Lefschetz section theorems for tropical hypersurfaces

We establish variants of the Lefschetz hyperplane section theorem for the integral tropical homology groups of tropical hypersurfaces of toric varieties. It follows from these theorems that the integral tropical homology groups of non-singular tropical hypersurfaces which are compact or contained in $\mathbb{R}^n$ are torsion free. We prove a relationship between the coefficients of the $\chi_y$ genera of complex hypersurfaces in toric varieties and Euler characteristics of the integral tropical cellular chain complexes of their tropical counterparts. It follows that the integral tropical homology groups give the Hodge numbers of compact non-singular hypersurfaces of complex toric varieties. Finally for tropical hypersurfaces in certain affine toric varieties, we relate the ranks of their tropical homology groups to the Hodge-Deligne numbers of their complex counterparts.

Tropical homology is a homology theory with non-constant coefficients for polyhedral spaces.Itenberg, Katzarkov, Mikhalkin, and Zharkov, show that under suitable conditions, the Q-tropical Betti numbers of the tropical limit of a family of complex projective varieties are equal to the corresponding Hodge numbers of a generic member of the family [IKMZ16].This explains the particular interest of these homology groups in tropical and complex algebraic geometry.
In this paper we consider the integral versions of tropical homology groups for hypersurfaces in toric varieties.The pp, qq-th tropical homology group of a polyhedral complex Z is denoted H q pZ; F Z p q and the Borel-Moore homology group is denoted H BM q pZ; F Z p q.To avoid ambiguity we will often also refer to H q pZ; F Z p q as a standard tropical homology group.When a polyhedral complex Z is compact then H q pZ; F Z p q " H BM q pZ; F Z p q.These homology groups are defined in Section 2 as the cellular tropical homology groups [MZ14] and [KSW17].For a comparison between cellular homology and singular homology, see Remark 2.19.Our main goal is to prove that these homology groups are torsion free for a compact non-singular tropical hypersurface in a compact non-singular tropical toric variety.The road to the proof of this statement is quite similar to the one followed to prove that the integral homology of a complex projective hypersurface is torsion free.Namely, in order to prove that the integral tropical homology groups are without torsion, we first establish a tropical variant of the Lefschetz hyperplane section theorem.Ultimately however, the techniques used in the proofs are quite different from the complex setting, since we are working with polyhedral spaces instead of algebraic varieties.Also notice that in the tropical version of the Lefschetz section theorem stated below the tropical hypersurface is not required to be compact.However, the tropical hypersurface is required to be combinatorially ample in the tropical toric variety, see Definition 2.3.For the notion of cellular pair see Definition 2.6.
Theorem 1.1.Let X be a non-singular and combinatorially ample tropical hypersurface of an n `1 dimensional non-singular tropical toric variety Y .Then the map induced by inclusion i ˚: H BM q pX; F X p q Ñ H BM q pY ; F Y p q is an isomorphism when p `q ă n and a surjection when p `q " n.
If additionally, the tropical hypersurface X has full-dimensional Newton polytope and the pair pY, Xq is a cellular pair, then the map induced by inclusion i ˚: H q pX; F X p q Ñ H q pY ; F Y p q is an isomorphism when p `q ă n and a surjection when p `q " n.
Tropical homology with real or rational coefficients is the homology of the cosheaf of real vector spaces F p b R or F p b Q, respectively.Theorem 1.1 holds in the case of tropical homology with real coefficients for a singular tropical hypersurface X in a tropical toric variety Y which is combinatorially ample and also proper, see Definition 2.2.Below we state the theorems in the case of real coefficients.
Theorem 1.2.Let X be a combinatorially ample tropical hypersurface of an n `1 dimensional non-singular tropical toric variety Y that is proper in Y .Then the maps induced by inclusion i ˚: H BM q pX; F X p b Rq Ñ H BM q pY ; F Y p b Rq are isomorphisms when p `q ă n and surjections when p `q " n.If additionally, the tropical hypersurface X has full-dimensional Newton polytope and the pair pY, Xq is a cellular pair, then the maps induced by inclusion i ˚: H q pX; F X p q Ñ H q pY ; F Y p q are isomorphisms when p `q ă n and surjections when p `q " n.
Adiprasito and Björner established tropical variants of the Lefschetz hyperplane section theorem in [AB14].Their theorems relate the tropical homology with real coefficients of a non-singular tropical variety X contained in a tropical toric variety to the tropical homology groups of the intersection of X with a so-called "chamber complex".A chamber complex is a codimension one polyhedral complex in a tropical toric variety whose complement consists of pointed polyhedra, in particular it need not to be balanced.Adiprasito and Björner first establish some topological properties of filtered geometric lattices and then use Morse theory to prove their tropical versions of the Lefschetz theorem.The proof we present here does not utilise Morse theory but instead proves vanishing theorems for the homology of cosheaves that arise in short exact sequences relating the cosheaves for the tropical homology of X and the ambient space.Furthermore, we relate the integral tropical homology groups of a non-singular tropical hypersurface with the integral tropical homology groups of the ambient toric variety.Another result which follows from the Lefschetz section theorem for the integral tropical homology groups of hypersurfaces is that under the correct hypotheses on the ambient space these homology groups are torsion free.At the end of the introduction we discuss the implications of torsion freeness to recent results on the Betti numbers of real algebraic hypersurfaces arising from patchworking.
The tropical (co)homology groups with integral coefficients of a nonsingular tropical hypersurface satisfy a variant of Poincaré duality [JRS17].Using this we deduce in Section 4 that the tropical homology groups of a non-singular tropical hypersurface in a non-singular tropical toric variety which satisfy the assumptions below are torsion free, as long as the homology of the toric variety is also torsion free.
Theorem 1.3.Let X be a non-singular tropical hypersurface in a nonsingular tropical toric variety Y such that pY, Xq is a cellular pair and the Newton polytope of X is full dimensional.If the tropical homology groups of Y are torsion free, then both the Borel-Moore and standard tropical homology groups of X are also torsion free.
Corollary 1.4.Let Y be a compact non-singular tropical toric variety and let X be a combinatorially ample non-singular tropical hypersurface in Y .If the complex toric variety Y C is projective, then all integral tropical homology groups of X are torsion free.
Corollary 1.5.Let Y be a non-singular tropical toric variety associated to a fan whose support is a convex cone and such that the complex toric variety Y C is quasi-projective.Let X be a combinatorially ample non-singular tropical hypersurface in Y such that pY, Xq is a cellular pair and the Newton polytope of X is full dimensional.Then both the standard and Borel-Moore integral tropical homology groups of X are torsion free.
Corollary 1.6.The tropical homology groups of a non-singular tropical hypersurface in R n`1 with full dimensional Newton polytope are torsion free.
The above proposition and corollaries follow from the tropical Lefschetz section theorems established here for hypersurfaces.That is why we require in Theorem 1.3 that pY, Xq be a cellular pair, that the hypersurface X is combinatorially ample in Y , and that the Newton polytope of X be full dimensional.We do not know if these assumptions are necessary, or if an alternate more direct proof of torsion freeness exists.
Question 1.7.Are the integral tropical homology groups of any non-singular tropical hypersurface of a tropical toric variety torsion free?
In Section 5, we first find that the Euler characteristics of the cellular chain complexes for Borel-Moore tropical homology of a non-singular tropical hypersurface give the coefficients of the χ y genus of a torically non-degenerate complex hypersurface with the same Newton polytope (see Definition 5.1) for the definition of torically non-degenerate.
Theorem 1.8.Let X be an n-dimensional non-singular tropical hypersurface in a non-singular tropical toric variety Y .Let X C be a complex hypersurface torically non-degenerate in the complex toric variety Y C such that X and X C have the same Newton polytope.Then and thus From the above theorem we obtain an immediate relation between the dimensions of the R-tropical homology groups of a tropical hypersurface and the χ y genus of corresponding complex hypersurface.Namely, in the situation of the above theorem we have Moreover, when the integral tropical homology groups are torsion free we also have q"0 e p,q c pX C q. (1.1) We combine the torsion freeness results from Section 4 and Equation 1.1 to calculate the ranks of the tropical homology groups of tropical hypersurfaces in compact toric varieties in Corollary 1.9.
