2.1 Polytopes and toric varieties

Let $M \cong \mathbb {Z}^{n+1}$ be a lattice and let $N = \operatorname {\mathrm {Hom}}(M, \mathbb {Z})$ be its dual lattice. We define $M_{\mathbb {R}} = M \otimes \mathbb {R}$ and $N_{\mathbb {R}} = N \otimes \mathbb {R}$ . A lattice polytope $\Delta $ in $M_{\mathbb {R}}$ is the convex hull of a finite number of points of M. In this paper, we will always assume that $\Delta $ is full dimensional. We will denote by greek letters ( $\sigma $ , $\tau $ , etc.) the faces of $\Delta $ . The normal fan $\Sigma _{\Delta }$ of $\Delta $ is the collection of convex rational cones $\{ C_{\sigma }\}_{\sigma \preceq \Delta }$ in $N_{\mathbb {R}}$ given by

$$\begin{align*}C_{\sigma} = \{ v \in N_{\mathbb{R}} \, | \, \langle {v}, {w-u} \rangle \geq 0, \quad \forall u \in \sigma \ \text{and} \ w \in \Delta \}. \end{align*}$$

We will work with complex, real and tropical toric varieties. Let us briefly recall how they are constructed. Let $\Sigma $ be a rational strongly convex polyhedral fan in $N_{\mathbb {R}}$ . For any cone $\sigma $ of $\Sigma $ , consider the dual cone

$$ \begin{align*}\sigma^{\vee} = \{ u \in M_{\mathbb{R}} \, | \, \langle {u}, {v} \rangle \geq 0, \quad \forall v \in \sigma\}. \end{align*} $$

By Gordon’s Lemma, the intersection $S_\sigma =\sigma ^{\vee }\cap M$ is a finitely generated semigroup. So for any semigroup S, one can consider the set $X_\sigma ^S:=\operatorname {\mathrm {Hom}}(S_\sigma ,S)$ of semigroup homomorphisms. If moreover the semigroup S has a topology, we equipp the set $X_\sigma ^S$ with the coarsest topology such that for any $\lambda \in S_\sigma $ , the corresponding evaluation map $X_\sigma ^S\rightarrow S$ is continuous. In fact, it is enough to do it for a set of generators of $S_\sigma $ (say of cardinality N), and the topology coincides with the induced topology from $S^N$ . If $\sigma \subset \tau $ , there is a continuous restriction map from $X_\sigma ^S$ to $X_\tau ^S$ .

Definition 2.1. The toric variety associated to the fan $\Sigma $ over the semigroup S is the direct limit

$$ \begin{align*}S\Sigma := \varinjlim X_\sigma^S. \end{align*} $$

Classically, the most common semigroups are either $\mathbb {R}$ , $\mathbb {C}$ or $\mathbb {R}_+$ (for multiplication). We will also consider the tropical semigroup $\mathbb {T}:=\mathbb {R}\cup {-\infty }$ with the tropical multiplication being the addition. The neutral element $1_{\mathbb {T}}$ is equal to $0$ , while the zero element $0_{\mathbb {T}}$ is $-\infty $ . All these semigroups are equipped with the euclidean topology. The torus $N\otimes S^*$ is open in $S\Sigma $ and acts on each $X^S_\sigma $ , and this action extends to all of $S\Sigma $ . The orbits are in correspondence with the cones of $\Sigma $ via the map $\sigma \rightarrow \mathcal {O}^S_\sigma $ , where $\mathcal {O}^S_\sigma $ is the orbit of the point $x_\sigma \in X^S_\sigma $ defined by

$$ \begin{align*}x_\sigma(m)=\left\lbrace \begin{array}{@{}c} 1_S \text{ if } -m\in S_\sigma, \\ 0_S \text{ otherwise }.\end{array}\right.\end{align*} $$

For $S=\mathbb {C}$ , we obtain the classical construction of complex toric varieties; see Section 1.3 of [Reference Fulton13]. If $\Sigma =\Sigma _\Delta $ is the normal fan of a lattice polytope, the variety $\mathbb {C}\Sigma $ is projective. A fan $\Sigma $ is said to be regular if every cone can be generated by a subset of a basis of N. Then the toric variety $\mathbb {C}\Sigma $ is smooth if and only if $\Sigma $ is regular. A polytope in $M_{\mathbb {R}}$ is called non-singular if its normal fan is regular. We say $\Sigma $ is complete if its support $|\Sigma |$ (i.e., the union of its cones) is equal to $N_{\mathbb {R}}$ . Completeness is necessary and sufficient for the toric variety $\mathbb {C} \Sigma $ to be compact. For any fan $\Sigma $ , there is a natural map from $\mathbb {C}\Sigma $ to $\mathbb {R}_+\Sigma $ induced by the absolute value from $\mathbb {C}$ to $\mathbb {R}_+$ (composition with the logarithm will identify $\mathbb {R}_+\Sigma $ with $\mathbb {T}\Sigma $ ). In the case where $\Sigma =\Sigma _\Delta $ , one has $\mathbb {R}_+\Sigma _{\Delta }=\Delta $ , and the map $\mathbb {C}\Sigma _{\Delta }$ to $\Delta $ is the moment map. In general, the real toric variety $\mathbb {R} \Sigma $ can be reconstructed combinatorially directly form the fan $\Sigma $ as follows (see, for example, [Reference Bihan, Franz, McCrory and van Bihan6]). For any group homomorphism $\xi :M\rightarrow \mathbb {F}_2$ , consider $\mathbb {R}_+\Sigma (\xi )$ a copy of $\mathbb {R}_+\Sigma $ . Then

(3) $$ \begin{align} \mathbb{R} \Sigma\simeq \bigsqcup_{\xi\in \operatorname{\mathrm{Hom}}(M,\mathbb{F}_2)}\mathbb{R}_+\Sigma(\xi) / \sim, \end{align} $$

where $(p,\xi )\sim (p',\xi ')$ if and only if $p=p'$ and $\xi \vert _{\sigma ^{\perp }}=\xi '\vert _{\sigma ^{\perp }}$ for the unique cone $\sigma $ such that $p\in \mathcal {O}^{\mathbb {R}_+}_\sigma $ .

2.2 Reflexive polytopes and mirror symmetry

A lattice polytope $\Delta $ is called reflexive if it is defined by a system of inequalities $ \langle {v}, {u} \rangle \leq 1$ with $v\in N$ and $u\in M_{\mathbb {R}}$ . In particular, it implies that $\operatorname {\mathrm {Int}}(\Delta ) \cap M = \{ 0 \}$ (i.e., $0$ is the unique interior lattice point of $\Delta $ ). The dual polytope $ \Delta ^{\circ } \subset N_{\mathbb {R}}$ is defined as

$$\begin{align*}\Delta^{\circ} = \{ u \in N_{\mathbb{R}} \, | \, \langle {u}, {v} \rangle \leq 1, \quad \forall v \in \Delta \}. \end{align*}$$

If $\Delta $ is reflexive, then $\Delta ^{\circ }$ is a lattice polytope which is also reflexive and $(\Delta ^{\circ })^\circ = \Delta $ .

A reflexive polytope $\Delta \subset M_{\mathbb {R}}$ defines another fan, in the same space $M_{\mathbb {R}}$ called the face fan of $\Delta $ denoted by $\Xi _{\Delta }$ whose cones are the cones over the faces of $\Delta $ . We have the following relations:

$$\begin{align*}\Xi_{\Delta} = \Sigma_{\Delta^{\circ}} \quad \text{and} \quad \Xi_{\Delta^{\circ}} = \Sigma_{\Delta}.\end{align*}$$

If $\Delta $ is a non-singular reflexive polytope, then the cones of $\Xi _{\Delta ^{\circ }}$ are regular. So all faces of $\Delta ^{\circ }$ are simplices with no integer points in the interior. Moreover, if $\sigma $ is a facet of $\Delta ^{\circ }$ , then the convex hull of the origin in N with $\sigma $ is a primitive simplex.

