Lefschetz (1,1)-theorem in tropical geometry

For a tropical manifold of dimension n we show that the tropical homology classes of degree (n-1, n-1) which arise as fundamental classes of tropical cycles are precisely those in the kernel of the eigenwave map. To prove this we establish a tropical version of the Lefschetz (1, 1)-theorem for rational polyhedral spaces that relates tropical line bundles to the kernel of the wave homomorphism on cohomology. Our result for tropical manifolds then follows by combining this with Poincar\'e duality for integral tropical homology.


Introduction
The classical Lefschetz (1,1)-theorem characterises the cohomology classes of complex projective varieties which arise as Chern classes of complex line bundles. The theorem asserts that these classes are precisely the integral classes in the (1,1)-part of the Hodge decomposition. It implies the Hodge conjecture (over Z) for the degree 2 cohomology classes of a complex projective variety. In this paper we establish analogous results for rational polyhedral and tropical spaces.
Tropical homology in the sense of Itenberg, Mikhalkin, Katzarkov, and Zharkov was introduced as an invariant of tropical varieties capable of providing Hodge theoretic information about complex projective varieties via their tropicalisations [IKMZ16]. Tropical homology groups with coefficients in a ring Q can be defined for any rational polyhedral space X, see Definition 2.1. The tropical homology groups with Q coefficients of a rational polyhedral space X are denoted by H p,q (X, Q). We also consider the tropical Borel-Moore homology groups, which are denoted by H BM p,q (X, Q). The corresponding Borel-Moore and usual tropical homology groups agree when X is compact.
A tropical cycle, synonymously a tropical space, is a rational polyhedral space that satisfies a balancing condition which is ubiquitous in tropical geometry, see Definition 4.12. Tropical cycles are the candidates for tropicalisations of classical algebraic cycles. To a tropical cycle Z of dimension k in a rational polyhedral space X, we can associate a tropical homology class which we call the fundamental class and denote by [Z] ∈ H BM k,k (X, Z). Tropical manifolds are tropical spaces which are locally modelled on matroidal fans, see Definition 5.1. In this paper, we determine exactly which tropical homology classes in H BM n−1,n−1 (X, Z) of a tropical manifold X of dimension n arise from codimension one tropical cycles. In order to characterise these tropical homology classes, we make use of the wave homomorphism φ : H BM p,q (X, Z) → H BM p+1,q−1 (X, R), introduced by Mikhalkin and Zharkov [MZ14], which is defined for any rational polyhedral space X. When X arises as the tropicalisation of a family of complex projective varieties and satisfies some additional assumptions, then the wave homomorphism is related to the monodromy operator on the mixed Hodge structure of the family [MZ14, Section 7]. Liu constructed an analogous operator on tropical Dolbeault cohomology of non-archimedean analytic spaces, which he relates to the monodromy operator in the weight spectral sequence [Liu17]. It was pointed out by Mikhalkin and Zharkov that the fundamental class of a tropical cycle in X is in the kernel ofφ. To prove Theorem 1.1, we first establish for rational polyhedral spaces, an analogue of the line bundle version of the Lefschetz (1, 1)-theorem. To do so, we consider the sheaf Aff Z of integral affine functions. These functions play the role of invertible regular functions in tropical geometry. We also consider tropical cohomology groups H p,q (X, Q), which are the cohomology groups of a sheaf F p Q on X. The tropical Picard group of X is defined to be Pic(X) := H 1 (X, Aff Z ) and there is a Chern class map c 1 : Pic(X) → H 1,1 (X, Z), see Definition 3.6. These notions in tropical geometry have also appeared in the context of curves [MZ08] and tropical complexes [Car13,Car15]. Definition 2.9 also describes the wave homomorphism on tropical cohomology namely, φ : H p,q (X, Z) → H p−1,q+1 (X, R).
Theorem 1.2. Let X be a rational polyhedral space with polyhedral structure, then the image of c 1 : Pic(X) → H 1,1 (X, Z) is equal to the kernel of the wave homomorphism φ : H 1,1 (X, Z) → H 0,2 (X, R).
For p = 0, the sheaf F 0 R is the constant sheaf R, so we can identify H 2 (X, R) and H 0,2 (X, R). In Proposition 3.5, we show that the boundary map δ : H 1,q (X, Z) → H q+1 (X, R) coincides up to sign with the wave homomorphism. For q = 1, this implies Theorem 1.2.
When X is an abstract tropical space of dimension n, the cap product with its fundamental class provides a map ∩[X] : H p,q (X, Z) → H BM n−p,n−q (X, Z). (1.1) This allows us to describe the kernel of the wave homomorphism on homology groups. Theorem 1.3. Let X be a tropical space of dimension n. α ∈ H 1,1 (X; Z) is such that φ(α) = 0, then α ∩ [X] ∈ H BM n−1,n−1 (X, Z) is the fundamental class of a codimension one tropical cycle in X.
To prove Theorem 1.3 we first show that any element L ∈ Pic(X) has a rational section in the sense of Definition 4.2. A tropical Cartier divisor is a tropical line bundle L ∈ Pic(X) together with a section s. We can then define a map div : CaDiv(X) → Z n−1 (X), where CaDiv(X) is the group of Cartier divisors on X and Z n−1 (X) is the group of dimension one tropical cycles in X. We then show that the map given by capping with the fundamental class (1.1) is an isomorphism when X is a tropical manifold. This extends the version of Poincaré duality with real coefficients of Smacka and the first and third authors [JSS15,Theorem 2]. Combining this statement with Theorem 1.3, we are able to prove Theorem 1.1.
The last section presents corollaries and examples of our main theorems. In particular, we consider tropical abelian surfaces and Klein bottles with a tropical structure. We also calculate the wave map for two combinatorial types of smooth tropical quartic surfaces. The Picard rank of a polyhedral space X is defined to be the rank of Pic(X). We prove the following statement regarding the Picard ranks of smooth tropical quartic surfaces.