Corollary 1.9.Let X be a non-singular and combinatorially ample compact tropical hypersurface in a non-singular compact toric variety Y and assume that X has Newton polytope ∆.Let X C be a torically non-degenerate complex hypersurface in the compact toric variety Y C also with Newton polytope ∆.Then for all p and q we have dim H p,q pX C q " rank H q pX; F X p q.In the situation of Corollary 1.5, if the toric variety is affine and constructed from a fan whose support is a convex cone of maximal dimension, we also determine the ranks of the Borel-Moore tropical homology groups of tropical hypersurfaces in terms of the Hodge-Deligne numbers with compact support of complex hypersurfaces.The Hodge-Deligne numbers of a complex variety X C are denoted by h p,q pH k c pX C qq (see for example [DK86]).
Corollary 1.10.Let Y be a non-singular tropical toric variety associated to a fan whose support is a convex cone of maximal dimension in R n`1 and such that the complex toric variety Y C is affine.Let X be a combinatorially ample non-singular tropical hypersurface in Y such that pY, Xq is a cellular pair and the Newton polytope of X is full dimensional.If X C is a torically non-degenerate complex hypersurface in Y C with the same Newton polytope as X, then Lastly we use again Equation 1.1 to calculate the ranks of the tropical homology groups of tropical hypersurfaces in the tropical torus R n`1 .
Corollary 1.11.Let X be a non-singular tropical hypersurface in R n`1 with full-dimensional Newton polytope.If X C is a non-singular torically nondegenerate complex hypersurface in pC ˚qn`1 with the same Newton polytope as X, then We point out that for X C a non-singular torically non-degenerate complex hypersurface in pC ˚qn`1 we have h p,p pH n`p pX C qq " `n`1 p`1 ˘.Our main motivation establishing torsion freeness of the tropical homology groups of tropical hypersurfaces and a relation between their ranks and the Hodge-Deligne numbers of complex hypersurfaces comes from a recently established relation between the Z 2 -tropical homology groups of tropical hypersurfaces and Betti numbers of patchworked real algebraic hypersurfaces.In the theorem below H q pX; F X,Z 2 p q denotes the tropical homology groups considered with coefficients in Z 2 :" Z{2Z.
Theorem 1.12.[RS18, Theorem 1.4]If V is a non-singular real algebraic hypersurface in a toric variety obtained from a primitive patchworking of tropical hypersurface X equipped with a real structure then for all q we have, When the integral tropical homology groups are torsion free then we have rank H q pX; F X p q " dim H q pX; F X,Z 2 p q for all p and q.This together with Corollaries 1.9 and 1.11 allow the bounds in Theorem 1.12 on the Betti numbers of the real points of a patchworked algebraic variety to written in terms of Hodge-Deligne numbers of the complexification.
The research of C.A. is supported by the DIM Math Innov de la région Ile-de-France.A.R. acknowledges support from the Labex CEMPI (ANR-11-LABX-0007-01).The research of K.S. is supported by the BFS Bergen Research Foundation project "Algebraic and topological cycles in complex and tropical geometry".

Tropical toric varieties.
In this text we will always use the standard lattice Z n`1 Ă R n`1 .The tropical numbers are T " r´8, `8q.We equip the set T with a topology so that it is isomorphic to a half open interval.Tropical affine space of dimension n is T n and is equipped with the product topology.Tropical manifolds are topological spaces equipped with charts to T n .For the general definitions of tropical varieties and manifolds see [MS15, Section 3.2], [MR, Section 7], [MR, Section 1.2].In algebraic geometry over a field a rational polyhedral fan in R n`1 produces an n `1 dimensional toric variety.The same fact is true in tropical geometry.Given a rational polyhedral fan in R n`1 we can construct a tropical toric variety, see [MS15, Section 6.2], [MR, Section 3.2].A tropical toric variety of dimension n`1 has charts to T n`1 .A rational polyhedral fan Σ is simplicial if each of its cones is the cone over a simplex.A simplicial rational polyhedral fan is unimodular if the primitive integer directions of the rays of each cone can be completed to a basis of Z n`1 .Just as in the case over a field, a tropical toric variety is non-singular if it is built from a simplicial unimodular rational polyhedral fan.The tropical toric varieties considered in this text are always nonsingular.A tropical toric variety is compact if and only if the corresponding fan is complete.
A tropical toric variety Y has a stratification and the combinatorics of the stratification is governed by its fan Σ.A stratum of dimension k of Y corresponds to a cone ρ of dimension n `1 ´k of Σ.We let Y ρ denote the strata in Y corresponding to the cone ρ.Therefore, when σ is of dimension k we have Y ρ -R n`1´k .For two cones ρ and ρ 1 of Σ we have Y ρ 1 Ă Y ρ if and only if ρ is a face of ρ 1 in Σ. Morover if ρ is a face of ρ 1 in Σ, we have a projection map denoted by π ρ,ρ 1 : Y ρ Ñ Y ρ 1 .We denote the vertex of the fan by ρ 0 and the corresponding open stratum of Y by simply Y 0 .For any point y P Y , the order of sedentarity of y, denoted sedpyq, is defined to be the codimension in Y of the stratum containing y.
Example 2.1.The tropical projective space TP n is the tropical toric variety constructed from the fan consisting of cones R ě0 e i 1 `¨¨¨`R ě0 e i k , for all ti 1 ¨¨¨i k u t0, ¨¨¨, nu, where e 1 , ¨¨¨, e n is the standard basis of R n and e 0 " ´řn k"1 e k .It can also be described as the quotient T n`1 zp´8, . . ., ´8q rx 0 : ¨¨¨: x n s " ra `x0 : ¨¨¨: a `xn s , where a P Tz ´8.The stratification of TP n can be described using homogeneous coordinates.For a subset I Ă t0, . . ., nu define TP n I " tx P TP n | x i " ´8 if and only if i P Iu.The set TP n I corresponds to the cone The order of sedentarity of a point x " rx 0 : ¨¨¨: A rational polyhedron in Y is the closure in Y of a rational polyhedron in some stratum Y ρ .Therefore the polyhedra in Y are always closed.A polyhedral complex Z in a tropical toric variety Y is a collection of polyhedra in Y such that Z X Y ρ is a polyhedral complex in Y ρ -R codimρ for every cone ρ of Σ and satisfying: (1) for a polyhedron σ P Z, if τ is a face of σ, which is denoted τ Ă σ, we have τ P Z; (2) for σ, σ 1 P Z, if τ " σ X σ 1 is non-empty then τ is a face of both σ and σ 1 .
For a polyhedron σ in Y we define sedpσq to be sedpyq for any y in the relative interior of σ.This is a generalization of the notion of sedentarity from [BIMS15, Section 5.5] to tropical toric varieties beyond TP n`1 .Two polyhedral complexes are combinatorially isomorphic if they are isomorphic as posets under inclusion.
If σ is a polyhedron in Y which is the closure of a polyhedron in Y 0 then σ X Y ρ ‰ H if and only if the recession cone of σ intersects intpρq [OR13, Lemma 3.9].The same lemma also shows that if σ X Y ρ ‰ H, then σ X Y ρ " π 0ρ pσ X Y 0 q.Therefore, if a polyhedral complex Z is proper in Y and σ is a face of Z such that int ρ Ă Y ρ where ρ ‰ 0, then there exists at most one face of sedentarity 0 of Z containing σ as a face.This is because such a face must be of dimension dim σ `dim ρ since Z is proper and the image of such a face under π 0,ρ is σ where dim Kerπ 0,ρ " dim ρ.
A polyhedral complex Z in Y is rational if Z X Y ρ is a rational polyhedral complex for every stratum Y ρ .For σ a cell of Z we use int σ to denote its relative interior.

Tropical hypersurfaces.