Given a pair of dual polytopes $\Delta $ and $\Delta ^{\circ }$ , we can consider anticanonical subvarieties X and $X^{\circ }$ inside $\mathbb {C}\Sigma _{\Delta }$ and $\mathbb {C}\Sigma _{\Delta ^{\circ }}$ , respectively. These are both Calabi-Yau varieties, and they constitute a so called mirror pair. Typically these varieties, as well as the ambient toric varieties, will be singular. Sometimes singularities can be resolved in the realm of Calabi-Yau varieties by a crepant resolution. The easiest way to do this, if possible, is to consider primitive convex central subdivisions.

Let T be a primitive central triangulation of $\Delta $ . Denote by $\Sigma _T$ the regular fan constructed by taking the cones from the origin of M over the simplices in the subdivision of $\Delta $ . This fan is a refinement of the fan $\Xi _{\Delta }=\Sigma _{\Delta ^{\circ }}$ . So it defines a desingularization $ \mathbb {C}\Sigma _{T}$ of $\mathbb {C}\Sigma _{\Delta ^{\circ }}$ .

In this article, we assume there exist primitive triangulations T and $T^{\circ }$ of $\Delta $ and $\Delta ^{\circ }$ , respectively, we do not always assume they are convex. In this case, generic anti-canonical hypersurfaces in $\mathbb {C}\Sigma _{T^{\circ }}$ or $\mathbb {C}\Sigma _{T}$ are smooth Calabi-Yau varieties. From now on, we will denote by X and $X^{\circ }$ smooth anticanonical hypersurfaces in $\mathbb {C}\Sigma _{T^{\circ }}$ and $\mathbb {C}\Sigma _{T}$ , respectively, instead of the singular ones in $\mathbb {C}\Sigma _{\Delta }$ and $\mathbb {C}\Sigma _{\Delta ^{\circ }}$ . When the triangulations are not convex, these may be non-projective. In the projective case (i.e., when T and $T^{\circ }$ are both convex), Batyrev and Borisov [Reference Batyrev and Borisov3] proved that the Hodge numbers of X and $X^{\circ }$ satisfy the following mirror identity (see also [Reference Cox and Katz12]):

$$\begin{align*}h^{p,q}(X) = h^{n-p,q}(X^{\circ}). \end{align*}$$

This relation is in fact proved in Theorem 4.15 of [Reference Batyrev and Borisov3] in the case where the Hodge numbers are replaced by the so called stringy-Hodge numbers, which only depend on the singular Calabi-Yau. Then, Proposition 1.1 of [Reference Batyrev and Borisov3] states that the stringy Hodge numbers coincide with the Hodge numbers of the smooth, projective crepant resolutions given by T and $T^{\circ }$ . We expect that the mirror identity of Hodge numbers should also hold in the non-projective case, although we could not find a proof of this in the literature.

2.3 Combinatorial mirror pairs

Let us introduce first some important notations we will use all along the paper. Given a simplex $\sigma \in T$ ,

• • denote by $C(\sigma ) \in \Sigma _{T}$ the cone over $\sigma $ ;

• • denote by $\sigma ^{\perp }$ the subspace of N orthogonal to the integral tangent space of $\sigma $ ;

• • if $\sigma \neq 0$ , let

  $$ \begin{align*}\sigma_{\infty} = \sigma \cap \partial \Delta.\end{align*} $$

Given a cone $\rho \in \Sigma _{T}$ ,

• • denote by $S(\rho )$ the simplex $\rho \cap \Delta $ ;

• • denote by $\hat {\sigma }$ the simplex $S(C(\sigma ))$ (i.e., the convex hull of $0$ and $\sigma $ ).

In particular, when $0 \in \sigma $ , then $\hat {\sigma }=\sigma $ and $\sigma _{\infty }$ is the unique facet of $\sigma $ contained in $\partial \Delta $ . If $0 \notin \sigma $ , then $\sigma _{\infty } = \sigma $ .

Given two fans $\Sigma $ and $\Sigma '$ such that $\Sigma $ is a refinement of $\Sigma '$ ,

• • for every cone $\rho \in \Sigma $ , denote by $\min (\rho )$ the smallest cone of $\Sigma '$ containing $\rho $ .

If $\Sigma $ is the normal fan of a polytope $\Delta $ ,

• • denote by $\rho ^{\vee }$ the face of $\Delta $ which is normal to $\rho $ , and by $F^\vee $ the cone of $\Sigma $ which is normal to the face F of $\Delta $ .

Assume for a moment that T is a primitive convex central triangulation of $\Delta $ . If ${\mu :\Delta \cap M\rightarrow \mathbb {Z}_{\geq 0}}$ is a convex function ensuring the convexity of T, one can consider its Legendre transform (defined over $N_{\mathbb {R}}$ ):

(4) $$ \begin{align} \mu^*(x)=\max_{y\in\Delta\cap M} ( \langle x,y\rangle -\mu(y)), \end{align} $$

and the set

$$ \begin{align*}V_\mu:=\left\lbrace x\in N_{\mathbb{R}} \mid \exists y_1\neq y_2, \:\: \mu^*(x) = \langle x,y_1\rangle -\mu(y_1)=\langle x,y_2\rangle -\mu(y_2)\right\rbrace. \end{align*} $$

In other words, the set $V_\mu $ is the set of points where the maximum in (4) is attained at least twice. It is the so-called tropical hypersurface associated to $\mu $ : it is a weighted rational balanced polyhedral complex, inducing a polyhedral subdivision of $N_{\mathbb {R}}$ which is dual (as a poset) to the triangulation T (see, for example, [Reference Brugallé, Itenberg, Mikhalkin and Shaw9] for more details). Moreover, the compactification of $V_\mu $ in the tropical toric variety $\mathbb {T}\Sigma _\Delta $ induces a subdivision of $\mathbb {T}\Sigma _\Delta $ with underlying poset the subposet of $\Sigma _\Delta \times T$ :

(5) $$ \begin{align} \Phi:=\left\lbrace (\rho,\sigma)\in\Sigma_\Delta\times T \mid \sigma\subset \rho^\vee \right\rbrace. \end{align} $$

The order on the poset is the inverse of the inclusion, meaning that $(\rho ,\sigma )\leq (\rho ',\sigma ')$ iff $\rho '\subset \rho $ and $\sigma '\subset \sigma $ (see [Reference Brugallé, López de Medrano and Rau8] Remark 2.3).

The tropical variety $X_{\text {trop}}$ induces a subdivision of $\mathbb {T}\Sigma _{T^\circ }$ with underlying poset the subposet of $T^\circ \times T$ defined by

$$ \begin{align*}\mathcal{P}(T^\circ,T):=\left\lbrace (\tau,\sigma)\in T^\circ\times T \: \mid \: 0\in\tau \text{ and } \sigma\subset \min(C(\tau))^\vee \right\rbrace .\end{align*} $$

Recall that $C(\tau )$ is the cone (in $\Sigma _{T^\circ }$ ) over $\tau $ . The fan $\Sigma _{T^\circ }$ is a refinement of the fan $\Xi _{\Delta ^\circ }$ , which is the normal fan of $\Delta $ . Then by definition, $\min (C(\tau ))^\vee $ is a face of $\Delta $ . The order on the poset $\mathcal {P}(T^\circ ,T)$ is also the inverse of the inclusion.