Preliminaries
We set T = [−∞, ∞) and equip this set with the topology whose basis consists of the intervals [−∞, b) and (a, b) for a, b −∞. We equip T r with the product topology. The set T r is a stratified space. For a subset

2.A. Abstract polyhedral spaces and tropical varieties
A rational polyhedron in R r is a subset defined by a finite system of affine (non-strict) inequalities w i , v ≥ c i with c i ∈ R and w i ∈ Z r . A face of a polyhedron σ is a polyhedron which is obtained by turning some of the defining inequalities of σ into equalities.
A rational polyhedron in T r is the closure of a rational polyhedron in R r . A face of a polyhedron σ in T r is the closure of a face of σ ∩ R J for some J ⊂ [r]. A rational polyhedral complex C in T r is a finite set of polyhedra in T r , satisfying the following properties: (1) For a polyhedron σ ∈ C, if τ is a face of σ (denoted τ ≺ σ ) we have τ ∈ C.
The maximal polyhedra, with respect to inclusion, are called facets. If all facets of C have the same dimension n, we say C is of pure dimension n. The support of a polyhedral complex C is the union of all its polyhedra and is denoted |C|. If X = |C|, then X is called a rational polyhedral subspace of T r and C is called a rational polyhedral structure on X.
The relative interior of a polyhedron σ in T r , denoted relint(σ ), is defined to be the set obtained after removing all of the proper faces of σ . Given a polyhedral complex C in T r , for σ ∈ C, the closed star of σ is St(σ ) := {τ ∈ C | ∃σ ∈ C such that τ, σ ⊂ σ }. The open star St(σ ) of σ is the open set which is the relative interior of the support of St(σ ). Also, let C I denote the union of polyhedra σ ∈ C for which relint(σ ) ⊂ R r I . For a rational polyhedron σ in T r , we denote σ ∩ R r I by σ I . A map f : M → N , where M ⊂ T m and N ⊂ T n , is an extended affine Z-linear map if it is continuous and there exist A ∈ Mat(n × m, Z), b ∈ R n such that f (x) = Ax + b for all x ∈ R m . Definition 2.1. A rational polyhedral space X is a paracompact, second countable Hausdorff topological space with an atlas of charts (ϕ α : U α → Ω α ⊂ X α ) α∈A such that: (1) The U α are open subsets of X, the Ω α are open subsets of rational polyhedral subspaces X α ⊂ T r α , and the maps ϕ α : U α → Ω α are homeomorphisms for all α; (2) for all α, β ∈ A the transition map are extended affine Z-linear maps.
Definition 2.2. Let X be a rational polyhedral space. A rational polyhedral structure on X is a finite family of closed subsets C such that the following conditions hold: (1) X = σ ∈C σ ; (2) for each σ there exists a chart ϕ σ :

2.B. Multi-(co)tangent (co)sheaves
Let C be a rational polyhedral complex in T r . For a face σ ∈ C I , denote by L(σ ) ⊂ R r I the subspace generated by the vectors in R r I tangent to σ . Set L Z (σ ) = L(σ ) ∩ Z r I ⊂ Z r I .
Definition 2.3. For σ ∈ C I , the p-th integral multi-tangent and integral multi-cotangent space of C at σ are the Z-modules respectively. If τ is a face of σ there are natural maps When Q = R we drop the use of the sup-and sub-scripts on F p (σ ) and F p (σ ), respectively.
From the Z-modules F p Z (σ ), it is possible to construct a sheaf on |C| ⊂ T r following [MZ14, Section 2.3]. For each open set Ω ⊂ |C|, consider the poset P (Ω) whose elements are the connected components σ of faces of C intersecting with Ω. The elements of P (Ω) are ordered by inclusion and if τ ≺ σ recall there are maps ρ τ,σ :

Definition 2.4. ([MZ14]) Let C be a rational polyhedral complex of T r . For an open set Ω ⊂ |C| define the vector space
The sheaves F p Q are constructible and do not depend on the polyhedral structure C but only on the support |C|. For a polyhedral space X, the sheaves F p are defined by gluing along charts. In fact, this definition does not require a polyhedral structure on X, see [JSS15].

2.C. Tropical (co)homology
In the following we always assume that X is a rational polyhedral space which admits a rational polyhedral structure C. In this case, the Z-modules F Z p (σ ) and F p Z (σ ) and the maps ι τ,σ , ρ τ,σ are well-defined for any τ ≺ σ ∈ C.
We let ∆ q denote an abstract q-dimensional simplex. Again Q will be a ring satisfying Z ⊂ Q ⊂ R.
Definition 2.6. The groups of tropical (p, q)-chains and cochains with respect to C and with Q-coefficients are respectively, (2.1) The complexes of tropical (p, •)-chains and cochains are respectively, where the ∂ and d are the usual singular differentials composed, when necessary, with ι τ,σ and ρ τ,σ respectively. The tropical homology and tropical cohomology groups with coefficients in Q are respectively, Definition 2.7. The tropical Borel-Moore chain groups C BM p,q (X, Q) consist of formal infinite sums of elements of C p,q (X, Q) with the condition that locally only finitely many simplices have non-zero coefficients.
The tropical Borel-Moore homology groups are denoted H BM p,q (X, Q). They are the homology groups of the complex (C BM p,• (X, Q), ∂).
Remark 2.8. For computations, we will often use the simplicial version of the tropical (co)homology groups defined above. It is possible to construct a locally finite simplicial structure D on X such that all simplices are C-stratified, see [MZ14, Section 2.2]. We call such a structure a C-stratified simplicial structure. Following the standard conventions for simplicial (co)homology, we obtain simplicial tropical homology and cohomology groups, as well as the Borel-Moore variants respectively. The equivalence of singular and simplicial homology with F Q p -coefficients is proved in [MZ14, Section 2.2]. The argument uses cellular homology as an intermediate step, which is introduced in Section 5 in this work. The argument, which uses barycentric subdivisions, still applies to our case thanks to the fact that D is a C-stratified simplicial structure. The discussion in [MZ14] is restricted to standard homology, but the arguments can be extended to the Borel-Moore case after noting that the cellular homotopy argument still applies and that the cellular chain complex can still be described in terms of relative singular homology Here, X q denotes the support of the q-skeleton of C and H p,q (X q , X q−1 ; Q) denotes relative homology. We use the identification of singular and simplicial homology throughout the rest of the text. We also use the same notation to denote both variants of the tropical (co)homology groups.

