i , i > 0. Let now y be another point of |Π|. Repeating the argument with the sequence of points y, x 1 , x 2 , . . . , we obtain a new continuous function g 1 , that agrees with f in the point y and agrees with g in a dense subset. Hence g(y) = f(y). Since y is arbitrary, we deduce that f = g. Therefore f is weakly concave and is a continuous Since all the functions f α are weakly concave, we can repeat again the previous argument to extract a sequence (f α i ) i>0 that converges to f. We conclude that the space of C-nef Weil-b-divisors is Fréchet-Urysohn. Hence the topology is determined by the convergent sequences. Using again [BBS,Theorem 6.23] a sequence in WDiv S Π C converges if and only if it converges uniformly in S Π . This concludes the proof.
2.2. The size of a Minkowski cycle. We have the following lemma which we will use later on.
To define the size of a positive Euclidean Minkowski cycle we need to choose an auxiliary function.
(2) Since ϕ 0 is concave, the function ϕ is strongly concave in the terminology of [BBS, Definition 4.2]. Therefore, by [BBS,Proposition 4.9], for every k dimensional positive cycle z and j k, the cycle (ϕ·) j · z is positive. In particular the size of z is positive.
Proof. The first statement follows directly from the definition. Let Π be a subdivision of Π ′ such that ϕ is piecewise linear on Π. By the definition of ϕ, for every τ ∈ Π(1), the inequality ϕ(v τ ) −1 is satisfied. Therefore The following is a Chern-Levine-Nirenberg type inequality for the Euclidean tropical intersection product and is a key step to prove the main result of this section.
Lemma 2.14. Let z ∈ Z + k (Π) and φ ∈ Ca-b-Div(Π) be a k-dimensional positive Euclidean tropical cycle and a Cartier b-divisor, respectively. Assume that the Euclidean tropical intersection product φ · z is a positive Euclidean tropical cycle. Then the inequality Proof. Let Π Π be a subdivision of Π such that ϕ is piecewise linear on Π and that the cycle z and the function φ are defined in Π. We define the positive real constant B by Then, for every τ ∈ Π(1) we have that Hence, since both φ and Bϕ are piecewise linear on Π, we conclude that φ Bϕ.
Therefore, using Lemma 2.10, the positivity of (ϕ ·) k−1 z (Remark 2.12 (2)), and the commutativity of the Euclidean tropical intersection product, we get as we wanted to show.

Weak convergence of Monge-Ampère measures.
We recall the definition of the total variation norm (see e.g. [AL06, Definition 4.2.5 and Proposition 4.2.5]).
Definition 2.15. Let X be a locally compact topological space and let M(X) be the space of finite Radon measures on X, i.e. the space of continuous linear forms on the space C 0 (X) of continuous real-valued functions on X with respect to its weak topology. The total variation norm · on M(X) is given by In order to prove the main result of this section (Theorem 2.20), we use the following version of Prokhorov's theorem which follows from [Bog06, Proposition 8.6.2].
Theorem 2.16. Let X be a compact metrized space. Then the following is satisfied.
We define the discrete measure µ z on S Π by where δv τ denotes the Dirac delta measure supported onv τ ∈ S Π . (Note that this does not depend on the choice of Π ′ .) (2) Let φ be a Cartier b-divisor on Π. The discrete Monge-Ampère measure µ φ on S Π is defined by The total variation of a discrete measure with finite support is given by the sum of the absolute value of the measures of the points in the support. Therefore, for z and Π ′ as in Definition 2.17 we have Remark 2.18. Although defined in a different setting, we note the similarity between the discrete measure µ φ and the Monge-Ampére measure M(g) given in [BFJ08b, Section 4.2], defined with respect to a piecewise-affine plurisubharmonic function g (see [BFJ08b, Proposition 4.9]).
The following proposition is a consequence of Lemma 2.14.
of measures on S Π has bounded total variation.
Proof. Since, by Theorem 2.9, the convergence By assumption, for each ℓ = 1, . . . , n − 1, there exist elements β 0 ℓ and β 1 ℓ in C such that γ ℓ = β 0 ℓ − β 1 ℓ . Since there are finitely many, we may choose a positive real number C such that Since the involved b-Cartier divisors are C-nef, we have that for every j ∈ N and for every tuple (i 1 , . . . , i n−1−k ) ∈ {0, 1} n−1−k , the 1-dimensional Euclidean tropical cycle Fix j ∈ N and let Π ′ be a subdivision of Π where the φ j and the functions γ i , i = 1, . . . , n− 1 − k, are defined. Then, using equation (2.4), Lemma 2.14 and the estimate (2.3), we get proving the proposition.
The following is the main result of this section. The proof is inspired in the classical proof of the existence of Monge-Ampère measures of [RT77, Proposition 3.1].
Theorem 2.20. Let D D D be a C-nef Weil b-divisor on Π, k ∈ {0, . . . , n−1} and γ 1 , . . . , γ n−1−k ∈ C−C. Moreover, let φ D D D be the function in S Π corresponding to D D D. Then the following holds true. ( for some Radon measures µ and ν (with respect to the weak- * topology). Then (2) The map from Div S Π C to Radon measures on S Π given by extends to a continuous operator from WDiv S Π C to Radon measures on S Π . This operator is also denoted as in (2.5).
Proof. The fact that statement (1) implies statement (2) is a standard consequence of Theorem 2.9, Proposition 2.19 and Theorem 2.16.
We prove the theorem by induction on k. If k = 0 there is nothing to prove. So we can assume that both statements of the Theorem are true for k − 1. By part (3) of Definition 2.3, in order to prove that µ = ν, it is enough to prove that µ(η) = ν(η) for η ∈ C − C.
By Proposition 1.27 we have that . Moreover, by Theorem 2.9 the sequence of functions φ j , j ∈ N, converge uniformly to the continuous function φ D D D . Therefore, the double limit lim exists and agrees with the diagonal limit i = j. Therefore . Hence, we get that µ(η) = ν(η). This concludes the proof of the theorem. . We obtain the following corollary which is the main result of this section.
exists, is finite and is given by Moreover, for any 1 i n, we have integral formulae