Example 2.5. Let

$$\begin{align*}\Delta = \operatorname{\mathrm{Conv}}((-1,-1), \ (-1,2), \ (2,-1)). \end{align*}$$

It is a reflexive polygon, and its dual is given by

$$\begin{align*}\Delta^\circ = \operatorname{\mathrm{Conv}}((1,1), (-1,0),(0,-1)).\end{align*}$$

Denote by T and $T^\circ $ their unique primitive central triangulations. They are both convex. We draw in Figure 1 the subdivision of $\mathbb {T}\Sigma _{T^\circ }$ induced by a tropical curve dual to T and some examples of cells in the underlying poset $\mathcal {P}(T^\circ ,T)$ .

Figure 1

The subdivision of $\mathbb {T}\Sigma _{T^\circ }$ induced by a tropical curve dual to T.

In fact, in this paper, we do not assume any convexity hypothesis. For any T and $T^\circ $ as above (not necessarily convex), one can still consider the posets $\mathcal {P}(T,T^{\circ })$ and $\mathcal {P}(T^{\circ },T)$ . Recall that to any poset P is associated an abstract simplicial complex, called the order complex, whose vertices are the elements of P and whose simplices are the (finite) nonempty chains of P. Moreover, to any simplicial complex, there is a topological space called the geometric realization defined by formal convex combinations of vertices belonging to a simplex. The geometric realization of a poset P will be denoted by $\vert P\vert $ . By definition, the geometric realization is subivided into simplices: for any chain ${x_\bullet =(x_1 < \cdots < x_n )}$ in P, we denote the corresponding simplex of $\vert P\vert $ by $\vert x_\bullet \vert $ . We will consider here a coarser subdivision, defined as follows. For any $x\in P$ , consider the set of chains whose elements are smaller than x:

$$ \begin{align*} x_\leq=\left\lbrace x_\bullet \text{ chains in } P \: \vert \: x_k\leq x \:\: \forall k \right\rbrace, \end{align*} $$

and

$$ \begin{align*} F(x)= \bigcup_{x_\bullet \in x_\leq}\vert x_\bullet \vert. \end{align*} $$

The coarser subdivision of $\vert P\vert $ we consider is defined by

(6) $$ \begin{align} \vert P\vert := \bigcup_{x\in P} F(x). \end{align} $$

The cells $F(\tau ,\sigma )$ associated to elements $(\tau , \sigma )$ in the poset $\mathcal {P}(T^\circ ,T)$ form a regular CW-structure on $\mathbb {T}\Sigma _{T^\circ }$ .

Proof. We can describe the cells $F(\tau , \sigma )$ induced by $\mathcal {P}(T^\circ ,T)$ as follows. Since $\Sigma _{T^\circ }$ is a refinement of $\Sigma _{\Delta }$ , we have a blow-down map

$$\begin{align*}\operatorname{Bl}: \mathbb{T}\Sigma_{T^\circ} \rightarrow \mathbb{T}\Sigma_{\Delta}.\end{align*}$$

If $\rho \in \Sigma _{T^\circ }$ , then $\operatorname {Bl}$ maps the cell $F(\rho )$ to the cell $F(\min \rho )$ . In the interior of these cells, the map is given by the quotient

$$\begin{align*}\operatorname{Bl}_{\rho}: N_{\mathbb{R}}/ \langle \rho \rangle \rightarrow N_{\mathbb{R}}/ \langle \min \rho \rangle. \end{align*}$$

It is shown in [Reference Brugallé, López de Medrano and Rau8] Section 3.2.1 that the cells $F(\rho , \sigma )$ associated to elements in the poset $\Phi $ (see (5)) define a regular CW-structure on $\mathbb {T}\Sigma _{\Delta }$ . Then, given $(\tau , \sigma ) \in \mathcal {P}(T^\circ ,T)$ , the cell $F(\tau , \sigma )$ can be identified with the preimage of the cell $F(\min C(\tau ), \sigma )$ with respect to $\operatorname {Bl}$ ; that is,

$$\begin{align*}F(\tau, \sigma) = \operatorname{Bl}_{C(\tau)}^{-1}(F(\min C(\tau), \sigma)).\\[-37pt] \end{align*}$$

Again, in the case of convex triangulations, this subdivision is the subdivision induced on $\mathbb {T}\Sigma _{T^\circ }$ by $X_{\text {trop}}$ . We will be intersted more specifically in the subposet of $\mathcal {P}(T^\circ ,T)$ defined by

$$ \begin{align*}\mathcal{P}^1(T^\circ,T):=\left\lbrace (\tau,\sigma)\in \mathcal{P}(T^\circ,T) \vert \dim(\sigma)\geq 1 \right\rbrace. \end{align*} $$

In the convex case, this poset parametrizes $X_{\text {trop}}$ . As it will appear later, this poset will be the support of the cosheaves we will consider. Denote by $X_{T^\circ ,T}$ the geometric realization $\vert \mathcal {P}^1(T^\circ ,T)\vert $ as defined above with the subdivision (6). Define the following subspace of $X_{T^\circ ,T}$ :

(7) $$ \begin{align} S_{T^\circ,T}=\left\lbrace F(0,\sigma)\in X_{T^\circ,T} \:\: \vert \:\: 0\in\sigma \right\rbrace. \end{align} $$

This is topologically a sphere, and in the case where $X_{T^\circ ,T}=X_{\text {trop}}$ , this is the union of all bounded faces of $X_{\text {trop}}$ in $N_{\mathbb {R}}$ .

Definition 2.8. Let T be a primitive central triangulation of a reflexive polytope $\Delta $ and $T^\circ $ a primitive central triangulation of its dual $\Delta ^\circ $ . The pair of CW-complexes

$$ \begin{align*}(X_{T^\circ,T},X_{T,T^\circ})\end{align*} $$

is called a combinatorial mirror pair.

2.4 Poset homology

We define the homology groups $H_{\bullet }(P; \mathcal F)$ of a cosheaf $\mathcal F$ defined over a poset P. See [Reference Brugallé, López de Medrano and Rau8] and the references therein for more details. View a poset P as a category, whose objects are its elements and whose morphisms are the ordered pairs $x\leq _P y$ . For any ring R, an R-cosheaf $\mathcal {F}$ on P is a contravariant functor $\iota _{\mathcal {F}}$ from P to the category of R-modules. Given additional assumptions on P, one can associate a differential complex $(C_\bullet (P;\mathcal {F}),\partial )$ to any cosheaf $\mathcal {F}$ . A cover relation is a pair $x\lessdot y$ such that there exists no $z\in P$ with $x< z<y$ . A grading on P is a function $\dim : P\rightarrow \mathbb {Z}$ such that $\dim (y)-\dim (x)=1$ for any cover relation $x\lessdot y$ . An interval of length $2$ is an interval $\left [ x,y \right ]$ such that $\dim (y)-\dim (x)=2$ . We say that P is thin if every interval of length $2$ contains exactly $4$ elements. A signature is a map s from the set of all cover relations of P to $\{\pm 1\}$ , and it is called balanced if any interval of length $2$ contains an odd number of $-1$ ’s. Given a graded, thin poset P with a balanced signature, the differential complex is defined by

$$ \begin{align*}C_q(P;\mathcal{F})=\bigoplus_{\dim(x)=q}\mathcal{F}(x), \:\:\:\: \partial: C_q(P;\mathcal{F})\rightarrow C_{q-1}(P;\mathcal{F}), \end{align*} $$

where for all $x\in P$ of dimension q, one has

$$ \begin{align*}\partial\vert_{\mathcal{F}(x)}(a)=\sum_{y\lessdot x}s(y\lessdot x)\iota(y\leq_P x)(a). \end{align*} $$

Since the poset is thin, we have $\partial ^2=0$ . The homology groups of $(C_\bullet (P;\mathcal {F}),\partial )$ are denoted as usual by $H_\bullet (P;\mathcal {F})$ . If a subset U of P is closed under taking larger elements (sometimes U is called open), one can restrict the differential complex and the homology groups to U. We denote this restriction by $C_\bullet (U;\mathcal {F})$ and $H_\bullet (U;\mathcal {F})$ . Also maps of cosheaves induce maps between homology groups, and short exact sequences of cosheaves induce long exact sequences of homology groups.