Tropical exponential sequence
Here we will prove Theorem 1.2 using the tropical exponential sequence (3.1). Throughout this section X is a rational polyhedral space with a rational polyhedral structure C.
Definition 3.1. The sheaf of real valued functions on X which are affine with integral slope in each chart is denoted by Aff Z .
Definition 3.2. Let x be a point in a polyhedral space X and ϕ : U → T r a chart such that x ∈ U and sed(ϕ(x)) ∅. Then v ∈ R r is a divisorial direction at x if there exists an x 0 ∈ U with sed ϕ(x 0 ) = ∅ such that for all t < 0 we have x t = ϕ(x 0 ) + tv ∈ ϕ(U ) and lim t→∞ ϕ(x 0 ) + tv = x.
Note that any affine function f ∈ Aff(U ) is constant along the divisorial directions to any x ∈ U since the value f (x) is a real number. Taking the differential of a real valued function provides a surjective map d : Aff Z → F 1 Z . The kernel is the sheaf of locally constant real functions R. The tropical exponential sequence is After passing to the long exact sequence in cohomology for all q there is the coboundary map, Recall that F 0 is the constant sheaf R for all X, therefore we identify H q (X, R) and H 0,q (X, R). The open set U is contractible, thus acyclic for R. Furthermore, the contraction can be chosen so that it respects the simplicial structure on U . Following the arguments in the proof of [JSS15, Proposition 3.11], we see that U is acyclic for F 1 Z . The long exact sequence associated to (3.1) implies that U is acyclic for Aff Z as well. The intersection of stars of vertices is the star of the minimal face containing these vertices. Therefore, all intersections of the cover are acyclic and U is a Leray cover of X.
Remark 3.4. Let U be the open cover given by stars of vertices of a C-stratified simplicial structure D on X. Then there is a canonical isomorphism between the tropical simplicial cohomology groups with respect to D and theČech cohomology of the sheaves F p Q with respect to the cover U . TheČech chain group Also the differential maps in both cases agree. We also use this identification of simplicial andČech cohomology groups throughout the following sections without using different notations.
Proof. Let D be a stratified simplicial structure on X. Let D q denote the simplicies of D of dimension q. Write [i 0 , . . . , i q ] for the q-simplex with vertices i 0 , . . . , i q ∈ D with the orientation induced by the ordering of the vertices. For a q-simplex We will compare the coboundary and the eigenwave maps usingČech cochains with respect to the cover Since the functions f i 0 ...i q are integer affine and the vertex i q has minimal sedentarity among all of i 0 . . . i q , each function f i 0 ...i q extends uniquely by continuity to the vertex i q . We normalise our choices in such a way that is a constant function. To compute this constant, we evaluate at i q+1 and find because of our normalisation. Note that if i q+1 has strictly lower sedentarity than i q , then f i 0 ...i q is constant when moving along the divisorial direction at i q+1 towards i q . Let π τ,σ be the projection (along the divisorial direction) between strata containing the relative interior of two faces τ and σ . In particular, if the relative interiors of τ and σ are contained in the same strata this map is the identity. Then, whether or not i q and i q+1 have the same sedentarity, we have , where τ and σ are the faces containing [i 0 , . . . , i q ] and [i 0 , . . . , i q+1 ], respectively. Therefore, Definition 3.6. The tropical Picard group is Pic(X) := H 1 (X, Aff Z ). The map from Pic(X) to H 1,1 (X, Z) provided by the tropical exponential sequence is called the Chern class map and is denoted by c 1 : Pic(X) → H 1,1 (X, Z).
Proof of Theorem 1.2. The kernel of the boundary map δ is Pic(X) = H 1 (X, Aff Z ) by the long exact sequence associated to (3.1) and by Propostion 3.5 this is also the kernel of the eigenwave homomorphism. This completes the proof.
Remark 3.7. There is also a version of Sequence (3.1) with real coefficients namely, where Aff denote the sheaf of functions which are affine in each chart, not necessarily with integral slopes. By the same argument the boundary map of the long exact sequence is equal to the eigenwave map extended to cohomology with R-coefficients φ : H 1,q (X, R) → H 0,q+1 (X, R). Mikhalkin and Zharkov conjecture that is surjective for all q. By the long exact sequence derived from the short exact sequence in (3.2) this happens if and only if H 0,q (X, R) → H q (X, Aff) is zero for all q ≥ 1, which would in turn imply that for all q the following sequence is exact A conjecture similar to the one of Mikhalkin and Zharkov was made by Liu [Liu17] for tropical Dolbeault cohomology, which is a cohomology theory of non-archimedean analytic spaces defined using superforms in the sense of Lagerberg [Lag12]. We refer the reader to [CLD12,Gub16] for the construction of these forms on analytic spaces and to [JSS15] for the relation between the cohomology of superforms and the tropical cohomology groups considered here.

Tropical cycle class map
In this section we prove Theorem 1.3. To do this, we first prove the existence of sections of tropical line bundles, and that the construction of the divisor of a section is compatible with the Chern class map combined with capping with the fundamental class.