T
In this section, we give the definition of a quasi-embedded rational conical polyhedral complex. In short, a rational conical polyhedral complex is a conical polyhedral complex endowed with a lattice structure. We recall the definition of toroidal embeddings and describe the rational conical polyhedral complex associated to a toroidal embedding (see [KKMS73] or [AMRT10] for further details). Following [Gro15], we also give a natural weak embedding of this complex. Moreover, we show that by adding boundary components one can modify the toroidal structure of a toroidal embedding in such a way that the rational conical polyhedral complex becomes quasi-embedded. Then we describe the proper toroidal birational modifications of a toroidal embedding which, on the combinatorial side, correspond to subdivisions of the corresponding rational conical polyhedral complex.
Definition 3.1. A rational conical polyhedral complex is a triple consisting of a topological space |Π| together with a finite covering by closed subsets σ α ⊆ |Π| and for each σ α , a finitely generated Z-module M α of continuous, R-valued functions on σ α satisfying the following conditions. Let N α := Hom(M α , Z) denote the dual lattice.
(2) The preimage under φ α of each face of φ α (σ α ) is a cone σ α ′ for some index α ′ ∈ Λ, and we have that The intersection of two cones is a union of common faces. The Z-modules M α give the complex a so called integral structure.
Most of the notations and terminology of Section 1 carry over to the case of rational conical polyhedral complexes, by taking into account the integral structure.
(1) Rational conical polyhedral complexes will be referred as rational conical complexes.
(2) As in Remark 1.2, the set of cones of dimension zero is in bijection with the set of connected components of |Π|.
(3) The terminology concerning cones, faces, interior, support and dimension is the same as in the non-rational case keeping in mind the compatibility between the integral structures. (4) The notion of a simplicial rational conical complex is the same. However, in the rational case we also have a notion of smoothness. A rational conical complex is called smooth if every cone σ ∈ Π is unimodular, i.e. if φ σ (σ) is generated by a Zbasis of N σ . Clearly, a smooth rational conical complex is automatically simplicial. (5) The notion of a morphism between rational conical complexes is the same except that we require the restriction to each cone to be integral. (6) The notion of subdivisions is the same except that we require them to be rational as well. We denote the set of smooth subdivisions of Π by R sm (Π). This has an induced structure of directed set. (7) The notions of weakly-embedded and quasi-embedded rational conical complexes are the same except that the co-domain of the weak-(respectively quasi-) embedding is an R-vector space N Π R with an integral structure N Π and the restriction of the weak (respectively quasi-) embedding to each cone is required to be integral.

A bridge between Euclidean and integral structures.
Following [Gro15], there is a rational tropical intersection product on quasi-embedded rational conical complexes. We compare the rational tropical intersection with the Euclidean one from Section 1 by means of the normalization of cycles.
We fix a rational conical complex Π and start with some definitions. These are adapted from [Gro15, Section 3.1].
Definition 3.2. Let k 0 be an integer and let τ ∈ Π(k − 1) be a cone. For every cone σ ∈ Π(k) with τ ≺ σ. We define the lattice normal vector v σ/τ of σ relative to τ to be the image in the quotient N Π R /N τ R of the unique generator of N σ /N τ that points in the direction of σ. For every pair of cones σ and τ as before we will chose a liftingṽ σ/τ ∈ N σ Usually, lattice Minkowski weights will be called Minkowski weights. The k-dimensional Minkowski weights form a real vector subspace, which is also denoted by M k (Π). Note the symbol M k (Π) denotes lattice Minkowski weights when Π is rational and Euclidean Minkowski weights when Π is Euclidean.
The condition (3.1) is called the (lattice) balancing condition around τ, while the condition (1.1) is called the Euclidean balancing condition. The balancing conditions (1.1) and (3.1) depend on the choice of the quasi-embedding.
The following notions carry over from the Euclidean to the lattice case directly.
• The definition of balanced rational conical complex is the same as in the Euclidean case. • The definition of the pull-back along a subdivision is the same.
• The definition of the group of (lattice) tropical cycles is analogous. This group is also denoted by Z k (Π). • The definition of a Cartier divisor and that of the pull-back of an Cartier divisor along a morphism of rational conical complexes is the same. Also here, we must have in mind that we are working with real coefficients. • Given a Cartier divisor φ on Π, for every cone σ ∈ Π we choose a linear function φ σ on N Π R such that φ σ | σ = φ| σ . By abuse of notation φ σ will also denote the linear function restricted to N σ R . • The definition of rational equivalence of Cartier divisors is also the same.
• The definition of Cartier and Weil b-divisors is the same except that now we only consider rational subdivisions and we can take the direct and inverse limit of Cartier divisors only with respect to smooth subdivisions. Also here, we must have in mind that we are working with real coefficients.
Remark 3.4. Let Π be a quasi-embedded rational conical complex and let Π be the Euclidean one obtained by choosing a metric on N Π R and forgetting the integral structure. Since the allowed subdivisions of Π are much less that the ones of Π, the spaces of b-divisors are different. In fact there is a commutative diagram The space Ca-b-Div( Π) can be identified with the space of all piecewise linear functions on |Π|, while Ca-b-Div(Π) is the space of piecewise linear functions whose linearity locus is defined over Q. The space W-b-Div( Π) can be identified with the space of conical functions on |Π|, while the space W-b-Div(Π) can be identified as the space of real valued conical functions on Π(Q) := |Π| ∩ ι −1 π (N Π Q ). With these identifications the arrows in diagram (3.3) are the obvious ones. In particular the upward arrow on the right of the diagram send a conical function on |Π| to its restriction to Π(Q).
The definition of the intersection product in the lattice case is different from the Euclidean case, because of the change in the definition of Minkowski weights and normal vectors. To avoid confusion we will use a different symbol.
Definition 3.5. Let φ be a Cartier divisor and c ∈ M k (Π) a Minkowski weight. Then the (lattice) tropical intersection product φ ⊙ c ∈ M k−1 (Π) is the Minkowski weight given, for τ ∈ Π(k − 1), by Note that this is well defined since c ∈ M k (Π) is a k-dimensional Minkowski weight and hence Moreover, ifṽ ′ σ/τ is another choice of liftings, then w σ/τ :=ṽ σ/τ −ṽ ′ σ/τ ∈ N τ R and therefore so the intersection product is independent of the choice of liftings.
As with Minkowski weights, lattice tropical cycles will be called just tropical cycles and lattice tropical intersection will be called tropical intersection.
The tropical intersection product extends to a bilinear pairing between classes of Cartier divisors and tropical cycles.
Remark 3.7. If Π is balanced, then the definition of the tropical top intersection numbers is the same as in the Euclidean case (Definition 1.29).
We are now ready to relate Euclidean and lattice structures. As before, we denote by Π the Euclidean conical complex induced by Π by forgetting the rational structure and by choosing a Euclidean metric , on N Π R . Note that if Π is smooth, then Π is simplicial. We introduce the following notation. For a cone σ ∈ Π we let vol(σ) := vol , (N σ R /N σ ) = det ( v i , v j ) i,j where {v 1 , . . . , v k } is an integral basis of N σ . Note that vol(σ) depends on both, the rational structure and the Euclidean one.
Recall that W * (Π) denotes the space of weights of Π We define a map : W * (Π) → W * (Π) given by c(σ) := vol(σ)c(σ), Proof. Let τ ∈ Π(k − 1). We have to show that c is a Euclidean Minkowski weight. For any σ ∈ Π(k) containing τ letṽ σ/τ ∈ N σ R be a lifting of the lattice normal vector as in Definition 3.2 and letṽ σ/τ = v σ,τ + v σ,τ ⊥ be an orthogonal decomposition ofṽ σ/τ with v σ,τ ∈ N τ R and v σ,τ ⊥ orthogonal to N τ R . The Euclidean normal vectorv σ/τ of Definition 1.12 is just the normalization of v σ,τ ⊥ , i.e. we havev σ/τ = v σ,τ ⊥ / v σ,τ ⊥ . If {v 1 , . . . , v k−1 } is an integral basis of N τ , then {v 1 , . . . , v k−1 ,ṽ σ/τ } is a basis on N σ . Therefore, We compute In the last equation we have used that, since c is a Minkowski weight then c(σ)ṽ σ/τ belongs to N τ , hence agrees with its orthogonal projection to N τ R which is c(σ)v σ,τ . We deduce that c is a Euclidean Minkowski weight. We make the following remark. The following proposition shows the compatibility between the tropical intersection product and the Euclidean one, allowing us to replace the integral structure by the Euclidean one in computations.
Proposition 3.11. Let φ ∈ Div(Π) be a Cartier divisor on Π and let c ∈ M k (Π) be a k-dimensional Minkowski weight. Then the following equality holds true.
Hence, also for a k-dimensional rational Minkowski cycle z ∈ Z k (Π) we have Proof. Let τ ∈ Π(k − 1). We use the same notation as in the proof of Lemma 3.8. Since c is a Minkowski weight, we have that We compute, using equation (3.4), hence the first statement of the proposition follows. The second statement clearly follows from the first.
3.3. The rational conical complex attached to a toroidal embedding. Throughout this section, k will denote an algebraically closed field of characteristic 0. All of the varieties appearing in this section will be defined over k even if not stated explicitly. We recall the definition of a toroidal embedding and describe its associated rational conical complex. The following definition is taken from [KKMS73, Definition 1, pg. 54].
Definition 3.12. Let X be an n-dimensional normal, algebraic variety over k and let U be a smooth Zariski open subset of X. An open immersion U ֒→ X is a toroidal embedding if for every closed point x ∈ X there exists an n-dimensional torus T, an affine toric variety X σ ⊇ T, a point x ′ ∈ X σ and an isomorphism of k-local algebras such that the ideal in O X,x generated by the ideal of X \ U corresponds under this isomorphism to the ideal in O X σ ,x ′ generated by the ideal of X σ \ T. Here, the hat " " denotes the completion of the local ring at a point. Such an isomorphism is called a chart at x and the pair (X σ , x ′ ) is called a local model at x.
If all the irreducible components of the boundary divisor X \ U of a toroidal embedding are normal, then it is called a toroidal embedding without self intersection.  (1) B J is normal.
Moreover, the sets S α , α ∈ Λ define a stratification of X, i.e. every point of X is in exactly one stratum and the closure of a stratum is a union of strata. Furthermore, if x ∈ X and (X σ , x ′ ) is a local model at x, then the closures S α of the strata S α such that x ∈ S α correspond formally to the closure of the torus orbits in X σ containing x ′ . In particular, if x ∈ S α , then S α corresponds formally to the torus orbit O(x ′ ) itself.