2.5 Tropical homology

Tropical homology is a homology theory with non-constant coefficients defined over tropical varieties [Reference Itenberg, Katzarkov, Mikhalkin and Zharkov16], [Reference Brugallé, López de Medrano and Rau8]. In our case, if T and $T^\circ $ are convex primitive central triangulations of $\Delta $ and $\Delta ^\circ $ , respectively, the tropical varieties $X_{\text {trop}}\subset \mathbb {T}\Sigma _{T^\circ }$ and $X_{\text {trop}}^\circ \subset \mathbb {T}\Sigma _{T}$ of Definition 2.4 are non-singular projective tropical varieties. The tropical homology groups are denoted by $H_q(X_{\text {trop}}; \mathcal {F}_p)$ , and it follows from [Reference Itenberg, Katzarkov, Mikhalkin and Zharkov16] that

$$ \begin{align*}\dim H_q(X_{\text{trop}};\mathcal{F}_p\otimes\mathbb{Q})=h^{p,q}(X) \text{ and } \dim H_q(X_{\text{trop}}^\circ;\mathcal{F}_p\otimes \mathbb{Q})=h^{p,q}(X^\circ). \end{align*} $$

Here, X and $X^\circ $ are any smooth anticanonical hypersurfaces in $\mathbb {C} \Sigma _{T^\circ }$ and $\mathbb {C} \Sigma _{T}$ , respectively. Tropical homology (in its cellular version) belongs to the realm of poset homology.

We fix T and $T^\circ $ not necessarily convex primitive central triangulations of $\Delta $ and $\Delta ^\circ $ , respectively. The poset $\mathcal {P}(T^\circ ,T)$ is graded by the formula

$$ \begin{align*}\dim(\tau,\sigma)=\operatorname{\mathrm{codim}}(\tau)-\dim(\sigma). \end{align*} $$

Note that $\dim (\tau ,\sigma )=\dim F(\tau ,\sigma )$ in the geometric realization of $\mathcal {P}(T^\circ ,T)$ . Moreover, the poset $\mathcal {P}(T^\circ ,T)$ is thin, and it admits a balanced signature, obtained by choosing orientations on each $F(\tau ,\sigma )$ . The p-multitangent cosheaves over $\mathcal P(T^\circ ,T)$ are defined by

$$ \begin{align*}\mathcal{F}_p(\tau,\sigma):=\sum_{ \tiny{\begin{array}{c} \lambda\subset\sigma \\ \dim(\lambda)=1 \end{array}}} \bigwedge^p (\lambda^\perp / \langle C(\tau) \rangle ) , \end{align*} $$

where $\langle C(\tau ) \rangle $ is the submodule generated by $C(\tau ) \cap N$ . The cosheaf maps of $\mathcal {F}_p$ are induced by the projections

(8) $$ \begin{align} N /\langle C(\tau) \rangle \rightarrow N /\langle C(\tau') \rangle \end{align} $$

if $\tau \subset \tau '$ (recall that by definition of $\mathcal P(T^\circ ,T)$ , the faces $\tau $ and $\tau '$ contain $0$ ). Note that the support of the cosheaves $\mathcal {F}_p$ is the poset $\mathcal P^1(T^\circ ,T)$ . In order to lighten the notations, if $\mathcal {G}$ is any cosheaf on $\mathcal P^1(T^\circ ,T)$ , we will denote by $H_\bullet (X_{T^\circ ,T};\mathcal {G})$ the homology groups $H_\bullet (\mathcal P^1(T^\circ ,T);\mathcal {G})$ .

3.1 A refinement of the poset $\mathcal {P}(T^\circ ,T)$

Notice that we do not consider the pair $(0,0)$ in $\mathcal {J}(T^\circ ,T)$ . In fact, by definition, the pair $(0,0)$ would be isolated (i.e., not comparable to any other pair inside $\mathcal {J}(T^\circ ,T)$ ).

It follows from the definition that the map from $\mathcal {J}(T^\circ ,T)$ to $\mathcal {P}(T^\circ ,T)\setminus (0,0)$ sending $(\tau ,\sigma )$ to $(0,\sigma )$ if $0\notin \tau $ , and to itself if $0\in \tau $ is surjective. The poset $\mathcal {J}(T^\circ ,T)$ is still a graded poset (with the same grading as before), is thin and still admits a balanced signature. In fact, its geometric realization (as defined above) is a subdivision of the one of $\mathcal {P}(T^\circ ,T)$ (minus the cell $F(0,0)$ ). Similarly, the geometric realization of $\mathcal {J}^S(T^\circ ,T)$ is a subdivision of $S_{T^\circ ,T}$ , and $\mathcal {J}^\infty (T^\circ ,T)=\mathcal {P}^\infty (T^\circ ,T)$ . As a corollary of Lemma 3.1, one obtains the following:

Lemma 3.4. The map

(10) $$ \begin{align} j_{T^\circ,T}: \mathcal{J}(T^\circ, T) \longrightarrow \mathcal{J}(T , T^\circ) \end{align} $$

sending

• • $(\tau ,\sigma )$ to $(\hat {\sigma },\tau _\infty )$ if $0\in \tau $ and $\sigma \in \partial T$ ,

• • $(\tau ,\sigma )$ to $(\sigma _\infty ,\hat {\tau })$ if $\tau \in \partial T^\circ $ and $0\in \sigma $ , and

• • $(\tau ,\sigma )$ to $(\sigma ,\tau )$ if $(\tau ,\sigma )\in \partial T^\circ \times \partial T$

is an isomorphism of posets.

Notice that $j_{T^\circ ,T}$ sends $\mathcal {J}^S(T^\circ ,T)$ to $\mathcal {J}^S(T,T^\circ )$ , sends $\mathcal {J}^\infty (T^\circ ,T)$ to $\mathcal {J}^\infty (T,T^\circ )$ and preserves the dimensions.

Example 3.5. In Figures 3 and 4, we represented the geometric realizations of the posets $\mathcal {J}(T,T^\circ )$ and $\mathcal {J}(T^\circ , T)$ , where T and $T^\circ $ are defined again as in Example 2.5. The cells depicted in the two figures are dual in the sense that they correspond under the map $j_{T^\circ ,T}$ .

Figure 3

The subdivision $\mathcal {J}(T,T^\circ )$ .

Figure 4

The subdivision $\mathcal {J}(T^\circ , T)$ .