A tropical hypersurface X in R n`1 is a weighted rational polyhedral complex of codimension one which satisfies the balancing condition well-known in tropical geometry.A tropical hypersurface in R n`1 is defined by a tropical polynomial f .As a polyhedral complex, a tropical hypersurface X is dual to a regular subdivision of the Newton polytope of f , and this subdivision is also induced by the polynomial f .A tropical hypersurface X in R n`1 is non-singular if it is dual to a primitive regular triangulation of its Newton polytope.For the definitions and properties of tropical hypersurfaces in R n`1 and the dual subdivisions of their Newton polytopes we refer the reader to [MS15, Chapter 3] and [BIMS15, Section 5.1].Recall that for Theorem 1.1 to hold for standard tropical homology, the tropical hypersurface is required to have full-dimensional Newton polytope.See Example 2.8 for a situation where this assumption is necessary.
If Y is a tropical toric variety of dimension n `1, the closure in Y of any tropical hypersurface is defined by a tropical polynomial f ρ and X ρ is dual to a primitive regular triangulation of the Newton polytope of f ρ .We always consider the polyhedral structure on X X R n`1 which is dual to the regular subdivision of its Newton polytope.If X is non-singular in Y , then X equipped with this polyhedral structure is proper in Y .When considering a tropical hypersurface X contained in a toric variety Y , we always use the polyhedral structure on Y obtained from refining by X.
Let γ be a polyhedron of dimension s and sedpγq " 0 in a tropical toric variety Y .For each cone ρ in the fan Σ defining Y , set γ ρ :" γ X Y ρ and define If we assume that γ intersects the boundary of Y properly, a face σ of γ o of dimension q is necessarily of sedentarity order sedpσq " dim γ ´q.
To prove the tropical version of the Lefschetz hyperplane section theorem we require the following additional assumption on X.With the exception of Theorem 1.8, we will always require that X is combinatorially ample in Y .
Definition 2.3.A tropical hypersurface X in an n `1 dimensional toric variety Y is combinatorially ample if for every face γ of dimension n `1 of Y , considered with the refinement given by X, the polyhedral complex γ o is combinatorially isomorphic to a product of copies of T and R.
Suppose that a tropical polynomial f defines a non-singular tropical hypersurface X 0 in R n`1 .If the Newton polytope of f is full dimensional and the dual fan of the polytope defines a non-singular tropical toric variety Y , then the compactification of X 0 in Y is non-singular and combinatorially ample.The following example shows that the assumption that the tropical hypersurface be combinatorially ample is necessary for Theorem 1.1 to hold.
Example 2.5.Here is a counter example to Theorem 1.1 when we drop the condition of combinatorial ampleness from Definition 2.3.Consider the standard tropical hyperplane X o Ă R n`1 .The case when n " 2 is depicted in the left of Figure 2. Let Σ be the fan for n `1 dimensional projective space blown up in a toric fixed point, and let Y be the tropical toric variety defined by Σ.Let X denote the compactification of X o in Y .Then it can be computed that rank H 1 pX, F X 1 q " 1 and rank The connected component of Y zX containing the stratum of Y dual to the ray of Σ corresponding to the exceptional divisor of the blow up does not satisfy the condition to be combinatorially ample.The complex geometric version of the same scenario also fails the Lefschetz hyperplane section theorem, since the hypersurface of the toric variety is not ample.
To prove Lefschetz hyperplane section theorem in the case of standard tropical homology, we also need the following assumption on the topological pair pY, Xq.
Definition 2.6.Let Y be a tropical toric variety and let X Ă Y be a tropical hypersurface.We say that the pair pY, Xq is a cellular pair if the cellular structure induced by X on the one-point compactification Ŷ of Y is a regular CW-complex.More precisely, for any cell σ of Ŷ of dimension k, the pair pσ, intpσqq is homeomorphic to the pair pB k , intpB k qq, where B k is the closed Euclidean ball of dimension k.
Requiring pY, Xq to be a cellular pair implies that X and Y equipped with the polyhedral structure induced by X are both cellular complexes in the sense of [She85] and [Cur14,Chapter 4].This topological condition is required to use the cellular description of cosheaf homology groups from [Cur14].
Example 2.7.There are examples of tropical hypersurfaces in toric varieties which are not cellular pairs.For example, consider X to be supported on the line x " 0 in R 2 .Then the one point compactification of pR 2 , Xq is not a CW-complex.In fact, if X is a tropical hypersurface in R n`1 , then pR n`1 , Xq is a cellular pair if and only if the Newton polytope of X is full dimensional.
There exist tropical hypersurfaces in T n`1 which do not intersect the boundary of T n`1 .For example, let X Ă T 2 be the tropical curve with three rays in directions p´2, 1q, p1, ´2q and p1, 1q.In this case, the pair pT n`1 , Xq is not a cellular pair, though X may be combinatorially ample in T n`1 .However, if Y is a compact tropical toric variety and X is a hypersurface which intersects the boundary of Y transversally then pY, Xq is a cellular pair.
The next example shows that full-dimensionality of the Newton polytope is an essential for the Lefschetz theorem to hold for standard homology.
Example 2.8.Consider the case when the Newton polytope of X is an interval of lattice length equal to 1. Then the tropical hypersurface X is a (classical) affine subspace of Y " R n`1 of dimension n, therefore X " R n .Upon subdividing X " R n and Y " R n`1 so that they form a cellular pair, or using singular tropical homology, we can compute the standard tropical homology groups to be: Whereas, the Borel-Moore homology groups are We see that the conclusion of the Lefschetz section theorem as stated in Theorem 1.1 does not hold for the standard tropical homology groups, however there is no contradiction for the Borel-Moore homology groups.

Tropical homology.
A polyhedral complex Z has the structure of a category.The objects of this category are the cells of Z and there is a morphism τ Ñ σ if the cell τ is included in σ.We use the notation Z op to denote the category that has the same objects as Z, and with morphisms corresponding to the morphisms of Z but with their directions reversed.Let Mod Z denote the category of modules over Z.We now define cellular sheaves and cosheaves of Z-modules on Z.
Definition 2.9.Given a polyhedral complex Z, a cellular cosheaf G is a functor In particular, a cellular cosheaf consists of a Z-module Gpσq for each cell σ in Z together with a morphism ι στ : Gpσq Ñ Gpτ q for each pair τ , σ when τ is a face of σ.Since G is a functor, for any triple of cells γ Ă τ Ă σ the morphisms ι commute in the sense that Dually, a cellular sheaf H is a morphism H : Z Ñ Mod Z .Therefore, for each σ there is a Z-module Hpσq and there are morphisms ρ τ σ : Hpτ q Ñ Hpσq when τ is a face of σ.
The cosheaves that we use throughout the text will always be free Zmodules unless it is otherwise stated.We will now define the integral multitangent modules.We refer the reader to [BIMS15], [KSW17], and [MZ14] for the definitions of the multi-tangent spaces with rational and real coefficients.
Let Y be the non-singular tropical toric variety corresponding to a fan Σ.Let ρ be a simplicial cone of Σ which has rays in primitive integer directions r 1 , . . ., r s .Then we define T pY ρ q :" R n`1 xr 1 , . . ., r s y and T Z pY ρ q :" Z n`1 xr 1 , . . ., r s y .
If Y ρ and Y η are a pair of strata such that Y η Ă Y ρ then the generators of the cone η contain the generators of the cone ρ and thus we get projection maps: π ρη : T pY ρ q Ñ T pY η q and π ρη : T Z pY ρ q Ñ T Z pY η q. (2.1) Let T pσq denote the tangent space to the relative interior of σ in T pY ρ q when int σ is contained in Y ρ .When σ is rational there is a full rank lattice T Z pσq Ă T pσq.Definition 2.10.Let Z be a rational polyhedral complex in a tropical toric variety Y .The integral p-multi-tangent space of Z is a cellular cosheaf F p of Z-modules on Z.For a face τ of Z such that int τ is contained in the stratum Y ρ we have For τ Ă σ, the maps of the cellular cosheaf i στ : F Z p pσq Ñ F Z p pτ q are induced by natural inclusions when intpσq and intpτ q are in the same stratum of Y .