Moreover, let
Then M S α is a free abelian group (a lattice) while M S α + has the structure of a sub-semigroup. For each stratum S α we denote by N S α = M S α ∨ the dual lattice of M S α and by , S α the induced pairing. Finally, let Then σ S α ⊆ N S α R is a strongly convex rational polyhedral cone of maximal dimension. The idea behind Proposition/Definition 3.15 is that given a stratum S, we have produced a maximal dimensional cone σ S in the finite-dimensional real vector space N S R which comes equipped with a canonical lattice N S .
We now see that these cones can be glued together into a rational conical complex. For a toroidal embedding U ֒→ X without self intersection, let |Π (X,U) | be the quotient topological space defined by where ∼ is the equivalence relation generated by isomorphisms which in turn is induced by the map M S α ′ → M S α given by restricting divisors from Star (S α ′ ) to Star (S α ) (see [KKMS73, Chapter II, Section 1]). We have the following proposition.

Proposition 3.16. If U ֒→ X is a toroidal embedding without self intersection, then the triple
is a rational conical complex in the sense of Definition 3.1.
Proof. The proof can be found in [KKMS73, Chapter II, pg. 71].
The collection of lattices M S α in the above proposition gives the integral structure of the toroidal embedding.
Lemma 3.17. Let U ֒→ X be a toroidal embedding without self intersection and let x ∈ X belonging to a stratum S. If (X σ , x ′ ) is a local model at x then where M(T) refers to the lattice of characters of the torus T ⊆ X σ and σ ⊥ is the set defined by In particular, the local model (X σ , x ′ ) is determined up to isomorphism by the stratum S.
Given a cone σ in Π (X,U) , we will denote by S σ the stratum corresponding to σ and by S σ its closure in X. Remark 3.19. As in the toric case, the set of rays Π (X,U) (1) of the rational conical complex associated to a toroidal embedding U ֒→ X is in bijection with the set of irreducible components of the boundary divisor B = X \ U. Indeed for every irreducible component B i , the corresponding ray in Π (X,U) (1), which we will denote by τ B i is the linear function τ B i : M S {i} → Z given by nB i → n. Conversely, one can show that any ray τ ∈ Π (X,U) (1) arises in this way (see [KKMS73,pg. 63]). For any such ray τ, we will denote by B τ the corresponding irreducible boundary component.
Before giving a more general class of examples of toroidal embeddings, we recall some definitions.
Definition 3.20. Let B ⊆ X be a divisor on a smooth variety X. We say that B is a normal crossing divisor (abbreviated nc) if the following condition hold: (1) For all x ∈ X we can choose local coordinates x 1 , . . . , x n and natural numbers ℓ 1 , . . . , ℓ n such that B = i x ℓ i i = 0 in a neighborhood of x. We say that B is a simple normal crossing divisor (abbreviated snc) if furthermore (2) Every irreducible component of B is smooth.
We can now give a large class of examples of toroidal varieties.
Example 3.21. Let (X, B) be a pair consisting of a smooth projective variety X of dimension n together with a snc divisor B ⊆ X. We denote by {B i } i∈I the irreducible components of B. Set U := X \ B. Then U ֒→ X is a toroidal embedding. The rational conical complex associated to the toroidal embedding U ֒→ X is smooth and is constructed by adding a k-dimensional cone for each subset J ⊆ I with #J = k and each irreducible component of j∈J B j . In particular, the zero dimensional cones correspond to the irreducible components of X = j∈∅ B j .