We extend the cosheaves $\mathcal {F}_p(\tau ,\sigma )$ to the poset $\mathcal {J}(T^{\circ }, T)$ by precomposing with the map from above sending $(\tau ,\sigma )$ to $(0,\sigma )$ if $0\notin \tau $ and to itself if $0\in \tau $ . Define as before

$$ \begin{align*}\mathcal{J}^1(T^\circ, T):=\left\lbrace (\tau,\sigma)\in \mathcal{J}(T^\circ, T) \vert \dim(\sigma)\geq 1 \right\rbrace. \end{align*} $$

The geometric realization of $\mathcal {J}^1(T^\circ , T)$ defines a subdivision of $X_{T^\circ ,T}$ , and it is the support of the cosheaf $\mathcal {F}_p$ . One has a canonical isomorphism

$$ \begin{align*}H_q(\mathcal{J}(T^\circ,T); \mathcal F_p) \cong H_q(X_{T^\circ,T}; \mathcal F_p). \end{align*} $$

3.2 The mirror cosheaf

Let $(\tau ,\sigma )\in \mathcal {J}^S(T^\circ ,T)$ . Since $\sigma _\infty \subset \min (C(\tau ))^\vee $ , in particular, the cone $C(\tau )$ is a subset of $\sigma _\infty ^\perp $ . Then $\langle C(\tau ) \rangle \wedge \mathcal {F}_{p-1}(0,\sigma _{\infty })$ is a submodule of $\mathcal {F}_p(0,\sigma )$ , and we can consider the quotient.

Notice that the support of $\mathcal {M}_p$ is $\mathcal {J}^S (T^\circ ,T)$ and that, moreover,

$$\begin{align*}\mathcal M_p(\tau, \sigma) = \mathcal F_p(0, \sigma) \quad \text{when } 0\in\sigma \text{ and } \dim \sigma = 1.\end{align*}$$

We have the following main result:

Theorem 3.7. Let A be a commutative ring. For all $q \geq 0$ , there are canonical isomorphisms

(13) $$ \begin{align} H_q(\mathcal{J}(T^\circ,T); \mathcal F_p\otimes A) \cong H_q(\mathcal{J}(T^\circ,T); \mathcal M_p\otimes A), \end{align} $$

(14) $$ \begin{align} H_q(\mathcal{J}(T^\circ,T); \mathcal M_p\otimes A) \cong H_q(\mathcal{J}(T,T^\circ); \mathcal M_{n-p}\otimes A). \end{align} $$

Moreover, they induce a canonical isomorphism

(15) $$ \begin{align} H_q(X_{T^\circ,T}; \mathcal F_p\otimes A) \cong H_q(X_{T,T^\circ}; \mathcal F_{n-p}\otimes A). \end{align} $$

Corollary 3.8. If T and $T^\circ $ are convex subdivisions, and $X_{\text {trop}}$ and $X^\circ _{\text {trop}}$ are some correponding tropical hypersurfaces as in Definition 2.4, then we have the mirror symmetry of the tropical Hodge diamonds

$$\begin{align*}\dim H_q( X_{\text{trop}}; \mathcal F_p\otimes \mathbb{Q}) = \dim H_q(X^{\circ}_{\text{trop}}; \mathcal F_{n-p}\otimes \mathbb{Q}). \end{align*}$$

3.3 Proof of Theorem 3.7

We begin by extending the support of $\mathcal {M}_p$ to the whole poset $\mathcal {J}(T^\circ ,T)$ .

Definition 3.10. Let $p\geq 0$ . The cosheaf $\mathcal {M}_p^{\Delta }$ is defined over $\mathcal {J}(T^\circ ,T)$ by

(16) $$ \begin{align} \mathcal{M}_p^{\Delta}(\tau, \sigma)=\dfrac{\mathcal{F}_p(0,\sigma )}{\langle C(\tau) \rangle \wedge \mathcal{F}_{p-1}(0,\sigma_\infty)} \end{align} $$

if $\tau \in \partial T^\circ $ , and by $\mathcal F_p$ otherwise. Notice that

$$\begin{align*}\mathcal{M}_p^{\Delta}(\tau, \sigma ) = \mathcal{M}_p(\tau, \sigma ) \quad \text{when} \ (\tau, \sigma) \in \mathcal{J}^{S}(T^\circ,T) \end{align*}$$

and

$$\begin{align*}\mathcal{M}_p^{\Delta}(\tau, \sigma ) = \mathcal{F}_p(\tau ,\sigma ) \quad \text{if} \ 0\in \tau. \end{align*}$$

By definition, we have an inclusion $\mathcal {M}_p \hookrightarrow \mathcal {M}_p^{\Delta }$ and a surjection $\mathcal F_p \rightarrow \mathcal M_p^{\Delta }$ . These give two short exact sequences of cosheaves

(17) $$ \begin{align} 0 \rightarrow \mathcal M_p \rightarrow \mathcal M_p^{\Delta} \rightarrow \mathcal Q_p \rightarrow 0, \end{align} $$

where $\mathcal Q_p$ is the quotient cosheaf, and

(18) $$ \begin{align} 0 \rightarrow \mathcal R_p \rightarrow \mathcal F_p \rightarrow \mathcal M_p^{\Delta} \rightarrow 0, \end{align} $$

where $\mathcal R_p$ is the kernel.

The quotient $\mathcal {Q}_p$ is given by $\mathcal {Q}_p(\tau , \sigma )=\mathcal {M}_p^{\Delta }(\tau , \sigma )$ if $\sigma \in \partial T$ and is equal to zero otherwise. In particular, $\mathcal Q_p$ is supported on

$$\begin{align*}\mathcal{J}^{ub}(T^\circ,T):=\mathcal{J}(T^\circ,T)\setminus \mathcal{J}^S(T^\circ,T). \end{align*}$$

The kernel $\mathcal {R}_p$ is given by $\mathcal {R}_p(\tau ,\sigma )=\langle C(\tau ) \rangle \wedge \mathcal {F}_{p-1}(0,\sigma _\infty )$ if $\tau \in \partial T^\circ $ and is equal to zero otherwise. The support of $\mathcal {R}_p$ is

$$\begin{align*}\mathcal{J}^{0}(T^\circ,T):=\mathcal{J}(T^\circ,T)\setminus \mathcal{J}^\infty(T^\circ,T). \end{align*}$$

We have the following:

Proposition 3.11. The cosheaves $\mathcal M_p$ , $\mathcal M_p^{\Delta }$ , $\mathcal {Q}_p$ and $\mathcal R_p$ are cosheaves of free $\mathbb {Z}$ -modules. Moreover, let $\sigma \in \partial T$ and $\tau \in \partial T^\circ $ such that $\sigma \subset \min (C(\tau ))^\vee $ . Then the cosheaf maps

$$\begin{align*}\begin{aligned} \mathcal Q_p(\tau, \sigma) & \rightarrow \mathcal Q_p (\hat{\tau}, \sigma), \\ \mathcal R_p(\tau, \sigma) & \rightarrow \mathcal R_p (\tau, \hat{\sigma} ) \end{aligned} \end{align*}$$

are isomorphisms.

Proof. It is easy to see that $\mathcal F_p$ is a cosheaf of free $\mathbb {Z}$ -modules and therefore also $\mathcal R_p$ . It is then enough to show that $\mathcal M^{\Delta }_p$ is a cosheaf of free modules.