Otherwise are induced by the quotients π ρη composed with inclusions when intpσq Ă Y ρ and intpτ q Ă Y η .
Example 2.11.Let Y be a toric variety.Consider the polyhedral structure on Y given by Y " Ť Y ρ induced by the toric stratification.One has Z codimρ , and the cosheaf maps are the maps induced by the projection maps π ρη defined in (2.1).
Example 2.12.Let H n Ă R n`1 denote the standard tropical hyperplane in R n`1 .Then H n is the tropical variety defined by the tropical polynomial function f px 1 , . . ., x n`1 q " maxt0, x 1 , . . ., x n`1 u.Its Newton polytope is the standard simplex in R n`1 .
The tropical hypersurface H n is a fan of dimension n, it has n `2 rays that are in the directions ´e1 , . . ., ´en`1 , and e 1 `¨¨¨`e n`1 .See the left hand side of Figure 2 for the standard hyperplane in R 3 .Every subset of the rays of size less than or equal to n spans a cone of H n .If v is the vertex of H n , then F Hn p pvq " Λ p Z n`1 , for 0 ď p ď n, and F Hn p pvq " 0 otherwise.Moreover, we have Example 2.13. Figure 3 shows a tropical line X contained in the tropical projective plane TP 2 from Example 2.1.The polyhedral structure on TP 2 induced by X has 7 vertices, 9 edges, and 3 faces of dimension 2.
For any face σ of this polyhedral structure on TP 2 , the rank of F TP 2 p pσq depends only on the dimension of the stratum of TP 2 which contains intpσq.If intpσq is contained in a stratum of TP 2 of dimension k then The directions of the rays of the fan for TP 2 are v 1 " p´1, 0q, v 2 " p0, ´1q, and v 3 " p1, 1q.
Referring to the labeling in Figure 3, we have and F X 1 pτ i q " 0. When p " 0, we have F X 0 pγq " Z for all γ in X and F X p pγq " 0 for all γ in X when p ě 2.
The following lemma about the structure of the cosheaves in the case of a non-singular tropical hypersurface will be useful later on.
Lemma 2.14.Let X be a non-singular tropical hypersurface in a tropical toric variety Y .If τ is a face of X of dimension q whose relative interior is contained in a stratum Y ρ of dimension m, then where H m´q´1 is the standard tropical hyperplane of dimension m ´q ´1 in R m´q and v denotes its vertex.
If τ is a codimension one face of σ in X and intpτ q and intpσq are contained in the distinct strata Y ρ and Y η , respectively, then the cosheaf map i στ : F X p pσq Ñ F X p pτ q together with the above isomorphisms commute with the map which is induced by the map id bπ ηρ on each factor of the direct sum, where π ηρ : Ź l T Z pσq Ñ Ź l T Z pτ q is from Equation 2.1.
Proof.Recall that T Z pτ q denotes the integral points in the tangent space of the face τ .Now let L be a m ´q dimensional affine subspace of R m -Y ρ defined over Z such that L intersects all faces of X ρ that contain intpτ q transversally and that together T Z pLq and T Z pτ q generate the lattice T Z pY ρ q.
By the above transversality assumption, the intersection L 1 " L X X has a single vertex v 1 contained in τ .
For every l there is a map given by taking the wedge product of the vectors in F L 1 p´l pvq and Ź l T Z pτ q.Taking the direct sum of the maps i l for all 0 ď l ď p gives a map p à l"0 If σ is a facet of X X Y ρ containing the face τ , then by our assumptions on L 1 , we have Therefore, Now since F X p pτ q is generated by all F X p pσq for σ a facet containing τ , the map in Equation 2.4 is an isomorphism.
By the assumption that X is non-singular and intersects the boundary of Y properly, every non-empty stratum X ρ " Y ρ X X is a non-singular tropical hypersurface in R m , where m " n `1 ´dim ρ.Therefore, the hypersurface X ρ is defined by a tropical polynomial f ρ and it is dual to a primitive regular subdivision of the Newton polytope of f ρ which is induced by f ρ .A face σ of X whose relative interior is contained in X ρ is dual to a face of the dual subdivision of ∆pf ρ q, and since this dual subdivision is primitive, the face dual to σ is a simplex.Therefore, near the vertex v 1 the polyhedral complex L 1 is up to an integral affine transformation the same as a neighborhood of the vertex v of the tropical hyperplane H m´q´1 and we have pvq.This proves the isomorphism stated in the lemma.
If τ is a face of σ, and τ and σ are contained in Y η and Y ρ respectively, then we can write T Z pY ρ q " T Z pL σ q ' T Z pσq and T Z pY η q " T Z pL τ q ' T Z pτ q, where L σ and L τ are the linear spaces chosen in the argument above to intersect σ and τ , respectively.Since the polyhedral structure on X is proper in Y , the map π ρη : T Z pY ρ q Ñ T Z pY η q restricts to an isomorphism between T Z pL σ q and T Z pL τ q.Therefore, it also restricts to an isomorphism between F LσXX p pv σ q and F Lτ XX p pv τ q for all p.The claim about the commutativity of the above isomorphisms with the maps in Equation (2.3) and i στ : F X p pσq Ñ F X p pτ q follows since i στ is induced by projecting along a direction π ρη .
Corollary 2.15.Let X be a non-singular tropical hypersurface of a tropical toric variety Y .Let σ be a face of X of dimension q whose relative interior is contained in stratum Y ρ of dimension m.Then the polynomial defined by is χ σ pλq " p1 ´λq m ´p1 ´λq q p´λq m´q .
Proof.Using the isomorphism in Lemma 2.14, together with the formula for the ranks of F H n´q p pvq from Example 2.12, we obtain χ σ pλq " p1 ´λq q rp1 ´λq m´q ´p´λq m´q s.
The statement of the corollary follows upon simplification.
In order to define the cellular tropical homology groups of a polyhedral complex Z we must first fix orientations of each of its cells.Let Z q denote the cells of dimension q of Z.We define an orientation map on pairs of cells, O : Z q ˆZq´1 Ñ t0, 1, ´1u by: 1 if the orientation of τ coincides with its orientation in Bσ, ´1 if the orientation of τ differs from its orientation in Bσ.
(2.5) Definition 2.16.Let Z be a polyhedral complex and G a cellular cosheaf on Z.The groups of cellular q-chains in Z with coefficients in G are Gpσq.
The boundary maps B : C q pZ; Gq Ñ C q´1 pZ; Gq are given by the direct sums of the cosheaf maps i στ for τ Ă σ composed with the orientation maps O στ for all τ and σ.The q-th homology group of G is H q pZ; Gq " H q pC ‚ pZ; Gqq.
Definition 2.17.Let Z be a polyhedral complex and G a cellular cosheaf on Z.The groups of Borel-Moore cellular q-chains in Z with coefficients in G are C BM q pZ; Gq " Gpσq.
The boundary maps B : C BM q pZ; Gq Ñ C BM q´1 pZ; Gq are given by the direct sums of the cosheaf maps i στ for τ Ă σ with the orientation maps O στ for all τ and σ.The q-th homology group of G is Definition 2.18.The pp, qq-th tropical homology group is H q pZ; F Z p q " H q pC ‚ pZ; F Z p qq. (2.6) The pp, qq-th Borel-Moore tropical homology group is Remark 2.19.Both the Borel-Moore and the standard tropical cellular homology groups of cosheaves are defined with respect to a fixed polyhedral structure.Let X be a hypersurface in a toric variety Y , and consider the polyhedral structure on X coming from the dual subdivision of its Newton polytope and the polyhedral structure on Y induced by X.When pY, Xq is a cellular pair in the sense of Definition 2.6 then the cellular homology groups from (2.6) of X or Y are isomorphic to singular tropical homology groups of X or Y , respectively [Cur14, Theorem 7.3.2].
On the other hand, even when pY, Xq is not a cellular pair, the Borel-Moore tropical cellular homology groups of X and Y are always isomorphic to the Borel-Moore singular homology groups of X and Y , respectively.In fact, one can always find a compactification of the pair pY, Xq such that pX, Xq, pY , Y q and pY , Xq are cellular pairs.The Borel-Moore homology groups of X are isomorphic to the relative homology groups of the pair pX, Xq, and similarly for Y and pY , Y q.