Remark 3.22.
It follows from the definition of a toroidal embedding U ֒→ X that the boundary X \ U is a divisor, however, it may not be snc. Nevertheless, by Hironaka's resolution of singularities [Hir64], we can always find an allowable modification of X ′ → X (Definition 3.30) such that the boundary divisor X ′ \ U is snc.
3.4. From weak embeddings to quasi embeddings. Following [Gro15], when the ambient variety is proper, there is a natural weak embedding of the rational conical complex associated to a toroidal embedding without self intersection. We show in this section that, in the projective case, by adding boundary components, one can modify the toroidal structure of a toroidal embedding in such a way that the rational conical complex becomes quasiembedded. Dualizing, we get a linear map σ S → N Π R . These maps glue to give a continuous function ι Π : |Π| −→ N Π R , which is integral linear on the cones of Π, i.e. we obtain a weakly embedded rational conical complex associated to U ֒→ X which we also denote by Π (X,U) .

Two of the following examples are taken from [Gro15, Example 2.2].
Example 3.24.
(1) Consider the toric setting X = X Σ from Example 3.18, and write Π = Π (X Σ ,T) . Here, we have the lattice M Π = Γ T, O × X Σ /k × , which we can identify with M via the isomorphism M ≃ M Π given by the assignment We see that the image of σ O(σ) in N R under the weak embedding ι Π is precisely σ. Hence, Π is a weakly embedded rational conical complex, naturally isomorphic to Σ. Note that in this case, the weak embedding is globally injective.
(2) For a non-toric example, consider X = P 2 with homogeneous coordinates (x 0 : x 1 : x 2 ) but with open part U given by where H i is the hyperplane given by {x i = 0}. This is a toroidal embedding with snc boundary divisor and we see that the rational conical complex Π = Π (X,U) is naturally identified with the non-negative orthant R 2 0 , whose rays R 0 (1, 0) and R 0 (0, 1) correspond to the divisors H 1 and H 2 , respectively. The lattice M Π is generated by x 1 /x 2 , and using that generator to identify M Π with Z, we see that the weak embedding ι Π sends (1, 0) to 1 and (0, 1) to −1. Note that in this case ι Π is not a quasi-embedding since for example the cone R 2 0 is two-dimensional while N Π R has dimension one. (3) Consider again P 2 with the same homogeneous coordinates, D 1 the line x 0 = 0 and D 2 the conic x 2 0 + x 2 1 + x 2 2 = 0. then D 1 ∪ D 2 is a snc divisor and the corresponding conical complex consist of two copies of the non-negative orthant glued together by the axes. The description of the quasi-embedding is similar to the previous one.
The following key proposition says that given a toroidal embedding with a snc boundary divisor, we can always modify the toroidal structure in such a way that the associated weakly embedded rational conical complex becomes quasi-embedded.
Proposition 3.25. Let U ֒→ X be a toroidal embedding with X smooth and projective, such that the boundary divisor B = X \ U is snc and let Π = Π (X,U) be its associated weakly embedded rational conical complex. Then there exists a snc divisor B ′ with |B| ⊆ |B ′ | such that, writing U ′ = X \ B ′ , the weakly embedded rational conical complex Π ′ = Π (X,U ′ ) is quasi-embedded, i.e. the restriction of the weak embedding Proof. Recall that n denotes the dimension of X. If n = 1 then B = {p 1 , . . . , p k } is a finite set of points. Choose rational functions f i such that ord p i (f i ) = 0 and write The corresponding polyhedral complex is given by the finite set of rays {τ q j }, such that τ q j is joined with τ q j ′ at zero if and only if q j and q j ′ are in the same irreducible component.
Let v j denote the primitive vector of τ q j and let x i denote the point of M Π ′ corresponding to f i . By construction, for each j there is an i such that ι(v j ), Assume now that n 2. Write B = B 1 ∪· · ·∪B r for the decomposition of B into irreducible components. There is a hypersurface C such that B i + C is very ample for i = 1, . . . , r. Moreover we can find hypersurfaces A i,j , and a second hypersurface C 1 = C such that C 1 ∼ C. Here the symbol ∼ means linear equivalence. Finally by Bertini's theorem we can assume that all the hypersurfaces C, C 1 and A i,j are different, smooth and irreducible and is a snc. Then there are rational functions f i,j and g such that As in the statement of the theorem, write U ′ = X \ B ′ and Π ′ = Π (X,U ′ ) . Let x i,j be the point of M Π ′ corresponding to f i,j and y the point corresponding to g. For an irreducible component E of B ′ , write v E for the primitive generator of the ray corresponding to the the divisor E. By construction we have From the above identities, it follows that any subset of n vectors contained in {v B i , v A k,j , v C , v C 1 } is linearly independent. This implies that the weak embedding ι Π ′ is a quasi-embedding.
Remark 3.26. Let U ֒→ X be a toroidal embedding with snc boundary divisor B = X \ U and X projective. Then, by modifying the toroidal structure in a similar way as we did in the proof of Proposition 3.25, we may assume that there exists an ample and effective divisor A with |A| ⊆ |B|. Moreover, by adding a small multiple of all components of B, we may assume that A has positive multiplicity along all components of B. This will be useful for the monotone approximation lemma in Section 5.2.
3.5. Toroidal modifications and subdivisions of rational conical complexes. Recall from the classical theory of toric varieties that given a toric variety X Σ corresponding to a fan Σ, there is a bijective correspondence between proper birational toric morphisms to X Σ and subdivisions of the fan Σ. Following [KKMS73, Chapter 2, Section 2], a similar phenomenon occurs in the toroidal case. In this section we describe the proper toroidal birational modifications of a toroidal embedding which, on the combinatorial side, correspond to subdivisions of the associated rational conical complex.
Definition 3.27. Let U X 1 ֒→ X 1 and U X 2 ֒→ X 2 be two toroidal embeddings and let f : X 1 → X 2 be a birational morphism mapping U X 1 to U X 2 . Then f is called toroidal if for every closed point x 1 ∈ X 1 there exist local models (X σ 1 , x ′ 1 ) at x 1 ∈ X 1 and (X σ 2 , x ′ 2 ) at f(x 1 ) ∈ X 2 , and a toric morphism g : X σ 1 → X σ 2 , with f(x ′ 1 ) = x ′ 2 , such that the following diagram commutes.
Here,f # andĝ # are the ring homomorphisms induced by f and g, respectively.
Remark 3.28. The following two properties are satisfied.
(1) The composition of two birational toroidal morphisms is again a birational toroidal morphism.
(2) A toroidal morphism f : (U 1 ֒→ X 1 ) → (U 2 ֒→ X 2 ) induces a morphism f Π : Π (X 1 ,U 1 ) → Π (X 2 ,U 2 ) of rational conical complexes. The restrictions of f Π to the cones of Π (X 1 ,U 1 ) are dual to pulling back Cartier divisors. From this, we see that f Π can also be considered as a morphism between weakly embedded rational conical complexes by adding to it the data of the linear map N Π (X 1 ,U 1 ) → N Π (X 2 ,U 2 ) dual to the pullback The following definition is taken from [KKMS73, Definition 1, pg. 73].
Definition 3.29. A toroidal birational morphism f : (U ֒→ Y) → (U ֒→ X) between two toroidal embeddings of the same open subset U is called canonical over X if the following conditions hold true: (2) For all x 1 , x 2 ∈ X in the same stratum S and for all morphisms ξ :Ô X,x 1 −→Ô X,x 2 (3.7) which preserve the strata (i.e. if S ⊆ S ′ for some stratum S ′ then ξ takes the ideal of S ′ at x 1 to the ideal of S ′ at x 2 ), we have that Spec(ξ) can be lifted to give an isomorphism Y × X Spec(Ô X,x 2 ) ≃ Y × X Spec(Ô X,x 1 ) preserving the strata, i.e. such that the following diagram commutes.
We can now define the class of toroidal birational morphisms which correspond to subdivisions of rational conical complexes. The following is [KKMS73, Definition 3, pg. 87]. (1) Y has an open covering for some stratum S i of X and V i is affine and canonical over Star(S i ).
Toroidal embeddings U ֒→ Y as above are called allowable modifications of the toroidal embedding U ֒→ X.
Theorem 3.31. Given a toroidal embedding without self intersection U ֒→ X, there is a bijective correspondence between subdivisions of the rational conical complex Π X and isomorphism classes of proper allowable modifications of X.