Let $(\tau , \sigma ) \in \mathcal {J}(T^\circ ,T)$ with $0 \in \sigma $ and $\tau \in \partial T^{\circ }$ . We begin by describing bases for $\mathcal F_p(0, \sigma )$ , $\mathcal F_p(0, \sigma _{\infty })$ and $\langle C(\tau ) \rangle \wedge \mathcal {F}_{p-1} (0,\sigma _{\infty })$ . The cell $\sigma $ is a primitive simplex; let $\dim \sigma = s+1$ . Choose a vertex $e_0$ of $\sigma _{\infty }$ and fix s edges of $\sigma _{\infty }$ containing this vertex. Denote by $e_1,\cdots ,e_{s}$ a choice of primitive vectors directing those edges. Complete the set $e_1, \ldots , e_{s}$ to a basis of $\langle C(\tau )\rangle ^\perp $ denoted by $e_1,\cdots ,e_m$ , and complete the set $e_0, \ldots , e_m$ to a basis of M denoted by $e_0,\cdots ,e_{n}$ . Consider the dual basis $e_0^*,\cdots ,e_{n}^*$ of N. We have

$$ \begin{align*}\langle C(\tau) \rangle = \langle e_0^*, e_{m+1}^*,\cdots,e_{n}^*\rangle.\end{align*} $$

We use multi-indices $I=\{ i_1, \ldots , i_p \} \subseteq \{0, \ldots , n \}$ with $i_1 < i_2 < \ldots < i_p$ , and we denote by $e_I^*$ the p-vector $e_{i_1}^* \wedge \ldots \wedge e_{i_p}^*$ . We also denote by $|I|$ the cardinality of I and by $I^c$ the complement of I in $\{0, \ldots , n \}$ . A basis of $\mathcal F_p(0, \sigma )$ consists of p-vectors of the type

$$ \begin{align*}e_I^*, \quad \ I^c \cap \{0, \ldots, s \} \neq \emptyset. \end{align*} $$

Similarly, a basis of $\mathcal F_{p-1}(0, \sigma _{\infty })$ is given by the $p-1$ -vectors

$$ \begin{align*}e_J^*, \quad J^c \cap \{1, \ldots, s \} \neq \emptyset. \end{align*} $$

Then a basis of $\langle C(\tau ) \rangle \wedge \mathcal {F}_{p-1}(0,\sigma _{\infty })$ is given by p-vectors of the type

$$ \begin{align*}e_I^*, \quad I \cap \{0, m+1, \ldots, n \} \neq \emptyset \ \text{and} \ I^c \cap \{1, \ldots, s \} \neq \emptyset. \end{align*} $$

This clearly shows that $\mathcal M_{p}^{\Delta }$ is free.

Assume now that $\tau \in \partial T^\circ $ and $\sigma \in \partial T$ (in particular, $\sigma = \sigma _{\infty }$ ). With the same notation as above, the quotient $N/ \langle C(\tau ) \rangle $ can be identified with $\langle e_1^*,\cdots ,e_m^*\rangle $ . By definition,

$$ \begin{align*}\mathcal Q_p (\hat{\tau}, \sigma) = \mathcal F_p(\hat{\tau}, \sigma)=\sum_{i=1}^{s} \bigwedge^p (e_i^\perp / \langle C(\tau) \rangle ),\end{align*} $$

so a basis of $\mathcal Q_p(\hat {\tau }, \sigma )$ can be identified with p-vectors of the type

$$ \begin{align*}e_I^*, \quad I \subseteq \{1, \dots, m \} \ \text{and} \ I^c \cap \{1, \ldots, s \} \neq \emptyset, \end{align*} $$

where now $I^c$ denotes the complement of I in $\{1, \dots , m \}$ . Now notice that also a basis of

$$\begin{align*}\mathcal Q_p (\tau, \sigma) = \mathcal M_p^{\Delta}(\tau, \sigma) = \dfrac{\mathcal{F}_p(0,\sigma )}{\langle C(\tau) \rangle \wedge \mathcal{F}_{p-1}(0,\sigma)}\end{align*}$$

can be identified by the same set of p-vectors, since multi-indices I such that $I \cap \{ 0, m+1, \ldots , n \} \neq \emptyset $ are killed by the quotient. This implies that $\mathcal Q_p(\tau , \sigma ) \rightarrow \mathcal Q_p (\hat {\tau }, \sigma )$ is an isomorphism. The case of $\mathcal R_p$ is trivial, indeed by definition $\mathcal {R}_p(\tau ,\sigma ) = \mathcal {R}_p(\tau , \hat {\sigma })$ .

We also have the following:

Corollary 3.12. For any commutative ring A, we have short exact sequences

$$\begin{align*}\begin{aligned} \ & 0 \rightarrow \mathcal M_p \otimes A \rightarrow \mathcal M_p^{\Delta} \otimes A \rightarrow \mathcal Q_p \otimes A \rightarrow 0, \\ \ & 0 \rightarrow \mathcal R_p \otimes A \rightarrow \mathcal F_p \otimes A \rightarrow \mathcal M_p^{\Delta} \otimes A \rightarrow 0, \end{aligned} \end{align*}$$

and isomorphisms

$$\begin{align*}\begin{aligned} \mathcal Q_p(\tau, \sigma) \otimes A & \rightarrow \mathcal Q_p (\hat{\tau}, \sigma) \otimes A, \\ \mathcal R_p(\tau, \sigma) \otimes A & \rightarrow \mathcal R_p (\tau, \hat{\sigma} )\otimes A, \end{aligned} \end{align*}$$

holding for all $\sigma \in \partial T$ and $\tau \in \partial T^\circ $ such that $\sigma \subset \min (C(\tau ))^\vee $ .

Proposition 3.13. For all $k \geq 0$ and any commutative ring A, we have

$$\begin{align*}H_k (\mathcal{J}(T^\circ,T); \mathcal Q_p \otimes A) \cong 0, \end{align*}$$

and

$$\begin{align*}H_k (\mathcal{J}(T^\circ,T); \mathcal R_p \otimes A) \cong 0. \end{align*}$$

Proof. For simplicity, we denote $\mathcal Q_p \otimes A$ and $\mathcal R_p \otimes A$ by $\mathcal Q_p$ and $\mathcal R_p$ . Recall that the support of $\mathcal {Q}_p$ is $\mathcal {J}^{ub}(T^\circ ,T)$ . Introduce

$$ \begin{align*}\mathcal{J}_0^{ub}=\{ (\tau, \sigma) \, | \, \sigma\in \partial T \ \text{and} \ \tau\in \partial T^\circ \} \end{align*} $$

and recall that

$$ \begin{align*}\mathcal{J}^\infty(T^\circ,T)=\{ (\tau, \sigma) \, | \, \sigma\in \partial T \ \text{and} \ 0\in \tau \}. \end{align*} $$

One has $\mathcal {J}^{ub}(T^\circ ,T)=\mathcal {J}_0^{ub}\cup \mathcal {J}^\infty $ , and there is a bijection between $\mathcal {J}_0^{ub}$ and $\mathcal {J}_\infty $ given by $(\tau ,\sigma ) \mapsto (\hat {\tau },\sigma )$ . Denote by $\mathcal {C}^{ub}$ the complex of chains $\mathcal {C}(\mathcal {J}^{ub}(T^\circ ,T),\mathcal {Q}_p)$ , and similarly by $\mathcal {C}_0^{ub}$ the set of chains supported on $\mathcal {J}_0^{ub}$ and by $\mathcal {C}_\infty $ the complex $\mathcal {C}(\mathcal {J}^\infty (T^\circ ,T),\mathcal {Q}_p)$ . Notice that $\mathcal {C}_0^{ub}$ is not closed by taking the boundary. As a module, we have a decomposition

$$\begin{align*}\mathcal C^{ub} = \mathcal C_{0}^{ub} \oplus \mathcal C_{\infty}, \end{align*}$$

and we denote by $\Pi _{0}$ and $\Pi _{\infty }$ the projections onto the respective component. Proposition 3.11 implies that