4.A. Tropical line bundles and sections
Throughout this section X is a rational polyhedral space with polyhedral structure C. The map U → M(U ) defines a sheaf on X. We consider the short exact sequence of sheaves Upon taking the long exact sequence in cohomology we obtain a map δ : H 0 (X, M/ Aff) → H 1 (X, Aff).
Recall that Pic(X) = H 1 (X, Aff) Let us assume that L can be represented by transition functions (f ij ) with respect to the open covering U = (U i ). Then a section s of L is equivalent to a collection of tropical rational functions s i ∈ M(U i ) which satisfies s i − s j = f ij for all i j. We use the notation CaDiv(X) = H 0 (X, M/ Aff) and call an element s ∈ CaDiv(X) a Cartier divisor of X.
In the remainder of this section we establish the existence of a section of a tropical line bundle on a polyhedral space. A version of this statement first appeared in the thesis of Torchiani in the case when X has no points of sedentarity [Tor10, Theorem 2.3.4]. We start with the following lemmas. Proof. We start with the case where σ ⊂ R r , so that σ does not contain points of higher sedentarity. We can assume without loss of generality that σ is of dimension r. For a codimension one face τ of σ , let H τ denote the hyperplane in R r containing τ. We can construct a rational function h τ : H τ → R which restricts to s on the face τ. To do this, notice that each point in H τ can be uniquely written as x + v where x ∈ δ ≺ τ and v lies in the normal cone of the face δ in the polyhedron τ with respect to the standard scalar product in R r . Then For each codimension one face τ of σ , choose a vector v σ ,τ ∈ Z r pointing from τ to σ such that We will show that for each τ there exists satisfies h| ∂σ = s, as required.
To find m τ for a fixed τ, we proceed as follows. Let D ⊂ H τ be a domain of linearity of h τ and δ ⊂ ∂σ be a domain of linearity of s. We will show that there exists an m such that Since there are only finitely many pairs D, δ to check we can find the desired m τ .
and take v 1 , . . . , v l to be generators of this cone. Notice that the differentials (df m,τ ) x (v i ) and ds y (v i ) are constant over all x ∈ π −1 τ (D) and all y ∈ δ. Then choose an m satisfying , and all y ∈ δ. Such a choice of m is possible since the left hand side can be made arbitrarily small except for when v i lies in the lineality space of cone(D, δ). In this case, both sides agree since f τ and s agree on D ∩ δ. By linearity it follows that Then by choosing such an m, it follows that for all . Now suppose that σ is a polyhedron in T r . We proceed by induction, with the base case being when all points in σ are of sedentarity zero. In this case, the above argument applies. Now asume that the statement There exists a constant c k and a function s k : where H − k is the closed half-space defined by x, e k ≤ c k . Let H k be the hyperplane defined by x, e k = c k and H + k be the closed half-space defined by x, e k ≥ c k . Now σ ∩ H + k is a rational polyhedron in T r which contains no points of sedentarity {k}. We define a rational function Hence, the rational functions s and s − k induce a function s : ∂(σ ∩ H + k ) → R. By the induction assumption there exists a rational function s + k on σ ∩ H + k which extends s . Then the function s : σ → R given by is a well-defined rational function that extends s, as required. Proof. Note that since s is rational function that is defined on an open neighborhood of K it extends to a compact polyhedral subset K 1 that satisfies K ⊂K 1 . Let K 2 be a compact polyhedral subset such that K 1 ⊂K 2 . We can assume that K 2 is contained in the same open star as K. We fix a polyhedral structure D on K 2 such that K 1 is the support of a polyhedral subcomplex. We construct s inductively on the skeleta of D. Assume that an extension of s is defined on the k-skeleton of D. Let σ be a k + 1 polyhedron in D.
If σ is contained in the boundary of K 2 , then s | σ = 0. If σ ⊂ K 1 , then s | σ = s| σ . Otherwise Lemma 4.3 provides an extension of the rational function s | ∂σ to a rational function s | σ . Therefore, we can extend the rational function s to all of K 2 . By construction the rational function s satisfies s | ∂K 2 = 0. By declaring s to be zero outside of K 2 we obtain a rational function on X. Since s |K 1 = s|K 1 we have s | K = s. Proof. Take an injective map from M to an acyclic sheaf F and denote the quotient sheaf by G := F /M. We claim that F (X) → G(X) is surjective. For a t ∈ G(X), let U = (U i ) i∈I be a cover of X such that there exist a collection s i ∈ F (U i ) that map to t| U i . Since X is paracompact we may assume that U is locally finite. For each U i take a locally finite cover by a collection of compact polyhedral subsets so that for each i there is a member of the cover K i which is contained in U i and (K i ) i∈I still cover X. For J ⊂ I set K J : = j∈J K j . Since the covering of X by the sets K i is locally finite, the union K J is closed. The set carries a partial order given by (J, s) ≤ (J , s ) whenever J ⊂ J and s | K J = s. By Zorn's lemma, E has a maximal element (J, s). We want to show J = I. Assume that there exists j ∈ I \ J. Then s j − s maps to zero in G(K j ∩ K J ) and hence is the image of an element r ∈ M(K j ∩ K J ). By Lemma 4.4, we can extend r to a section r ∈ M(X). Let s ∈ F (X) denote the image of r and consider the section s j − s ∈ F (K j ). By construction, this section agrees with s on K J ∩ K j . Therefore we can glue s j − s and s to a section of F over K J∪j . But this is contradiction to the maximality of (J, s).
Thus F (X) → G(X) is surjective. Using the long exact sequence in cohomology associated to 0 → M → F → G → 0 and the fact that F is acyclic, we conclude that H 1 (X, M) = 0. By definition, we need to show that δ is surjective. This follows from the vanishing of H 1 (X, M) established in Lemma 4.5.

4.B. Tropical spaces and the fundamental class
Throughout this section X is a rational polyhedral space of pure dimension n with a polyhedral structure C and D is a C-stratified simplicial structure on X.
A We can extend Definition 2.3 to simplicial structures. For any ∆ ∈ D k whose relative interior is contained in σ ∈ C we define L(∆) to be the minimal linear subspace of L(σ ) which is defined over Q and has the property that ∆ is contained in a translate of L(∆). Note that in general k ≤ dim L(∆). We set L Z (∆) := L(∆) ∩ L Z (σ ). For ∆ ∈ D k with rank L Z (∆) = k, we define Λ ∆ to be the unique generator of k L Z (∆) Z compatible with the orientation of ∆. Then for a simplex ∆, we define F 1 (∆) in the same way as for polyhedra in Definition 2.3. We call X an (abstract) tropical space if ch(X) is closed. In this situation, we call [X] := [ch(X)] ∈ H BM n,n (X) the fundamental class of X.
When X is a tropical space, it is straightforward to check that the class [X] does not depend on the choice of simplicial structure on X.
Remark 4.9. The more conventional definition of a tropical space refers to the so-called balancing condition [MS15, Definition 3.3.1], [MR, Section 6.1]. To formulate this condition in our context, first let us use the notation ∆ ∆ to indicate pairs D n−1 ∆ ≺ ∆ ∈ D n of the same sedentarity. Let ∆ ∈ D n−1 be such that L Z (∆ ) has rank n − 1. A primitive generator of a pair ∆ ∆ is an integer vector where ε ∆,∆ is the sign with which ∆ appears in ∂∆. Primitive generators are unique up to adding an element in L Z (∆ ). The rational polyhedral space X is called balanced at ∆ if where the v ∆,∆ are primitive generators. The space X is called balanced if it is balanced at all ∆ ∈ D n−1 such that L Z (∆ ) has rank n − 1. It follows from [MZ14, Proposition 4.3] that ch(X) is a closed (n, n)-cycle if and only if X is balanced. In particular, whether or not X is a tropical space does not depend on the choice of simplicial structure D.
Definition 4.10. Given l ∈ F p Z (σ ) and v ∈ F Z p (σ ) with p ≤ p , the contraction l; v ∈ F Z p −p (σ ) is induced by the usual contraction map ; : p (Z r I ) * × p Z r I → p −p Z r I . More generally, given τ, τ ≺ σ and l ∈ F p Z (τ), v ∈ F Z p (σ ), the contraction l; v is given by Definition 4.11. The cap product with the fundamental class of X is the map where ; denotes the contraction introduced in Definition 4.10.
The definition of the cap product can be extended to ∩σ for arbitrary simplicial chains σ ∈ C BM p ,q (X, Z). Also the Leibniz formula d(α ∩ σ ) = (−1) q+1 (δα ∩ σ − α ∩ dσ ) holds on the chain level. If X is a tropical space then d ch(X) = 0, and it follows that the map ∩[X] described above descends to to a map between cohomology and Borel-Moore homology groups . To see that ∩[X] does not depend on the simplicial structure on X, note that the cap product can also be described on the level of singular chains.