I -
In this section k still denotes an algebraically closed field of characteristic zero and U ֒→ X will denote a toroidal embedding with X a smooth and projective n-dimensional k-variety with snc boundary divisor B = X \ U. We further assume that the corresponding smooth, weakly embedded rational conical complex Π := Π (X,U) is quasi-embedded and that there is an effective ample divisor with support B. By Chow's lemma, resolution of singularities, Proposition 3.25 and Remark 3.26, if we start with a proper variety X ′ , an open dense subset U ′ ⊆ X ′ and B ′ = X ′ \ U ′ , we can always find X, U and B as above with a birational map π : X → X ′ such that π −1 (B ′ ) ⊆ B. Therefore for many questions, the above assumptions are harmless.
The goal of this section is to show that nef toroidal b-divisors have well defined top intersection products (Definitions 4.14 and 4.19 and Theorem 4.25). The idea of the construction is, first, following [Gro15], to relate the geometric intersection product of toroidal divisors with the rational tropical intersection product on quasi-embedded rational conical complexes (Theorem 4.6). Second to use the convergence results of Section 2 in order to extend the top intersection product to nef b-divisors. However, note that the Monge-Ampére measures of Section 2 are defined in an Euclidean setting (no integral structure). Therefore we will use the comparison in section 3.2 to relate the rational tropical intersection product with the Euclidean one.

Intersection products of toroidal divisors.
We will give the definition of R-toroidal divisors and give a bijection between the set of R-toroidal divisors on (X, U) and the set of R-Cartier divisors on Π. Moreover, following [Gro15], we recall the tropicalization of an algebraic cycle and relate algebraic and tropical intersection numbers. We end this section by showing that one can compute tropically the top intersection numbers of divisors.
Definition 4.1. Let Div(X) R be the vector space of R-Cartier divisors on X. We define the subspace Div(X, U) R ⊆ Div(X) R consisting of R-Cartier divisors which are supported on the boundary B = X \ U. It is a finite dimensional R-vector space and it is endowed with a canonical topology. Elements in Div(X, U) R are called R-toroidal Cartier divisors (of (X, U)).
To simplify notation, we will usually omit the coefficient ring R from the notation and call R-toroidal Cartier divisors simply toroidal Cartier divisors.
Recall from Remark 3.19, that we have a bijective correspondence between the set of rays of Π and the set of irreducible components of the boundary divisor B = X \ U. For a ray τ ∈ Π(1) we denote by B τ the corresponding component and by v τ = v τ/0 τ the primitive lattice normal vector spanning the ray τ.
Definition 4.2. Let D ∈ Div(X, U) be a toroidal Cartier divisor on X. The corresponding tropical Cartier divisor where, for a cone σ ∈ Π, we denote by S σ the corresponding stratum of X. Since we are assuming that the toroidal embedding is smooth, we can give an alternative description going through Weil divisors. The function φ D is linear on each cone and, for τ ∈ Π(1), By Remark 3.19, any Cartier divisor φ in Π defines a toroidal Cartier divisor D φ by setting D| B τ = −φ(v τ ) for any τ ∈ Π(1). These constructions are clearly inverses of each other. We summarize the above in the following proposition, which can be seen as a special case of [KKMS73, Theorem 9*].