(19) $$ \begin{align} \Pi_{\infty} \circ \partial: \mathcal C_0^{ub} \rightarrow \mathcal C_{\infty} \end{align} $$

is an isomorphism. Denote by $L: \mathcal C_{\infty } \rightarrow \mathcal C_0^{ub}$ its inverse. Suppose now that $\gamma \in \mathcal C^{ub}$ is closed and decompose it as

$$\begin{align*}\gamma = \gamma_0 + \gamma_{\infty}, \end{align*}$$

its $\mathcal C_0^{ub}$ and $\mathcal C_{\infty }$ components. Then

$$\begin{align*}\partial \gamma_0 = - \partial \gamma_{\infty}. \end{align*}$$

Let

$$\begin{align*}\beta = L(\gamma_{\infty}). \end{align*}$$

We claim that

$$\begin{align*}\partial \beta = \gamma. \end{align*}$$

First of all, decomposing $\partial \beta $ and using the fact that L is the inverse of $\Pi _{\infty } \circ \partial $ , we have

$$\begin{align*}\partial \beta = \Pi_{0}(\partial \beta) + \gamma_{\infty}. \end{align*}$$

Taking the boundary again, we obtain

$$\begin{align*}\partial ( \Pi_{0} (\partial \beta)) = - \partial \gamma_{ \infty} = \partial \gamma_0; \end{align*}$$

that is,

$$\begin{align*}\partial ( \Pi_{0} (\partial \beta) - \gamma_0) = 0. \end{align*}$$

Now the fact that (19) is an isomorphism implies that

$$\begin{align*}\Pi_{0} (\partial \beta) = \gamma_0. \end{align*}$$

Hence,

$$\begin{align*}\partial \beta = \gamma, \end{align*}$$

and so any closed chain in $\mathcal {C}^{ub}$ is exact. The proof for $\mathcal {R}_p$ goes along the exact same lines, by considering the decomposition of $\mathcal {J}^0$ into $\mathcal {J}^{ub}_0 \cup \mathcal {J}^S$ .

Then, from the long exact sequences in homology associated to the above short exact sequences, we obtain canonical isomorphisms

$$\begin{align*}H_q (\mathcal{J}(T^\circ,T); \mathcal M_p \otimes A) \cong H_q (\mathcal{J}(T^\circ,T); \mathcal M^{\Delta}_p \otimes A) \cong H_q (\mathcal{J}(T^\circ,T); \mathcal F_p \otimes A) \end{align*}$$

for all $q \geq 0$ . This proves (13).

Example 3.14. Suppose $\gamma _{\infty } \in \mathcal {C}(\mathcal {J}^\infty (T^\circ ,T),\mathcal {F}_p)$ is a cycle (supported at infinity) defining a class in $H_{\ast }(X_{T^\circ ,T}, \mathcal F_p)$ . Let

$$\begin{align*}\gamma_{\infty} = \sum_{(\tau, \sigma) \in \mathcal{J}^\infty (T^\circ,T)} (\tau, \sigma) \otimes v_{(\tau, \sigma)}. \end{align*}$$

Consider the chain in $\mathcal {C}(\mathcal {J}^{ub}_{0} (T^\circ ,T), \mathcal M_p^{\Delta })$ given by

$$\begin{align*}C = \sum (\tau_{\infty}, \sigma) \otimes \tilde v_{(\tau, \sigma)}, \end{align*}$$

where $\tilde v_{(\tau , \sigma )} \in \mathcal M_p^{\Delta }(\tau _{\infty }, \sigma )$ is the unique element corresponding to $v_{(\tau , \sigma )}$ under the isomorphism of Proposition 3.11. Using the notation from Proposition 3.13, we have $C = L (\gamma _{\infty })$ . In particular, the boundary of C is given by

$$\begin{align*}\partial C = \gamma_{\infty} + \gamma_{S}, \end{align*}$$

where

$$\begin{align*}\gamma_{S} = \sum (\tau_{\infty}, \hat \sigma) \otimes \tilde v_{(\tau, \sigma)}\end{align*}$$

is a cycle in $\mathcal {C}(\mathcal {J}^S (T^\circ ,T),\mathcal {M}_p)$ defining the class $[\gamma _S] \in H_\ast ( (\mathcal {J} (T^\circ ,T), \mathcal M_p)$ corresponding to $[\gamma _{\infty }]$ under the isomorphism (13).

Let us now prove (14). It will be enough to prove it over $\mathbb {Z}$ . Recall that the support of $\mathcal {M}_p$ is the subposet $\mathcal {J}^S(T^\circ ,T)$ and that the bijection $j_{T^\circ ,T}$ (10) restricts to a bijection from $\mathcal {J}^S(T^\circ ,T)$ to $\mathcal {J}^S(T,T^\circ )$ which sends $(\tau , \sigma ) \in \mathcal {J}^S(T^\circ ,T)$ to $(\sigma _{\infty }, \hat {\tau }) \in \mathcal {J}^S(T, T^\circ )$ . We now prove a stronger statement than (14) – namely, that the pairs $(\mathcal {J}^S(T^\circ ,T), \mathcal M_p)$ and $( \mathcal {J}^S(T, T^\circ ) , \mathcal M_{n-p})$ are canonically isomorphic. By definition, we have

$$\begin{align*}\mathcal M_p(\tau,\sigma) = \frac{\mathcal{F}_p(0,\sigma)} { \langle C(\tau) \rangle \wedge \mathcal{F}_{p-1}(0,\sigma_{\infty})}\end{align*}$$

and

$$\begin{align*}\mathcal M_{n-p}(\sigma_\infty, \hat{\tau}) = \frac{\mathcal{F}_{n-p}(0,\hat{\tau})} { \langle C(\sigma) \rangle \wedge \mathcal{F}_{n-p-1}(0,\tau)}. \end{align*}$$

Given an orientation on $N_{\mathbb {R}}$ , there exists a unique integral $(n+1)$ -form $\Omega _{N}\in \bigwedge ^{n+1} M$ compatible with the orientation. Given $z \in \mathcal {F}_p(0,\sigma )$ , we denote by $[z]$ its class in $\mathcal M_p(\tau ,\sigma )$ . Let $v \in N$ be a vertex of $\tau $ . Consider the map

(20) $$ \begin{align} \begin{aligned} \mathcal M_p(\tau, \sigma) & \rightarrow \mathcal M_{n-p}(\sigma_\infty, \hat{\tau}) \\ [z] & \mapsto [\iota_{v \wedge z} \Omega_N ], \end{aligned} \end{align} $$

where $\iota $ denotes the contraction operator. We have the following:

Proposition 3.15. After fixing an orientation of $N_{\mathbb {R}}$ , for all $(\tau , \sigma ) \in \mathcal J^S(T^\circ ,T)$ , the map (20) is well defined and gives an isomorphism

$$\begin{align*}\mathcal M_p(\tau, \sigma) \cong \mathcal M_{n-p}(\sigma_\infty, \hat{\tau}) \end{align*}$$

independent of the choice of v. Moreover, it is compatible with cosheaf maps; that is, for any inclusion $(\tau _1,\sigma _1) \hookrightarrow (\tau ,\sigma )$ , the following diagram commutes:

where the horizontal maps are the cosheaf maps. The same is true after tensoring with a commutative ring A.