4.C. Subspaces, Divisors, and the Chern class map
Throughout this section X is a rational polyhedral space of pure dimension n with a polyhedral structure C.

Definition 4.12. A subset Z ⊂ X is a rational polyhedral subspace if it is closed and the restrictions of the charts of an atlas of X provide an atlas as a rational polyhedral space for Z.
Given a rational polyhedral subspace Z, there exists a C-stratified simplicial structure D for X such that Z a union of cells of D. We call such a structure fine enough for Z. It can be constructed inductively as a simplicial structure of |C k |. Indeed, for each σ ∈ C k , the intersection Z ∩ σ is a closed polyhedral subset of the polyhedron σ and any simplicial structure on ∂σ fine enough for Z ∩∂σ can be extended to a simplicial structure on σ fine enough for Z ∩ σ .
Assume further that Z is of pure dimension k and is equipped with a weight function ω. We define It is a rational polyhedral subspace of dimension n − 1. Next we define weights on this set to turn it into a tropical cycle following [AR10, Section 3] and [MR, Section 5.2].
Definition 4.14. The divisor map is div : CaDiv(X) → Z n−1 (X), where div(s) is a tropical codimension one cycle supported on D(s). The weight of a generic point x ∈ D(s) gen (which is of sedentarity zero) is given as follows. Fix i such that x ∈ U i and choose • a neighbourbood x ∈ U ⊂ U i and simplicial structure D on U such that D(s) ∩ U ⊂ |D n−1 | and x ∈ relint ∆ , ∆ ∈ D n−1 , • primitive generators v ∆,∆ for any pair ∆ ∆, and set The weights ω div(s) (x) may happen to be zero; such parts of D(s) are tacitly removed from div(s). For details on this construction and proof of its well-definedness we refer to [ where ε ∆,∆ = ±1 is the relative orientation of ∆ and ∆ and i ∆ is chosen so that relint ∆ ⊂ U i ∆ .
Proof. Since D is a simplicial structure on X fine enough for D(s), the simplices {∆ ∈ D|∆ ⊂ D(s)} form a simplicial structure for D(s). In particular, it follows that for any ∆ ∈ D n−1 with ∆ ⊂ D(s) the lattice L Z (∆ ) is of rank n − 1. For any ∆ ∆ we choose primitive generators v ∆,∆ (satisfying (4.2)) and use the notation v ∆ ∈ L Z (∆ ) for the balanced sum in (4.3). We now want to show that for any ∆ ∈ D n−1 its coefficients on the right hand and left hand side of (4.6) agree. First assume sed(∆ ) = 0 and ∆ ⊂ D(s). Then for any facet ∆ ≺ ∆ the rules of contracting wedge products along 1-forms applied to equation (4.2) provide (4.7) Moreover, since v ∆ ∧ Λ ∆ = 0, we have Comparing with (4.4), this proves the equality of coefficients.
Let us now consider ∆ D(s). Then s i ∆ is affine linear in a neighbourhood of relint(∆ ) and therefore ds i ∆ defines an element in F 1 (∆ ) (independent of a choice of facet ∆ ≺ ∆). The coefficient of ∆ in (4.6) is therefore equal to But the sum on the right hand side is exactly equal to the coefficient of ∆ in ∂ ch(X), so it is zero since ch(X) is closed.  is Since the section s satisfies s i 1 = f i 0 i 1 + s i 0 we obtain The first summand is precisely the sum which appears in Equation (4.8) and the second summand is homologous to zero (namely, equal to ∂( ds i 0 ; Λ ∆ ⊗ ∆)). Therefore the diagram is commutative.
Proof of Theorem 1.3. Suppose that α ∈ Ker(φ : H 1,1 (X, Z) → H 0,2 (X, R)). By Theorem 1.2 there exists L in Pic(X) such that c 1 (L) = α. By Proposition 4.6 there exists a section s of L. By the commutativity of the diagram in Theorem 4.15, we have α ∩ [X] = c 1 (L) ∩ [X] = cyc(div(s)). The statement of the theorem now follows since cyc(div(s)) is the fundamental class of a codimension one tropical cycle.

Poincaré duality with coefficients in Z
In this section we prove Theorem 1.1. To do so, we restrict to tropical manifolds and establish a version of Poincaré duality over Z. We first introduce tropical manifolds which are tropical spaces locally modelled on matroidal fans. We do not describe these fans here but refer the reader to the literature for their definition and properties [AK06,Sha13b,FR13]. The above theorem is a extension of the version of Poincaré duality with real coefficients for tropical manifolds previously proved in [JSS15]. This version related tropical cohomology and tropical cohomology with compact support with real coeffcients via a pairing given by integration.