Proposition 4.3. The map
Div(X, U) −→ Div(Π) (4.1) given by the assignment D −→ φ D is an isomorphism of finite dimensional real vector spaces.
We recall the definition of the tropicalization of an algebraic cycle class on X as is explained in [Gro15, Section 4.2].
For 0 k n we denote by Z k (X) = Z k (X) R the group of algebraic k-cycles on X with real coefficients. For any C ∈ Z k (X), the assignment given by σ −→ deg C · S σ , is a k-dimensional Minkowski weight. Moreover, this Minkowski weight is compatible with taking refinements and thus the following definition makes sense. We make the following remarks.
(2) The tropicalization map factors through the group of numerical classes of k-cycles on X (with real coefficients), which is denoted by N k (X) R = N k (X), and hence we get a well defined tropicalization map which we also denote by trop.
The following theorem relates algebraic intersection numbers and tropical intersection numbers in the quasi embedded case.
Theorem 4.6. Let D ∈ Div(X, U) be a toroidal divisor. Then for every k-dimensional cycle class [C] in N k (X) the following tropical cycle classes agree where on the right hand side, the class [φ D ] is seen as an element in Cl(Π).
Proof. Recall that we are assuming that the rational conical complex Π is quasi-embedded. Hence, using Remark 1.30, this follows from [Gro15, Proposition 4.17].
As a consequence, we can compute top intersection numbers of arbitrary divisors using the tropical intersection product.
Then, the algebraic top intersection number deg(D 1 · · · D n ) can be computed tropically on the quasiembedded, balanced rational conical complex Π 1 as If we choose an Euclidean norm in N Π 1 R and denote by " "the map between lattice Minkowski cycles and Euclidean Minkowski cycles, then the algebraic top intersection number can also be computed as Proof. The pair (X ′ , B ′ ) exists thanks to resolution of singularities [Hir64], Proposition 3.25 and Theorem 3.31. By Theorem 4.6 and the functoriality of the intersection product, we get proving the first statement. Moreover, by Proposition 3.11 and Remark 3.10, we get φ π * D 1 ⊙ · · · ⊙ φ π * D n ⊙ [Π 1 ] = φ π * D 1 · · · · · φ π * D n · [Π 1 ], which proves the second one. Since the map " trop " between algebraic cycles on X and Minkowski weights on Π is neither surjective nor injective, the positivity notions in the algebraic and tropical worlds do not correspond exactly. Recall that N k (X) denotes the space of R-cycles of codimension k up to numerical equivalence. The cone Peff n−k (X) = Peff k (X) of pseudo-effective cycles is the closure of the cone generated by classes of effective cycles. The dual cone is the cone of numerically effective cycles The space of nef toroidal Cartier divisors is denoted by Div + (X, U).
The first relation between positivity in the algebraic and the tropical worlds is the following.
(2) If D ∈ Div + (X, U) is a nef toroidal Cartier divisor then φ D is weakly concave in the sense of [BBS,Definition 4.6].
Proof. For any cone σ ∈ Π(k), because α is nef and S σ is an effective cycle. The second statement follows from the first and the equality We next see several examples that show that the above lemma is almost all we can expect.
Example 4.9. Let X be the blow up of P 2 at a point. Let B be a snc divisor such that the exceptional divisor E is contained in the support of B and such that the associated complex is quasi-embedded. Then [E] is an effective cycle but trop([E]) is not positive. So the statement (1) of the above lemma can not be extended to pseudo-effective cycles.
This easily implies that the complex associated to the toroidal embedding X \ B ֒→ X is quasi-embedded. Let E denote again the exceptional divisor. Then trop([E]) 0 because E is not contained in B. Nevertheless E is not nef. Therefore the converse of statement (1) is not true. Put D = ℓ 1 − r 1 . Since D ∼ E, we see that φ D is weakly concave but D is not nef. Therefore the converse of statement (2) does not hold.
Example 4.11. We put ourselves in the situation of Example 4.10 and let B ′ = ℓ 1 ∪ ℓ 2 ∪ ℓ 3 . Then the obtained conical complex is still quasi-embedded, but the map trop satisfies trop(±[E]) = 0. Therefore trop(α) 0 does not even imply that α is effective. Therefore the conical complex associated to X \ B ֒→ X consists of three rays τ O , τ P and τ Q with lattice generators v O , v P and v Q . The lattice N Π X is one dimensional and can be identified with Z. Then the quasi-embedding is given by For simplicity a Minkowski weight c ∈ M 1 (Π X ) will be denoted as a triple of real numbers (c O , c P , c Q ). The balancing condition [X] is the Minkowski weight (1, 1, 1). The Minkowski weight (1, 2, 0) is also positive but it does not come from the geometry of X. Consider the divisor D = −P + 2Q. This is a nef divisor because it has positive degree. Nevertheless Hence it is not true that nef divisors give rise to concave functions in the sense of [BBS,Definition 4.6].
We end this section showing that algebraic nef Cartier divisors in allowable modifications of X provide an admissible family of nef Cartier b-divisors on Π. Proof. We have to show that C satisfies the three properties given in Definition 2.3. To show property (1), let D 1 , . . . , D r be nef Cartier toroidal b-divisors on the allowable modifications X Π 1 , . . . , X Π r , respectively. Let Π ′ be a smooth common refinement of Π 1 , . . . , Π r and denote by π i : X Π ′ → X Π i the corresponding toroidal modifications. Then, by Theorem 4.6 and Proposition 3.11, we get ) 0, where the last inequality uses the fact that the pullback of a nef divisor under a proper map is nef and Kleiman's criterion for nefness.
Property (2) is clear. To prove (3) we first recall that the set of piecewise linear functions on |Π| with rational slopes is dense in the set of continuous conical functions on |Π| with the topology of uniform convergence on compacts. Thus it is enough to show that any piecewise linear function on |Π| with rational slopes belongs to C − C.
A piecewise linear function φ with rational slopes on |Π| defines a Cartier Q-divisor D φ on an allowable modification X ′ of X. Since we are assuming that there is an ample divisor on X whose support is contained in B there is an allowable modification π : X ′′ → X ′ and an ample toroidal divisor A on X ′′ . We can choose an integer r > 0 such that C = π * D φ + rA is also ample. Therefore completing the proof.