If G is a cellular sheaf on a polyhedral complex Z, then the group of q cochains and q cochains with compact support of G are respectively, C q pZ; Gq " à dim σ"q σ compact Gpσq and C q c pZ; Gq " The complex of cochains and cochains with compact support of G are formed from the cochain groups together with the restriction maps r τ σ combined with the orientation map O as in the case for a cosheaf.The cohomology groups of G are defined as the cohomology of these complexes.
Definition 2.20.Let Z be a polyhedral complex and G a cellular sheaf on Z.The cohomology groups and cohomology groups with compact support of G are respectively, H q pZ, Gq :" H q pC ‚ pZ; Gqq and H q c pX, Gq :" H q pC ‚ c pZ; Gqq.Remark 2.21.Since the multi-tangent modules are free Z-modules we have C q pZ; F p Z q " HompC q pZ; F Z p q, Zq and C q c pZ; F p Z q " HompC BM q pZ; F Z p q, Zq.Therefore for Z a non-singular tropical toric variety or a non-singular tropical hypersurface of a toric variety, the tropical cohomology groups and cohomology groups with compact support are respectively, H q pX, F p q :" H q pHompC ‚ pZ; F Z p q, Zqq and H q c pX, F p q :" H q pHompC BM ‚ pZ; F Z p q, Zqq.

Tropical Lefschetz hyperplane section theorem
A tropical hypersurface X in a tropical toric variety Y induces a polyhedral structure on Y .Unless it is explicitly mentioned we will use this polyhedral structure on Y to compute its cellular tropical homology groups.Following Remark 2.19, we obtain the same homology groups using this polyhedral structure as if we chose the polyhedral structure from the stratification of Y dual to the polyhedral fan defining it, see Example 2.11.Notice that if σ is a face of X whose relative interior is contained in Y ρ then we have To prove Theorems 1.1 and 1.2, we consider two exact sequences of cosheaves.The first is the exact sequence of cosheaves on Y given by, The second one consists of cosheaves on X and is given by, Since we consider the polyhedral structure on Y induced by X, each face σ of X is also a face of the polyhedral structure on Y and there is a Zmodule F Y p pσq.The cosheaf F Y p | X considered as a cosheaf on Y assigns the Z-module F Y p pσq when σ is a face of X and it assigns 0 if σ is a face of Y but not of X.When we consider F Y p | X as a cosheaf on X, then it simply assigns F Y p pσq for all σ P X Ă Y .The injective maps on the left hand side of both cosheaf sequences are both natural inclusions on the stalks over faces.The cosheaves Q p and N p are defined as the cokernel cosheaves in both short exact sequences.The cosheaves F Y p | X , F Y p , and F X p are all free Z-modules.Moreover, since X is a non-singular tropical hypersurface, the cosheaves Q p and N p are also cosheaves of free Z-modules.
Example 3.1.Consider again the tropical line X in TP 2 from Example 2.13 and Figure 3. Then the cosheaf Q p on TP 2 assigns the trivial Z-module to any face of TP 2 which is also a face of X.For σ a face of TP 2 and not a face of X, then Q p pσq " F TP 2 p pσq.The inclusion maps Q p pσq Ñ Q p pτ q are either 0 or equal to ι στ : F TP 2 p pσq Ñ F TP 2 p pτ q.For x the unique vertex of sedentarity 0 of X, the cosheaf N p assigns N p pxq " 0, for all p ă 2. When p " 2, we have N p pxq " Ź 2 Z 2 .For an edge τ i of X the Z-module N p pτ i q is a free module of rank 1, similarly for the three other vertices x i of X that have non-zero sedentarity.
To prove the Lefschetz section theorem for hypersurfaces, we prove some statements about the vanishing of both the standard and Borel-Moore homology with coefficients in Q p and with coefficients in N p .Proposition 3.2.Let X be a combinatorially ample non-singular tropical hypersurface of an n `1 dimensional non-singular tropical toric variety Y .Then H BM q pY ; Q p q " 0 for all q ă n `1, and therefore the map pY ; F Y p q is an isomorphism when q ă n and a surjection when q " n.
If in addition pY, Xq is a cellular pair, then H q pY ; Q p q " 0 for all q ă n `1, and therefore the map H q pX; F Y p | X q Ñ H q pY ; F Y p q is an isomorphism when q ă n and a surjection when q " n.
Proposition 3.3.Let X be a combinatorially ample non-singular n-dimensional tropical hypersurface in a non-singular toric variety Y .Then H BM q pX; N p q " 0 for all p `q ď n, and therefore the map q is an isomorphism when p `q ă n and a surjection when p `q " n.
If in addition pY, Xq is a cellular pair and the Newton polytope of X is full dimensional, then H q pX; N p q " 0 for all p `q ď n, and therefore the map H q pX; F X p q Ñ H q pX; F Y p | X q is an isomorphism when p `q ă n and a surjection when p `q " n.
We recall the definition of γ o for a face γ of X of dimension s and sedpγq " 0. For each cone ρ in the fan Σ defining Y , set γ ρ :" γ X Y ρ and define the polyhedral complex γ ˝:" If X is a hypersurface in R n`1 , then for every face γ of X the complex γ o consists of a single open cell intpγq.See Figure 4   Example 3.4.Let X be a tropical hypersurface in a 3-dimensional toric variety Y .We describe the polyhedral complexes γ o for some faces γ of X.If γ is a face of X which does not intersect any of the strata Y ρ for ρ ‰ 0 then γ ˝consists of a single cell which is simply intpγq.Therefore γ is combinatorially isomorphic to R q where q is the dimension of γ.Suppose that γ is a 2-dimensional face of X and γ X Y ρ ‰ H for some 1-dimensional stratum Y ρ .There must be two strata Y ρ 1 and Y ρ 2 of Y which contain Y ρ , moreover γ has non-empty intersection with both Y ρ 1 and Y ρ 2 .Therefore, γ o consists of four open cells and is combinatorially isomorphic to T 2 , see the left hand side of Figure 4.If γ is 2-dimensional and intersects only a single 2-dimensional stratum Y ρ , then γ o consists of two open cells and is combinatorially isomorphic to R ˆT.
Suppose γ is a 1-dimensional face of X of sedentarity 0 such that γ X Y ρ is non-empty for some stratum Y ρ of codimension 1.Such a situation is depicted on the right hand side of Figure 4. Then γ o consists of two open cells, the 1-dimensional cell γ 0 " γ X R 3 and the point γ ρ :" γ X R 2 ˆt´8u.The face γ t3u is the minimal face of γ 0 .Moreover, we have γ 0 -H 2 Ă R 3 and γ t3u Ă R 2 ˆt´8u is the boundary of X and is a combinatorially isomorphic to a tropical line.
Lemma 3.5.Let X be a non-singular and combinatorically ample tropical hypersurface in a non-singular tropical toric variety Y .Then for every face γ of X the polyhedral complex γ o has a unique minimal face.
Proof.We can suppose without loss of generality that γ is of sedentarity 0. Let Σ be the fan defining Y .Suppose that ρ and ρ 1 are cones of Σ such that γ X Y ρ ‰ H and γ X Y ρ 1 ‰ H.We will show that there exists a cone η of Σ such that γ X Y η ‰ H and η contains ρ and ρ 1 .Since there are a finite number of faces in γ o , it will follow that there is a unique minimal face.
We consider the polyhedral structure on Y induced by X.Then there is a face γ of Y of dimension n `1 such that γ is a face of γ.Since X intersects the boundary of Y properly, we have γo X Y ρ , γo X Y ρ 1 ‰ H.Moreover, since X is combinatorially ample in Y , the face γo is combinatorially isomorphic to R r ˆTn`1´r for some r.Therefore, there exists a cone η of Σ such that ρ and ρ 1 are faces of η and γo X Y η ‰ H. Then γ X Y η is also non-empty since the recession cone of γ intersects the cone η, for example the faces ρ and ρ 1 are contained in the intersection.This completes the proof.