Definition 5.1. A tropical manifold X is a tropical space whose weight function is equal to one and which is equipped with an atlas of charts {ϕ
We first prove this version of Poincaré duality for matroidal fans using a cellular description of tropical (co)homology. Let X be a polyhedral subspace in T r ×R s and C a polyhedral structure on X such that every cell contains a vertex. Then there are descriptions of tropical (Borel-Moore) homology and cohomology in terms of cellular chain complexes with respect to C, For a justification of these identifications see Remark 2.8. Let V be a fan in R s and C a polyhedral fan structure on V such that 0 is a vertex of C. We then find using (5.2) and (5. Let X be a tropical variety in R r and f ∈ M(X) be a tropical rational function. The graph Γ X (f ) of f is a polyhedral complex in R r+1 . For every face τ of div(f ) denote by τ ≤ the polyhedron {(x, y)|x ∈ τ, y ≤ f (x)} ⊂ R r+1 . The union Y := Γ X (f ) ∪ {τ ≤ |τ ∈ div(f )} is a tropical cycle in R r if we define the weights on Γ X (f ) to be inherited from X and the weight on a face τ ≤ is defined to be equal to the weight of τ in div(f ). More details on this construction can be found in [AR10, Construction 3.3] as well as [MR, Chapter 5]. We provide some notation useful for modifications. Let δ : V → W be an open tropical modification of fans along a function f ∈ M(W ) with divisor D and let C be a polyhedral structure consisting of cones which contains a polyhedral structure for D. We always assume that δ is induced by the projection π : R r ×R → R r with kernel e r+1 . Denote by V the closure of V in R r × T . Then the polyhedra form a polyhedral structure on V . The intersection of the first two types of cones with V form a polyhedral structure on V .
Surjectivity: Let C = a σ σ be a closed chain in C BM p,q (W , Q). We can obviously find a lift of C of the form C 1 := a σ σ . Its boundary is of the form ∂C 1 = b τ τ. Since Ψ (∂C 1 ) = 0, we get π(b τ ) = 0 and hence b τ ∈ F Z p (τ ≤ ) by Lemma 5.7. Hence by adding b τ τ ≤ , we obtain C 2 with ∂C 2 = ρ c ρ ρ ≤ , where ρ runs through cells of dimension q − 2. Let us compute c ρ . By construction it is a sum over the flags ρ ⊂ τ ⊂ σ , each contributing ± a σ . But for each σ there are exactly two such flags, and they contribute with opposite sign, which implies c ρ = 0. Hence we get Ψ (C 2 ) = Ψ (C 1 ) = C and ∂C 2 = 0, as required.
Proof. The statement follows from the fact that the sequence is exact. This sequence is obtained by combining a similar short exact sequence for Orlik-Solomon algebras from [OT92, Theorem 3.65] together with the relation between the Orlik-Solomon algebras and F p from [Zha13].
Proof. Let C be a polyhedral structure on Y . Given a face σ ∈ C, denote σ ∞ := σ × {−∞} and σ := σ × T . The collection of all these polyhedra forms a polyhedral structure on Y ×T. The statement for cohomology now follows directly from (5.3) since the compact cells for the polyhedral structure on Y × T are precisely of the form σ × {−∞} for compact cells σ of C.
For Borel-Moore homology, we prove the claimed isomorphism by constructing an explicit homotopy equivalence on the cellular chain complexes with respect to these polyhedral structures. Let us first look at the behaviour of the multi-tangent spaces. There are projection and lifting maps The map π is induced by the linear projection R r × R → R r forgetting the last coordinate. The second map is given by v → v ∧ e r+1 := v ∧ e r+1 where v ∈ π −1 (v) and e r+1 denotes the kernel of the map π. Note that the wedge product does not depend on the choice of preimage. Let w : R r × R → R be the linear form given by projecting onto the last factor, regarded as an element w ∈ F 1 (σ ), as in the proof of Proposition 5.6. We define where ; denotes the contraction from Definition 4.10. It easy to check Φ • Ψ = id since π( w; v ∧ e r+1 ) = v. We define where v is the map F p (σ ) → F p ( σ ) induced by mapping each vector v ∈ F 1 (σ ) to the unique preimage v ∈ π −1 (v) with w; v = 0. Then h provides a chain homotopy between id and Ψ • Φ, thus the lemma is proven.
Corollary 5.9. Let V be a matroidal fan in R s and set Y = V × T r . Then Y satisfies Poincaré duality.
Proof. This follows from Proposition 5.5 and Lemma 5.8.
Remark 5.10. In the following proof we use a local gluing argument. To do so, we need to slightly extend our terminology. Let X be a polyhedral space with polyhedral structure C. Let U ⊂ X be an open subset. A C-stratified q-simplex in U is a C-stratified q-simplex δ in X such that δ(∆) ⊂ U . Using this convention, Definitions 2.6 and 2.7 can be carried over to the open set U instead of X. In particular, we obtain groups H p,q (U , Z) and H BM n−p,n−q (U , Z). Moreover, if X is a tropical space any C-stratified simplicial structure on U gives rise to a fundamental class [U ] ∈ H BM n,n (U , Z) and a map ∩[U ] : H p,q (U , Z) → H BM n−p,n−q (U , Z) which do not depend on the simplicial structure. Again we say that U satisfies Poincaré duality if ∩[U ] is an isomorphism for all p, q.
Proof of Theorem 5.3 . Let C be a polyhedral structure for X. The proof is completed in two steps.
Step 1: Open stars of faces satisfy Poincaré duality: A star U σ of a face σ ∈ C of a tropical manifold is isomorphic as a tropical manifold to a connected neighbourhood U of (0, (∞) r ) in a polyhedral complex of the form Y = V × T r , where V is a matroidal fan. Note that C induces a polyhedral structure C on Y .
There is a homeomorphism f : U → Y which preserves the stratification given by C and C . Hence if δ is a C-stratified simplex in U , the push-forward f * (δ) = f • δ is a C -stratified simplex in Y . We obtain the following commutative diagram.
It is straightforward to check that the two vertical arrows are isomorphisms. Since ∩[Y ] is an isomorphism by Corollary 5.9, the map ∩[U ] is also an isomorphism. Step where in each degree we take the cap product with the appropriate fundamental class. Now the claim follows from the five lemma, since by our assumption U , V and U ∩ V satisfy Poincaré duality.
Corollary 5.11. If X is a compact tropical manifold of dimension n, then H p,q (X, Z) H n−q,n−q (X, Z). The last step in order to prove Theorem 1.1 is to relate the wave homomorphism on cohomology to its variant on homology.
Lemma 5.13. The following diagram is commutative: Proof. This follows on the level of individual simplices by the definition of φ (see Definition 2.9).