Toroidal b-divisors.
We give the definition of the toroidal Riemann-Zariski space X U of (X, U) and define Cartier and Weil toroidal b-divisors on X U . Extending the bijection between toroidal divisors on X and Cartier divisors on Π (Proposition 4.3), we give a bijection between Cartier (respectively Weil) toroidal b-divisors and Cartier (respectively Weil) b-divisors on Π. Moreover, extending the results in Section 4.1 we show that the top intersection product of Cartier toroidal b-divisors can be computed tropically on the rational conical complex.
Consider the directed set R sm (Π) of smooth subdivisions of Π. For every smooth subdivision Π ′ ∈ R sm (Π), we denote by X Π ′ the corresponding smooth proper allowable modification from Theorem 3.31. The toroidal Riemann-Zariski space of (X, U) is defined formally as the inverse limit with maps given by the proper toroidal birational morphisms X Π ′′ → X Π ′ defined whenever Π ′′ Π ′ . Toroidal b-divisors can be viewed as divisors on X U : Definition 4.14. The group of R-Cartier toroidal b-divisors on X is defined to be the injective limit with maps given by the pull-back map of R-toroidal divisors. It is endowed with its inductive limit topology, called the strong topology.
The group of R-Weil toroidal b-divisors on X is defined to be the projective limit with maps given by the push-forward map of R-toroidal divisors (note that the X Π ′ 's are smooth, hence we can identify Cartier and Weil divisors). It is endowed with its projective limit topology, called the weak topology.
We will write R-Cartier and R-Weil toroidal b-divisors in bold notation D D D to distinguish them from classical R-toroidal divisors D.
We make the following remarks.
(1) As before, to simplify notation, we will usually omit the coefficient ring R from the notation (real coefficients being always implicit) and refer to Rtoroidal b-divisors simply as toroidal b-divisors.
(2) Since the set of smooth subdivisions is directed, a toroidal Cartier b-divisor can be represented by a pair (X Π ′ , D), where X Π ′ is the allowable modification of X given by the refinement Π ′ . Two pairs represent the same Cartier b-divisor if there is a common refinement and the pull back of both divisors to the corresponding modification agree.
(3) We can view a Weil toroidal b-divisor as a family where for each Π ′ ∈ R sm (Π), we have that D Π ′ ∈ Div(X Π ′ , U), and these elements are compatible under push-forward.
(4) We can view a Cartier toroidal b-divisor as a Weil toroidal b-divisor for which there is a model X Π for some Π ∈ R sm (Π) such that for every other model Hence, we have the inclusion and we may refer to a Weil toroidal b-divisor just as a toroidal b-divisor. (5) A net (Z Z Z i ) i∈I converges to a b-divisor Z Z Z in W-b-Div(X U ) if and only if for each Π ′ ∈ R sm (Π) we have that (Z i,Π ′ ) i∈I converges to Z Π ′ coefficient-wise.
between toroidal b-divisors on X and b-divisors on Π. Thus, the space W-b-Div(X U ) is homeomorphic to the space of conical functions Π(Q) → R with the topology of pointwise convergence.
Proof. This follows from the definition of Weil b-divisors on a rational conical complex and Proposition 4.3.
As in the case of functions, one can define the top intersection product of a collection of toroidal b-divisors when there is at most one Weil b-divisor involved (all the other must be Cartier).
Definition 4.17. Let D D D 1 , . . . , D D D n be toroidal b-divisors on X and assume that at most one of the D D D i 's is not Cartier. Without lost of generality, assume that D D D 1 is not Cartier. Let Π ′ ∈ R sm (Π X ) be a subdivision such that all of the D D D i 's for i = 2, . . . , n are determined on X Π ′ . The top intersection product D D D 1 · · · D D D n is defined by D D D 1 · · · D D D n := deg(D 1,Π ′ · D 2,Π ′ · · · D n,Π ′ ).
By the projection formula in algebraic geometry, this is independent of the choice of the common refinement Π ′ .
Remark 4.18. It follows from Corollary 4.7 that the top intersection product of a collection of toroidal b-divisors where at most one of them is Weil can be computed tropically.

Top intersection product of nef toroidal Weil b-divisors.
Let X, U, B and Π be as at the beginning of section 4. Chose a Euclidean metric on N Π R and denote Π the induced Euclidean conical complex.
We start by defining nef Cartier and nef Weil toroidal b-divisors. Using Lemma 4.13 we deduce that the set of Cartier b-divisors on Π which are induced from nef Cartier toroidal b-divisors forms an admissible family C of nef Cartier b-divisors in the sense of Definition 2.3. Using the results of Section 2, we deduce that the top intersection product of Cartier b-divisors can be extended continuously to nef Weil toroidal b-divisors and, by linearity, to differences of nef Weil toroidal b-divisors.
is nef for some (hence any) determination E Π ′ of E E E. The set of nef toroidal Cartier b-divisors forms a cone in Ca-b-Div(X U ), denoted by Ca-b-Div + (X U ). The cone of nef toroidal Weil b-divisors is the closure in W-b-Div(X U ) of Ca-b-Div + (X U ). It is denoted by W-b-Div + (X U ).
Remark 4.20. By Proposition 4.3 and Remark 3.4 the inclusion between Cartier and Weil b-divisors Ca-b-Div(X) → W-b-Div(X) can be factored as By Lemma 4.13 the image of Ca-b-Div + (X U ) in Ca-b-Div Π forms an admissible family of nef b-divisors. By Theorem 2.9 the elements in the closure of Ca-b-Div + (X U ) in W-b-Div Π are continuous functions. Since a continuous function is determined by its values on a dense subset and the topologies on W-b-Div Π and W-b-Div(Π) are both that of pointwise convergence, we deduce that the closure of Ca-b-Div + (X U ) in W-b-Div Π is naturally homeomorphic to its closure in W-b-Div(Π). The last one can be identified with the cone W-b-Div + (X). As a consequence, in order to work with nef toroidal Weil b-divisors, there is no difference between working on Π or in Π.

Remark 4.21.
A consequence of the preceding Remark and Theorem 2.9 is that the closure of Ca-b-Div + (X U ) in W-b-Div(X U ) agrees with its sequential closure. It follows that if E E E ∈ W-b-Div + (X U ), then there is a sequence (D D D k ) k∈N of nef toroidal Cartier b-divisors converging to E E E. Moreover, when we view E E E and the D D D k as functions on |Π|, the convergence is uniform on compacts.
From now on we fix C = image of Ca-b-Div + (X U ) in Ca-b-Div Π as the admissible family of nef Cartier b-divisors.
The following theorem describes nef toroidal b-divisors combinatorially. In view of Remark 4.20, it is a direct consequence of Theorem 2.9. Since we can view Cartier b-divisors as Weil b-divisor, there is a potential ambiguity when we say that a Cartier b-divisor is nef. Lemma 4.24 below shows that this potential ambiguity is not a real ambiguity. Before stating it we make the following remark.
Remark 4.23. Let Π ′′ ∈ R sm (Π) and let D be a nef divisor on X Π ′′ . Then for any Π ′ Π ′′ in R sm (Π) we have that D π * π * D, where π : X Π ′′ → X Π ′ denotes the corresponding proper birational morphism. Indeed, the divisor D − π * π * D is π-nef (i.e. has non-negative intersection with every curve contracted by π) and is π-exceptional. Hence, from the wellknown Negativity Lemma (see e.g. [KM08, Lemma 3.39]), we have that D − π * π * D 0 and thus D π * π * D. Proof. Let Π ′ ∈ R sm (Π) be a determination of D D D. We have to show that D Π ′ is nef on X Π ′ . For this, let C ⊆ X Π ′ be an irreducible curve. It suffices to show that the intersection product D Π ′ · C is non-negative.
Let B ′ i i ∈ I ′ be the irreducible components of the boundary divisor B ′ = X Π ′ \ U and for any subset J ′ ⊆ I ′ , denote by B J ′ the boundary intersection j∈J ′ B j (in particular, B ∅ = X Π ′ ). Let K ′ ⊆ I ′ such that B K ′ is the minimal boundary intersection containing C.
If codim (B K ′ ) 2, we can find a subdivision Π ′′ Π ′ in R sm (Π) and a curveC ⊆ X Π ′′ such that the following two conditions are satisfied: (1) π * C = aC for some natural number a > 0.
Let {D D D i } i∈N be a sequence of nef toroidal Cartier b-divisors converging to D D D. We view them as toroidal Weil b-divisors. In particular, on X Π ′ , we have that component-wise, and by continuity of the intersection product, Now, for each i ∈ N, let Π i ∈ R sm (Π) be a determination of D D D i . We may assume that Π i Π ′ . Also, we let π i : X Π i → X Π ′ denote the corresponding proper birational morphism. Let C i be the strict transform of the curve C under π i . Note that this is well defined by the assumption that the minimal boundary intersection that contains C has codimension less or equal than one.
Using the projection formula, we compute Indeed, the first summand is non-negative since it follows from Remark 4.23 that both the terms π * i π i * D i,Π i − D i,Π i and C i are effective and intersect properly. The second summand is non-negative since D i,Π i is nef and C i is effective.
By (4.2), D Π ′ · C is a limit of non-negative real numbers. Hence it is itself non-negative. This concludes the proof.
The next is the main result of this paper.  One has to be careful that the continuity condition (4.3) is only true when the sequences (D D D j,k ) k∈N consist of nef toroidal Cartier b-divisors. Namely, one can construct sequences of toroidal Cartier b-divisors (D D D ′ j,k ) k∈N such that, for each j, the sequence of functions (φ D D D j,k ) k∈N converges uniformly on compacts to φ D D D j and nevertheless the continuity condition (4.3) does not hold.
Remark 4.27. Since the intersection product is multilinear (for the semigroup law of W-b-Div + (X U )), it can be extended by multilinearity to the space W-b-Div + (X U ) − W-b-Div + (X U ) of toroidal Weil b-divisors that are differences of nef ones.