Before the proof of Proposition 3.2 we state a useful lemma that will be used throughout this section.By [JRS17], the singular Borel-Moore homology groups of γ o with coefficients in F γ o p vanish except in degree dimpγq.Combining this result with Remark 2.19 we obtain the next lemma which asserts the analogous statement for the cellular Borel-Moore homology groups of γ o .Lemma 3.6.[JRS17, Proposition 5.5] Let X be a non-singular and combinatorially ample tropical hypersurface of an n `1 dimensional non-singular tropical toric variety Y .Consider the polyhedral structure on Y obtained by refinement by X.Let γ be a face of Y of sedentarity 0. Then for any p and all q ‰ dim γ Proof of Proposition 3.2.We consider the polyhedral structure on Y given by refinement by X.For any face σ of Y which is also a face of X we have Q p pσq " 0. Therefore we have the following isomorphism of cellular chain complexes, When pY, Xq is a cellular pair, the cellular chain groups compute the standard homology by Remark 2.19 and we also have the isomorphism The complement Y zX consists of connected components each of dimension n `1.Each such connected component is equal to γ o where γ is a n `1 dimensional face of Y with polyhedral structure induced by X.For γ a face of Y of dimension n `1, there is the equality of cosheaves Moreover, the boundary of the face σ contained in γ o is also contained in γ o .Therefore, the cellular chain complexes for Q p split and we have the following isomorphisms, This produces the isomorphisms It follows from Lemma 3.6 that H BM q pγ o ; F γ o p q " 0 if q ‰ n `1, and we obtain that H q pY ; Q p q " H BM q pY ; Q p q " 0 for all q ă n `1.A direct comparison of the respective chain complexes gives isomorphisms Lastly, combining this with the long exact sequence in homology associated to the short exact sequence (3.1) and the vanishing of H BM q pY ; Q p q and H q pY ; Q p q for all q ă n `1 proves the statement of the proposition.
To prove the statement about the vanishing of the homology of the cosheaf N p from Proposition 3.3 we first establish a sequence of lemmas.
Lemma 3.7.Let X be a n-dimensional non-singular tropical hypersurface in a toric variety Y .For σ a face of X of dimension q and sedentarity sedpσq, we have N p pσq " 0 if p ď n ´q ´sedpσq.
Proof.The Z-modules N p pσq, F Y p | X pσq, and F X p are all free, so it suffices to show that the ranks of F Y p | X pσq and F X p are equal when p ď n ´q ´sedpσq.
So that rank F X p pσq " `n`1´sedpσq p ˘if p ď n´q´sedpσq.Therefore N p pσq " 0 when p ď n ´q ´sedpσq, and the proof is completed.
Lemma 3.8.Let X be a combinatorially ample n-dimensional non-singular tropical hypersurface in a toric variety Y .For a face γ of X of sedentarity 0 we have Proof.Denote by γ m the unique minimal face of γ ˝and suppose it is contained in the stratum Y ρm .Let Γ denote the star of γ m in X ρm , that is, Then as a polyhedral complex Γ Ă R n`1´sedpγmq is, up to GL n`1´sedpγmq pZq, equal to a basic open set of Γ 1 ˆRdim γm where Γ 1 " H n´dim γm is the standard tropical hyperplane in R n`1´dim γm .Again, for the notion of basic open set see [JSS15, Definition 3.7].
Moreover, the star of any other face γ ρ in γ o is, up to GL n`1´sedpγmq pZq, equal to a basic open set of Γ 1 ˆRdim γρ .Let v be the vertex of Γ 1 , then by Lemma 2.14, for any face γ ρ of γ o we have This isomorphism follows from the tensor product formula for the Z-module F X p pγ ρ q in Lemma 2.14.
For each l from 0 to p, let C l,p ‚ denote the chain complex whose terms are We define the boundary maps of the complex on the direct summands.If γ ρ 1 is a face of γ ρ then the map on the direct summand is Following the description of the cosheaf maps from Lemma 2.14, there are isomorphisms of chain complexes By distributivity of tensor products we also have the isomorphisms Moreover, the homology of the chain complex C BM ‚ pγ o , F γ o l q vanishes except in degree q " dim γ by Lemma 3.6, so we also have H BM q pγ o , F γ o l q " 0 for all q ă dim γ.Because the tensor product is right exact, we have H q pC p,l ‚ q " 0 for q ă dim γ and all l and p.It now follows that H BM q pγ ˝, F X p | γ ˝q " 0 for q ă dim γ.Lemma 3.9.Let X be a combinatorially ample n-dimensional non-singular tropical hypersurface in a toric variety Y .For a face γ of X of sedentarity 0 we have Proof.To prove the vanishing of the homology of the cosheaf N p | γ ˝we return to the short exact sequence from (3.2) but restricted to γ o , namely Consider the polyhedral complex γ o alone in Y .Now let γ be a polyhedron in Y of dimension n `1 obtained by taking an arbitrarily small polyhedral neighborhood of γ in Y .Notice that for every face σ of γ o of dimension q there is a face σ of γo of dimension n `1 ´dim γ `q.Moreover, for every face σ of γ o and its corresponding face σ of γ we have F Y p pσq -F γ p pσq.Therefore, the cellular chain complex C BM ‚ pγ ˝; F Y p | γ ˝q is isomorphic to the chain complex C BM ‚´n´1`dim γ pγ ˝; F γ p q.By Lemma 3.6, we have pγ ˝; F γ p q " 0 for q ‰ n `1.This implies that H BM q pγ ˝; F Y p | γ ˝q " 0 for q ă dim γ.By Lemma 3.8 we have H BM q pγ ˝; F X p | γ ˝q " 0 for q ă dim γ.By considering the long exact sequence in homology from the sequence (3.2) restricted to γ o proves that H q pγ o ; N p | γ o q " 0 for all q ă dim γ.
Proof of Proposition 3.3.Using Lemma 3.7, we have the following description of the Borel-Moore cellular chain groups with coefficients in N p , C BM q pX; N p q :" n à m"maxtq,n´p`1u à dim σ"q sedpσq"m´q N p pσq. (3.7) If in addition pY, Xq is a cellular pair, by Remark 2.19 the cellular chain complexes compute the standard homology of X and we also have the isomorphism C q pX; N p q :" n à m"maxtq,n´p`1u à dim σ"q sedpσq"m´q σ compact N p pσq. (3.8) Notice that in the two sums in Equations 3.7 and 3.8 there are no faces of X of dimension q and which have order of sedentarity strictly greater than n ´q.
We now filter the cellular chain complex for N p using the order of sedentarity of faces.Set, C BM q,m pX; N p q :" à dim σ"q sedpσqďm´q N p pσq and C q,m pX; N p q :" à dim σ"q sedpσqďm´q σ compact N p pσq Since X intersects the boundary of Y properly, the boundary operator can only increase the order of sedentarity by at most 1.Therefore, BC ‚ q,m pX; N p q Ă C ‚ q´1,m pX; N p q, where the ‚ in the exponent denotes either Borel-Moore or standard homology.Then there is a filtration of the chain complex C ‚ ‚ pX; N p q: C ‚ ‚ pX; N p q " C ‚ ‚,n pX; N p q Ą C ‚ ‚,n´1 pX; N p q Ą ¨¨¨Ą C ‚ ‚,n´p`1 pX; N p q.The spectral sequence associated to this filtration for the Borel-Moore complex has first page consisting of the terms E 1 q,m -à dim σ"q sedpσq"m´q N p pσq. (3.9) The differentials B 1 : E 1 q,m Ñ E 1 q´1,m are induced by the usual cellular differentials.Notice that the terms E 1 q,m are only non-zero when m satisfies n ´p `1 ď m ď n.In general the differentials are B r : E r q,m Ñ E r q´1,m`r´1 .