Corollaries and Examples
In this final section we deduce some corollaries of the main theorems and present some explicit examples. In the case of tropical abelian surfaces and Klein bottles with a tropical structure, we show how to represent (1, 1)-classes in the kernel of the wave map as fundamental classes of tropical cycles. We also calculate the wave map for two combinatorial types of smooth tropical quartic surfaces. We start with some interesting consequences of Theorem 1.1. Proof. We haveφ(α) = 0 ∈ H BM n,n−2 (X, R). Thus the corollary follows again from Theorem 1.1.

Remark 6.3.
In some instances the image of the wave homomorphism is contained in the appropriate cohomology group with rational instead of real coefficients. Then the dimension of the kernel of the extension φ : H 1,1 (X, Z) ⊗ Q → H 0,2 (X, Q) gives the rank of the free part of the kernel of φ : H 1,1 (X, Z) → H 0,2 (X, Q). For example, for the Q-tropical projective varieties as introduced in [IKMZ16] the wave homomorphism is always defined over Q.

6.A. Tropical structures on the Klein bottle
Let K be a Klein bottle obtained from gluing a parallelogram P ⊂ R 2 with edges a, b, c, d as follows. The edges b and d are glued using the translation in the direction of a, and the edges a and c are glued using an (orientation-reversing) affine transformation h which sends a to c with flipped orientation. Let H ∈ GL(2, R) be the linear part of h. Then the tropical structure given by Z 2 ⊂ R 2 extends to K if and only if H ∈ GL(2, Z). Note that det(H) = −1 and one of its eigenvalues is −1, hence the second eigenvalue is +1. It follows that the eigenvectors have rational directions and a computation shows that the two primitive eigenvectors either form a lattice basis or generate a sublattice of index 2. We can normalise the two cases to the following matrices: Correspondingly, the parallelogram P has vertices 0, l 1 e 1 , l 1 e 1 + l 2 v 2 , and l 2 v 2 , where v 2 is either e 2 or 1 2 (see Figures 1, 2). We denote the two Klein bottles by K 1 and K 2 . Note that H 2,0 (K, Z) = Z 2 so that H 2,0 (K, R) = 0. Hence we are in the situation of Corollary 6.1 which says that any (1, 1)-class can be represented by a tropical cycle of dimension one. Let a = [0, l 1 e 1 ] and b = [0, l 2 v 2 ] denote two oriented edges of P . Any (1, 1)-class can be represented by v ⊗ a + λv 2 ⊗ b, v ∈ Z 2 , λ ∈ Z.
As boundaries of (1, 2)-chains we obtain Hence we find H 1,1 (K 1 , Z) = Z 2 ⊕ Z ⊕ Z = e 2 ⊗ a, e 1 ⊗ a, e 2 ⊗ b and H 1,1 (K 2 , Z) = Z ⊕ Z = 1 1 ⊗ a, 1 2 ⊗ b . Among these generators, the chains e 1 ⊗ a and v 2 ⊗ b can obviously be represented by tropical cycles. Such representations are less obvious for the torsion class e 2 ⊗ a and the class 1 1 ⊗ a. Explicit representations by tropical cycles are depicted for the two cases in Figures 1, 2. Here the chains are drawn in thin red lines with framing and orientation given by simple and double arrows respectively. The homologous tropical cycles are drawn in thick red lines and labelled with their respective weights if not equal 1.
For the sake of completeness let us briefly discuss the full classification of tropical Klein bottles. Instead of just a translation, we may glue the edges b and d via the affine transformation x → T x + t, where t is the translation along a and T ∈ GL(2, Z) (before we assumed T = id). Depending on H, the possible matrices T are of the following two types (see [Sep10]) T 1,n = 1 0 n 1 , n ∈ Z T 2,n = 1 + 2n −n 4n 1 − 2n , n ∈ Z. (6. 3) The obtained Klein bottles K 1,n and K 2,n give a full list of Klein bottles with a tropical structure. Analogous to the case n = 0, we can compute the homology groups for n 0 as H 1,1 (K 1,n , Z) = Z/2Z ⊕ Z/nZ = e 2 ⊗ a, e 2 ⊗ b and H 1,1 (K 2,n , Z) = Z/2nZ = 1 2 ⊗ b . Again, it is clear that v 2 ⊗ b can be represented by tropical cycles, while for e 2 ⊗ a the same trick as for K 1 , Figure 1, is needed.

6.B. Tropical abelian surfaces
A tropical abelian surface is S = R 2 /Λ where Λ is a rank two lattice equal to w 1 , w 2 Z for w 1 , w 2 ∈ R 2 . Therefore S S 1 × S 1 . The sheaf F p Z is the constant sheaf p Z 2 for p = 0, 1, 2, and tropical homology groups H p,q (S, Z) are free Z modules whose ranks are given by the follow tropical Hodge diamond, We can choose a basis of H 1,1 (S, Z) as α ij = e i ⊗ σ j where σ 1 , σ 2 form a basis of H 1 (S, Z) and e 1 , e 2 are a lattice basis of Z 2 . Furthermore suppose that σ i is the quotient of the oriented line in R 2 in direction w i . Then the eigenwave homomorphism is given byφ(α ij ) = e i ∧ w j .
Suppose that v 1 and v 2 are linearly independent. Consider the triangle T in R 2 with vertices 0, w 1 , and w 2 . Firstly, we claim that if v 1 ∧ w 1 + v 2 ∧ w 2 = 0, then the lines in directions v 1 + v 2 , v 1 , and v 2 drawn from the vertices 0, w 1 , and w 2 , respectively, are concurrent.
Since v 1 ∧ v 2 0, it forms a R-basis of 2 R 2 and hence there exists an α ∈ R such that αv 1 ∧ v 2 = v 1 ∧ w 1 = −v 2 ∧ w 2 . Then the three lines mentioned above intersect at the point p = α(v 1 + v 2 ) in R 2 . To see this notice that (x − w i ) ∧ v i = 0 is the defining equation for the line from w i . Then a (1, 2)-simplex based on the triangle τ i = [0w i p] (with given orientation). Then α is the fundamental class of a tropical 1-cycle and it is homologous to α, see Figure 3.
If v 1 and v 2 are linearly dependent and α ∈ ker(φ) then w 1 + w 2 = αv 1 for some α ∈ R. In particular, w 1 + w 2 is a rational direction and α can be represented by a parallel cycle supported on the diagonal of a fundamental domain for S.
Let us make a few remarks about general n-dimensional tropical tori S.