A
Let U ֒→ X be a toroidal embedding as in the beginning of Section 4. That is, we assume that X is smooth and projective, B = X \ U is a snc divisor, that there is an effective ample divisor with support B and that the corresponding smooth rational conical complex Π is quasi-embedded. We also assume that we have chosen an auxiliary Euclidean metric on N Π . A toroidal b-divisor on X is big if it has enough global sections (Definition 5.10). In this section, as an application of our results, we show a Hilbert-Samuel type formula for nef and big toroidal b-divisors on X relating the degree of a nef toroidal b-divisor both with the volume of the b-divisor and with the volume of the associated convex Okounkov body (Definitions 5.3 and 5.6 and Theorem 5.15). As a corollary, we obtain a Brunn-Minkowski type inequality (Corollary 5.17).

Volumes and convex Okounkov bodies of toroidal b-divisors.
We start with the definition of the space of global sections of a toroidal b-divisor.
and where ⌊D D D⌋ denotes the integral divisor whose coefficients are given by the biggest integer numbers which are less than or equal to the corresponding coefficients of D D D.

Remark 5.2.
(1) By definition, we have that H 0 (X U , D D D) is an intersection of finite-dimensional vector spaces H 0 (X Π ′ , D Π ′ ) .
(2) We have a well defined map We associate a convex Okounkov body to a toroidal b-divisor. This is a graded sub-k-algebra of F[t].
One of the fundamental problems of algebraic geometry is the question about finite generation of divisorial algebras. It is clear that, in general, the b-divisorial algebra associated to a toroidal b-divisor is not finitely generated. However, the next proposition shows that it satisfies the weaker condition of being of almost integral type, which nevertheless, following [KK12], allows us to associate a convex Okounkov body to it.
Recall that a graded subalgebra R ⊆ F[t] is of integral type if it is a finitely generated kalgebra and is a finite module over the algebra generated by R 1 , while it is of almost integral Since the pullback of a nef divisor is again nef, D Π ′′′ = γ * D Π is nef.
By Remark 4.23, D Π ′′′ β * β * D Π ′′′ and we conclude that In the general case, choose a sequence {D D D i } i∈N of nef Cartier b-divisors converging to D D D. Then, by what was shown above, for each i ∈ N we have that Hence, taking limits at both sides we deduce as we wanted to show.
The following is a monotone approximation lemma for nef toroidal b-divisors. Recall that we are assuming that there is an effective ample divisor A whose support is B. The conditions on A imply that the function φ A | S Π is strictly negative.  Moreover, we have Hence, we get that This concludes the proof. Now, recall that a divisor on an algebraic variety is said to be big if it has strictly positive volume. We define big toroidal b-divisors analogously. Remark 5.11. If a toroidal b-divisor D D D = (D Π ′ ) Π ′ ∈R sm (Π X ) is big and nef, then by the monotonicity property of Lemma 5.8, it follows that D Π ′ is big for all Π ′ ∈ R sm (Π).
For any algebraic variety Y we denote by N 1 (Y) R = N 1 (Y) the real vector space of numerical equivalence classes of real divisors on Y.
Definition 5.12. We define the space of numerical equivalence classes of toroidal b-divisors on X as the inverse limit b-N 1 (X U ) : with maps given by the proper push-forward map f * : X Π ′′ → X Π ′ of numerical classes of divisors whenever Π ′′ Π ′ in R sm (Π X ).
Let b-Big(X U ) and b-Nef(X U ) be the cones of big, respectively nef toroidal b-divisors in b-N 1 (X U ).
Remark 5.13. Since N 1 (X) is finite dimensional, after enlarging B if needed, we can assume that the map Div(X, U) → N 1 (X) is surjective. If this is the case, then the map is surjective.
From now on we assume that B has been enlarged in such a way that the map (5.2) is surjective. We have the following lemma. Proof. Since for each Π ′ ∈ R sm (Π), the volume function vol is continuous on the big cone Big(X Π ′ ) ⊆ N 1 (X Π ′ ) (see e.g. where the third equality follows since both the sequence indexed by Π ′ ∈ R(Π X ) and the one indexed by i ∈ N are decreasing, hence taking limits is the same as taking the infimum. This concludes the proof.
As a consequence, we get the following Hilbert-Samuel type formula. Proof. Recall that we are assuming that the conditions at the beginning of Section 4 are satisfied and that the map (5.2) is surjective. By Theorem 5.15 the volume function agrees with the degree function on big and nef divisors. This last function descends to numerical equivalence classes and, by Theorem 4.25, it is continuous.
As a corollary we obtain the following Brunn-Minkowski type inequality.
Corollary 5.17. Let D D D and F F F be two nef and big toroidal b-divisors. Then the following Brunn-Minkowski type inequality holds true.
Proof. Consider the associated Okounkov bodies ∆ D D D and ∆ F F F , respectively. Then, using Theorem 5.15, the inequality follows from a standard result in convex geometry about volumes of convex sets (see e.g. [Sch93]).