Let t ďn´p`1 E 1 ‚,m be the brutal truncation of the complex E 1 ‚,m in dimension n ´p `1.This is the complex (3.10)Moreover, we have H q pt ďn´p`1 E 1 ‚,s q -H q pE 1 ‚,m q for q ď n ´p.We claim that the complex in (3.10) is isomorphic to the brutal truncation of the direct sum of complexes: To see this fact, notice that if σ is a face of dimension q and sedpσq " m ´q, then there is a unique face γ of X of dimension q and sedentarity 0 which contains σ.Using the isomorphism in (3.9), when q ď n ´p `1 and maxtq, n ´p `1u ď m ď n we have the splitting of vector spaces The differentials B 1 of the first page of the spectral sequence are compatible with the above splitting of the vector spaces E 1 q,m .By Lemma 3.9, for a face γ of dimension s and sedentarity 0, we have H BM q pγ ˝; N p | γ ˝q " 0 for q ă s.Therefore, the second page of the spectral sequence associated to the filtration under consideration satisfies E 2 q,m " 0 if q ‰ m and q ď n ´p.Since E 1 q,m " 0 if s ď n ´p `1, we conclude that E 2 q,m " 0 for q ď n ´p and any 0 ď m ď n.Therefore, the spectral sequence E ‚ ‚,‚ satisfies E r q,m " 0 for any r ě 2 and q ď n ´p.Since E ‚ ‚,‚ converges, we conclude that H BM q pX; N p q " 0 for p `q ď n.To obtain the analogous statement for H q pX; N p q, consider the spectral sequence associated to the filtration of the chain complex for the standard homology.The first page of this spectral sequence has terms like in Equation (3.9), except that the sum is taken over the faces σ which are compact.In order to proceed with the same argument as for standard homology, we require that if σ is compact then the unique face γ of X of sedentarity 0 which contains σ is also compact.This is guaranteed by the assumption that the Newton polytope of X is full dimensional.Then the rest of the argument is the same as in the case of the Borel-Moore homology except we restrict to only compact faces of X.
To complete the proof of the proposition, consider the long exact sequence in homology associated to the short exact sequence in (3.2).Applying the vanishing statements for H BM q pX; N p q gives the isomorphisms H BM q pX; F X p q -H BM q pY ; F Y p q for all p `q ă n.This completes the proof.
Proof of Theorem 1.1.The proof of the theorem follows by combining the statements in Propositions 3.2 and 3.3.
We now present the proof of the Lefschetz section theorem for the tropical homology groups with real coefficients of singular tropical hypersurfaces intersecting the boundary of a non-singular tropical toric variety transversally.
Proof of Theorem 1.2.The proof follows the same strategy as the proof of the Lefschetz theorems for the integral tropical homology groups.First we tensor the two exact sequences of Z-module cosheaves from (3.1) and (3.2) with R to obtain two exact sequences of cosheaves of R-vector spaces.Proposition 3.2 holds over R, since the tropical homology with real coefficients of basic open subsets of tropical manifolds satisfies the vanishing theorems used to prove the proposition in the integral case by [JSS15, Section 4].
We claim that a variant of Proposition 3.3 holds for the cosheaf N p b R. In order to prove this we describe the dimensions of the vector spaces F p pσq when X is a tropical singular hypersurface.Consider the polyhedral decomposition of Y induced by X, and let v be a vertex of X of sedentarity 0. Then v is contained in some n `1 dimensional face γ of this polyhedral decomposition of Y .For p ď n we have where in the last sum the faces of σ are faces of X.For any p ď n we have F X p pvq b R " Ź p R n`1 , and for p ą n we have F X p pvq b R " 0. Therefore, we have χ R v pλq :" n ÿ p"0 p´1q p dimpF X p pvq b Rqλ p " p1 ´λq n`1 ´p´λq n`1 .
We can repeat the above argument for vertices of X of non-zero sedentarity and also apply the same argument for the Künneth type formula from Lemma 2.14.Therefore, if τ is a face of X of dimension q whose relative interior is contained in a stratum Y ρ of dimension m, we obtain so that, as in Corollary 2.15, we have χ τ pλq " p1 ´λq s ´p1 ´λq q p´λq m´q .
This description enables us to conclude that Lemma 3.7 holds for N p b R.
Similarly the proofs of Lemmas 3.8 and 3.9 as well as Proposition 3.3 hold for a singular hypersurface when using R coefficients.Then the proof of the theorem is completed in the same way as the proof of Theorem 1.1.

The tropical homology of hypersurfaces is torsion free
We start this section with the proof of Theorem 1.3, which using the Lefschetz section theorem for the integral homology of a non-singular tropical hypersurface.This proposition establishes that the integral tropical homology groups of the hypersurface are also torsion free if the integral tropical homology groups of the toric variety are as well.
Proof of Theorem 1.3.Let X be a non-singular tropical hypersurface of a tropical toric variety Y such that the standard tropical homology of Y is torsion free.Suppose that p `q ě n, then by the universal coefficient theorem for cohomology [Hat02, Theorem 3.2] for every p and q we have the following short exact sequence: 0 Ñ ExtpH n´q´1 pX; F X n´p q, Zq Ñ H n´q pX; F n´p X q Ñ HompH n´q pX; F X n´p q, Zq Ñ 0.
If p `q ě n, then ExtpH n´q´1 pX; F X n´p q, Zq " 0 since and H n´q´1 pY ; F Y n´p q is a free Z-module by hypothesis.Also the Z-module HompH n´q pX; F X n´p q, Zq is free since it consists of maps to a free module.Therefore, for all p `q ě n we have H n´q pX; F n´p X q -HompH n´q pX; F X n´p q, Zq and H n´q pX; F n´p X q is torsion free.The tropical hypersurface X is a nonsingular tropical manifold, so by Poincaré duality for tropical homology with integral coefficients from [JRS17] we have for all p, q.This combined with the above argument proves that H BM q pX; F X p q is torsion free for all p and q.By the universal coefficient theorem for cohomology with compact support, for every p and q we have the following 0 Ñ ExtpH BM q´1 pX; F X p q, Zq Ñ H q c pX; F p X q Ñ HompH BM q pX; F X p q, Zq Ñ 0.
Since the Z-modules H BM q pX; F X p q are torsion free for all p and q, the Zmodules H q c pX; F p X q are also torsion free for all p and q.Applying again Poincaré duality, we have H q c pX; F p X q -H n´q pX; F X n´p q, and H q pX; F X p q are also torsion free for all p and q.
We now establish that the integral tropical homology groups of a compact tropical toric variety are torsion free.For a non-singular compact complex toric variety Y C , we let h p,q pY C q denote its pp, qq-th Hodge number.Recall that h p,q pY C q " 0 if p ‰ q and the numbers h p,p pY C q form the toric h-vector of the simple polytope ∆ whose normal fan is the fan defining Y C [Ful93, Section 5.2].
Proposition 4.1.The integral tropical homology groups of a non-singular compact tropical toric variety Y are torsion free.Moreover, we have rank H q pY ; F Y p q " h p,q pY C q where Y C is the corresponding non-singular compact complex toric variety.In particular, we have H q pY ; F Y p q " 0 unless p " q.