6.C. Tropical hypersurfaces
A tropical hypersurface in R n+1 or in an n + 1-dimensional tropical toric variety is the divisor (as in Definition 4.14) of a tropical polynomial function. It is an n-dimensional polyhedral complex which is dual to a regular subdivision of a lattice polytope. This implies the following statement. Figure 3: A (1, 1)-cycle on a tropical abelian surface in red and green on the left, and a representative of its class by the fundamental class of a tropical 1-cycle.

Proposition 6.4. A tropical hypersurface is homotopic to a bouquet of spheres.
A tropical hypersurface is non-singular if it is a tropical manifold. This is the case if and only if it is dual to a regular subdivision of a lattice polytope which is primitive, i.e. if every top dimensional polytope in the subdivision is of lattice volume 1.
Corollary 6.5. If X is a non-singular tropical hypersurface in an n + 1-dimensional tropical toric variety for n ≥ 3 then every class in H BM n−1,n−1 (X, Z) is the fundamental class of a tropical cycle in X.
Proof. It follows from Proposition 6.4 that H 0,2 (X, R) = 0, so the statement follows from Corollary 6.1.
The first example of non-trivial wave maps for tropical hypersurfaces is in the case of smooth tropical quartic surfaces. We now look at two specific examples. Definition 6.6. A smooth tropical quartic surface is dual to a primitive regular triangulation of a size 4 tetrahedron.
There is one 2-dimensional polytopal sphere P contained in smooth tropical quartic surface. It is dual to all cells of the regular subdivision of the size 4 tetrahedron which contain the unique interior lattice point of the size 4 simplex of dimension 3. The Betti diamond of a smooth tropical quartic surface is: 1 0 0 1 20 1 0 0 1 so the wave map sends a Z-module of rank 20 to a 1-dimensional real vector space. The Picard rank of smooth tropical quartic surface X is the rank of Pic(X). Since H 0,1 (X, Z) = 0 for a smooth tropical quartic surface X the map c 1 : Pic(X) → H 1,1 (X, Z) is injective by the long exact sequence obtained from (3.1). Hence the Picard rank is equal to the rank of the kernel of the wave homomorphism by Proposition 3.5.    Figure 5: On the left the branched path which is the support of a (2, 1)-cycle whose boundary is τ 1 +τ 2 +τ 3 . On the right a depiction of the polytope P and the cycles β and γ from Example 6.7. Figure 6: A tropical quartic surface and its bounded polytope from Example 6.8.
A floor decomposed tropical quartic surface is determined (up to choice of constants regulating the height of the floors) by a collection of non-singular planar tropical curves C 1 , C 2 , C 3 , C 4 where each C d is of degree d (i.e. dual to a primitive triangulation of the size d lattice triangle).
Given a floor decomposed surface X a basis of its H 1,1 (X, Z) tropical homology was described in [Sha13a]. On a floor given by the curves C i and C i+1 , there are i(i + 1) − 1 independent "floor cycles". This produces 11 + 5 + 1 = 17 independent (1, 1)-cycles. They can be chosen such that their support, after projecting to the plane, forms a minimal loop in C i ∪ C i+1 not contained in C i or C i+1 . By our particular choice of curves C 1 , . . . , C 4 , any such cycle is disjoint from the cycle of C 3 . Hence we can assume the floor cycles do not intersect the polytopal sphere. Additionally, there is a cycle h, a multiple of which is the hyperplane section, which can also be made disjoint from P . Together with the cycles α and β which are illustrated on the right hand side of Figure 5 and completely described in [Sha13a], these cycles form a basis of H 1,1 (X, Z).
We now describe how (2, 0) cells behave when passing to homology depending on their supporting point. Orient each face of P so that it is the boundary of the 3-dimensional polytope. At any edge γ of X of sedentarity ∅ there are three faces adjacent to it: σ 1 , σ 2 , and σ 3 . If τ i is an appropriately oriented generator of F Z 2 (σ i ) we can find a (2, 1)-cell whose support is the branched path on the left of Figure 5 and whose boundary is τ 1 + τ 2 + τ 3 . Moreover, for any point x not on the polytopal sphere we can find a branched path in X whose endpoints are x and points of positive sedentarity, thus showing that a (2, 0)cycle supported on x is homologous to 0. In addition, for two faces σ 1 , σ 2 of the polytopal sphere, we have τ 1 ∼ τ 2 where τ i are appropriately oriented generators of F Z 2 (σ i ) and such an τ i generates H 2,0 (X, Z). We denote the class of these τ i by τ.
This implies thatφ(h) =φ(γ) = 0 for all floor cycles γ. Moreover, using this description of H 2,0 (X, R) we can explicitly computeφ(α) = lτ andφ(α) = hτ, where l is the lattice length of the unique cycle in C 3 (i.e., the j-invariant of C 3 ) and k is the lattice height of the pentagonal prism P (or, the distance between the floors connected by C 3 ). Varying the coefficients of the defining tropical polynomial, these two parameters can be controlled independently. In particular, we can arrange both l k ∈ Q and l k Q. We conclude that Ker(φ) has rank 19 or 18, depending on our choice and thus any tropical quartic surface with the combinatorial type of the one chosen above must have Picard rank equal to 18 or 19 by Theorem 1.1.
Example 6.8. (A smooth tropical quartic surface with Picard rank 1) The second example is dual to a cone triangulation. Fix a primitive regular triangulation of each of the four two dimensional faces of the size 4 tetrahedron. We obtain a unique primitive regular triangulation by considering the cone over this triangulation with the cone point being the unique interior lattice point (1, 1, 1) of the size 4 simplex. See Figure 6 for an example.