Proof.We now switch to using the cellular homology groups of Y using the polyhedral structure on Y which is dual to the polyhedral structure on the defining fan Σ.Notice that every stratum Y σ is compact.Let us first show that H q pY ; F Y p q " 0 for all p ą q.With this cellular structure on Y , a face Y σ of dimension q has sedentarity order n `1 ´q where dim Y " n `1.By Definition 2.10, we have that F Y p pY σ q " Ź p F Y 1 pY σ q where dim F Y 1 pY σ q " q.Therefore, we have F Y p pY σ q " 0 if p ą q.Hence the chain groups C q pY ; F Y p q are equal to zero for any q ă p, which implies that H q pY ; F Y p q " 0 for q ă p. Recall by Remark 2.21 that the tropical cohomology groups are the cohomology of the complex dual to the tropical cellular complexes.Therefore we can apply the universal coefficient theorem for cohomology [Hat02, Theorem 3.2] to get the exact sequence 0 Ñ ExtpH q pY ; F Y p q, Zq Ñ H q`1 pY ; F p Y q Ñ HompH q`1 pY ; F Y p q, Zq Ñ 0. (4.1) When q ă p we have H q pY ; F Y p q " 0, so there is the isomorphism H q`1 pY ; F p Y q -HompH q`1 pY ; F Y p q, Zq.The tropical toric variety Y is a tropical manifold, thus Poincaré duality for tropical homology with integral coefficients from [JRS17] states that H q`1 pY ; F p Y q -H n´q pY ; F Y n`1´p q.If q ě p, then n ´q ă n `1 ´p and applying the isomorphism above we obtain H q`1 pY ; F p Y q " H n´q pY ; F Y n`1´p q " 0. This means that TorpH q pY ; F Y p qq " ExtpH q pY ; F Y p q, Zq " 0, and so H q pY ; F Y p q is torsion-free for all q ě p and thus for all p, q.We also see from the sequence in (4.1) that H q pY ; F Y p q " 0 for all q ‰ p.All of the chain groups for the cellular tropical homology of Y are also free so we have χpC ‚ pY ; F Y p qq :" n`1 ÿ q"0 p´1q q rank C q pY ; F Y p q " p´1q p rank H p pY ; F Y p q.
Let f q denote the number of strata of Y of dimension q.Then pf 0 , . . ., f n`1 q is the f -vector of a polytope P Y whose normal fan is the fan defining Y .Then for every p and q we have rank C q pY ; F Y p q " `q p ˘fq .Therefore, χpC ‚ pY ; F Y p qq :" n`1 ÿ q"0 p´1q q ˆq p ˙fq " p´1q p h p , where ph 0 , . . ., h n`1 q is the h-vector of the simple polytope P Y .By [Ful93, Section 5.2], we have h p " dim H 2p pY C q " h p,p pY C q which completes the proof.
Proof of Corollary 1.4.By Proposition 4.1, if Y is compact, all its integral tropical homology groups are torsion free.Then by Theorem 1.3, all the integral tropical homology groups of X are torsion free.
Proof of Corollary 1.5.Assume that the convex cone supporting the fan of Y is full dimensional in R n`1 .We will first show that the tropical toric variety Y equipped with the polyhedral structure dual to the polyhedral structure on its defining fan is a regular CW-complex.Thus the cellular tropical chain complexes can compute the standard and Borel-Moore homology groups of Y .To prove this claim, consider Y C , the quasi-projective toric variety associated to a fan Σ.Let D be any ample Cartier divisor on Y C and consider the associated polyhedron P (see for example [Ful93, Chapter 3]).The hypothesis on the support of Σ implies that it is the normal fan of P ([Mus04, Chapter 6]).Therefore, the polyhedron P is combinatorially isomorphic to Y , the tropical toric variety associated to Σ. Since P is a polyhedron, it is a cell-complex in the sense of [Cur14, Chapter 4], and one can use the cellular description to compute the standard homology groups of Y .As in the proof of Proposition 4.1, both standard and Borel-Moore tropical homology groups of Y vanishes if p ą q.It follows again from Poincaré duality and universal coefficient theorem that both standard and Borel-Moore tropical homology groups of Y are torsion free.The statement for X follows again from Theorem 1.1.Now suppose that the convex cone supporting the fan Y is of codimension s in R n`1 .Then the tropical toric variety Y is a product R s ˆY 1 where Y 1 is a tropical toric variety of dimension n `1 ´s satisfying the assumptions above.The tropical toric variety Y 1 is then combinatorially isomorphic to a polyhedron P 1 .By the Künneth formula for Borel-Moore tropical homology [GS, Theorem B] we have H BM q pY ; F Y p q " à i`j"p k`l"q H BM k pR s ; F R s i q b H BM l pY 1 ; F Y 1 j q.
Therefore, the Borel-Moore tropical homology groups of Y are all torsion free and thus so are the standard tropical homology groups.This completes the proof.

Betti numbers of tropical homology and Hodge numbers
The k-compactly supported cohomology group of a complex hypersurface X C Ă pC ˚qn`1 carries a mixed Hodge structure, see [DK86].The numbers e p,q c pX C q are defined to be e p,q c pX C q :" ÿ k p´1q k h p,q pH k c pX C qq, where h p,q pH k c pX C qq denote the Hodge-Deligne numbers of X C .The numbers e p,q c pX C q are the coefficients of the E-polynomial of X C , EpX C ; u, vq :" ÿ p,q e p,q c pX C qu p v q .
the Euler characteristic of the Borel-Moore complexes and the χ y genus are both additive.Therefore, we obtain rank H BM p pY ; F Y p q " h p,p pH 2p c pY C qq.Notice that since Y C is affine, the Andreotti-Frankel theorem imply that h p,p pH 2p c pY C qq " 0 if 2p ă n, and thus rank H BM p pY ; F Y p q " 0 if 2p ă n.Combining the tropical Lefschetz theorem and Poincaré duality, we obtain that if p `q ‰ n rank H BM q pX; F X p q " # rank H BM p`1 pY ; F Y p`1 q if p " q ą n 2 , 0 otherwise.
Since X C is affine, one has again that h p,q pH k c pX C qq " 0 if k ă n.By the Lefschetz-type theorems for the Hodge Deligne numbers on H n c pX C q [DK86, Section 3], we get h p,q pH k c pX C qq " 0 if k ą n and p ‰ q and that if 2p ą n h p,p pH 2p c pX C qq " h p`1,p`1 pH 2p`2 c pY C qq.
Therefore, e p,q c pX C q " $ ' & ' % p´1q n h p,q pH n c pX C qq if p `q ď n h p`1,p`1 pH 2p`2 c pY C qq if p " q ą n 2 0 otherwise.
Then by applying Theorem 1.8 and using the fact that the Borel-Moore tropical homology groups of X are torsion free by Corollary 1.5, we obtain the statement of corollary.
Proof of Corollary 1.11.The proof follows exactly the same lines as the proof of Corollary 1.10.It follows from [DK86] that h p,q pH k c ppC ˚qn`1 qq " $ & % ˆn `1 p ˙if p " q and k " n `1 `p 0 otherwise .
The Borel-Moore tropical homology groups satisfy H BM q pR n`1 ; F p q " 0 if q ‰ n `1 and rank H BM n`1 pR n`1 ; F p q " ˆn `1 p ˙.
Combining Theorem 1.1 and Poincaré duality for the tropical homology of X, when p `q ‰ n we have rank H BM q pX; F X p q " # `n`1 p`1 ˘if q " n, 0 if q ‰ n.
The hypersurface X C is a non-singular affine variety, so the Andreotti-Frankel theorem and Poincaré duality imply H k c pX C q " 0 if k ă n.By the Lefschetz-type theorems for the Hodge Deligne numbers on H n c pX C q [DK86, Section 3], if k ą n one has h p,q pH k c pX C qq " f p " q and k " n `p 0 otherwise.Therefore e p,q c pX C q " $ ' ' ' ' ' ' & ' ' ' ' ' ' % p´1q n h p,q pH n c pX C qq if p `q ď n and p ‰ q p´1q n h p,q pH n c pX C qq `p´1q n`p ˆn `1 p `1˙i f p `q ď n and p " q p´1q n`p ˆn `1 p `1˙i f p `q ą n and p " q 0 otherwise.
Then by applying Theorem 1.8 and using the fact that the Borel-Moore tropical homology groups of X are torsion free by Corollary 1.6, we obtain the statement of corollary.
tropical homology of hypersurfaces is torsion free 24 5. Betti numbers of tropical homology and Hodge numbers 27

Figure 1 .
Figure 1.The tropical projective plane TP 2 on the left and its normal fan on the right.

Figure 2 .
Figure 2. The standard tropical hyperplane in R 3 on the left its closure in the tropical toric variety described in Example 2.5 on the right.

Figure 3 .
Figure 3.The tropical line X in TP 2 from Example 2.13.
and Example 3.4 for illustrations of γ o .

Figure 4 .
Figure 4.A depiction of the polyhedral complexes γ o for two faces γ from Example 3.4