Proof. Recall that $(\tau , \sigma ) \in \mathcal {J}^S(T^\circ ,T)$ implies that $\tau \in \partial T^\circ $ and $0 \in \sigma $ . We will use the following essential properties of the contraction operator. First of all, it satisfies the Leibniz property; that is, for any vector v and forms $\alpha $ and $\beta $ , we have

$$\begin{align*}\iota_{v}(\alpha \wedge \beta) = (\iota_v \alpha) \wedge \beta + (-1)^{\deg \alpha} \alpha \wedge (\iota_v \beta).\end{align*}$$

Next, for any vector $v \in N$ , we have

(21) $$ \begin{align} \bigwedge^p v^{\perp} = \left \{ \omega \in \bigwedge^p M\, | \, \iota_v \omega = 0 \right \}. \end{align} $$

Let us first prove that the map is well defined. Notice that

$$\begin{align*}\iota_v (\iota_{v \wedge z} \Omega_N) = 0. \end{align*}$$

In particular, $ \iota _{v \wedge z} \Omega _N \in \bigwedge ^{n-p} v^{\perp }$ . Since v is a vertex of $\tau $ , it is also tangent to a one-dimensional edge of $\hat \tau $ . This proves that

$$\begin{align*}\iota_{v \wedge z} \Omega_N \in \mathcal F_{n-p}(0, \hat \tau). \end{align*}$$

Now let $z \in \langle C(\tau ) \rangle \wedge \mathcal {F}_{p-1}(0,\sigma _{\infty })$ . We need to show that $ \iota _{v \wedge z} \Omega _N \in \langle C(\sigma ) \rangle \wedge \mathcal {F}_{n-p-1}(0,\tau )$ . Let $f_1, \ldots , f_q \in M$ be the vertices of $\sigma _{\infty }$ . We can complete them to form an oriented basis $\{f_1, \ldots , f_{n+1} \}$ of M, so that

$$\begin{align*}\Omega_N = f_1 \wedge \ldots \wedge f_{n+1}. \end{align*}$$

We denote by $\{f_1^*, \ldots , f_{n+1}^* \}$ the dual basis of N. For every multi-index $I =(i_1, \ldots , i_p)$ , define the following sign:

$$\begin{align*}(-1)^{I} := \prod_{k \in I} (-1)^{k+1}. \end{align*}$$

Let $e_1, \ldots , e_s \in N$ be the vertices of $\tau $ . We can assume

$$\begin{align*}v = e_1. \end{align*}$$

Since $\sigma _\infty \subset \min (C(\tau ))^\vee $ and by reflexivity, we have

(22) $$ \begin{align} \langle f_i, e_j\rangle = 1 \quad \text{for all} \ \ 1\leq i \leq q \ \text{and} \ 1 \leq j \leq s. \end{align} $$

By definition,

$$\begin{align*}\mathcal F_p(0 , \sigma_{\infty})=\sum_{1 \leq i <j \leq q} \bigwedge^p (f_i - f_j)^\perp. \end{align*}$$

Without loss of generality, we can consider

$$\begin{align*}\omega \in \bigwedge^{p-1} (f_1 - f_2)^\perp \end{align*}$$

and let

$$\begin{align*}z = e_2 \wedge \omega. \end{align*}$$

Then $z \in \langle C(\tau ) \rangle \wedge \mathcal {F}_{p-1}(0,\sigma _{\infty })$ . Observe that

$$\begin{align*}(f_1 - f_2)^\perp = \langle f_1^* + f_2^*, f_3^*, \ldots, f_{n+1}^* \rangle.\end{align*}$$

Therefore, by linearity, it is enough to consider two cases for $\omega $ : either $\omega = f^*_I$ , with $I=\{i_1, \ldots , i_{p-1} \} \subset \{3, \ldots , n+1 \}$ , or $\omega = (f_1^* + f_2^*) \wedge f_I^*$ , with $I=\{i_1, \ldots , i_{p-2} \} \subset \{3, \ldots , n+1 \}$ . In these two cases, we compute $\iota _{e_1 \wedge z} \Omega _N = \iota _{e_1 \wedge e_2 \wedge \omega } \Omega _N$ . In the first case, we have

$$\begin{align*}\iota_{e_1 \wedge e_2 \wedge \omega} \Omega_N = \iota_{e_1}( \iota_{e_2} (\iota _{f_I^*} \Omega_N )) = (-1)^I \iota_{e_1}( \iota_{e_2} (f_1 \wedge f_2 \wedge f_{I'} )), \end{align*}$$

where $I'$ is the complement of I in $\{3, \ldots , n+1\}$ . Using the Leibnitz rule for $\iota $ and relations (22), we compute that

$$\begin{align*}\begin{aligned} \ & \iota_{e_1}( \iota_{e_2} (f_1 \wedge f_2 \wedge f_{I'} )) = \iota_{e_1}((f_2 - f_1) \wedge f_{I'} +f_1 \wedge f_2 \wedge (\iota_{e_2} f_{I'})) = \\ \ &=(f_1 - f_2) \wedge (\iota_{e_1} f_{I'}) + (f_2 - f_1) \wedge (\iota_{e_2} f_{I'}) + f_1 \wedge f_2 \wedge (\iota_{e_1 \wedge e_2} f_{I'})) = \\ \ &= (f_1 - f_2) \wedge (\iota_{e_1-e_2} f_{I'}) + f_1 \wedge f_2 \wedge (\iota_{e_1 \wedge e_2} f_{I'}). \end{aligned} \end{align*}$$

We now have that $f_1$ and $f_1 - f_2$ are both in $C(\sigma )$ and $\iota _{e_1-e_2} f_{I'} \in \bigwedge ^{n-p-1} (e_1 - e_2)^{\perp }$ by (21). In particular, since $e_1 - e_2$ is tangent to a one-dimensional edge of $\tau $ , we have $\iota _{e_1-e_2} f_{I'} \in \mathcal F_{n-p-1}(0, \tau )$ . Now observe that

$$\begin{align*}\iota_{e_1 - e_2} (f_2 \wedge (\iota_{e_1 \wedge e_2} f_{I'})) = \iota_{e_1 - e_2} (f_2 \wedge (\iota_{(e_1 - e_2)\wedge e_2} f_{I'})) = 0, \end{align*}$$

so that also $f_2 \wedge (\iota _{e_1 \wedge e_2} f_{I'}) \in \mathcal F_{n-p-1}(0, \tau ) $ . Hence, this proves, for this case, that $\iota _{v \wedge z} \Omega _N \in \langle C(\sigma ) \rangle \wedge \mathcal {F}_{n-p-1}(0,\tau )$ .

In the second case (i.e., $\omega = (f_1^* + f_2^*) \wedge f_I^*$ ), a similar computation shows that

$$\begin{align*}\iota_{e_1 \wedge e_2 \wedge \omega} \Omega_N = (-1)^I(f_2 - f_1) \wedge (\iota_{e_1 \wedge e_2} f_{I'}). \end{align*}$$

Again, also in this case, $\iota _{v \wedge z} \Omega _N \in \langle C(\sigma ) \rangle \wedge \mathcal {F}_{n-p-1}(0,\tau )$ . This concludes the proof that (20) is well defined.

Now consider two different vertices of $\tau $ , v and $v'$ . We prove that $\iota _{v \wedge z}\Omega _N - \iota _{v' \wedge z} \Omega _N$ is in $\langle C(\sigma ) \rangle \wedge \mathcal {F}_{n-p-1}(0,\tau )$ . This would show that the map (20) is independent of v. It is enough to check this for $z \in \bigwedge ^p f_1^{\perp }$ . In particular, we can assume $z = f_{I}^*$ for some multi-index such that $1 \notin I$ . Then

$$\begin{align*}\begin{aligned} \iota_{v \wedge z}\Omega_N - \iota_{v' \wedge z} \Omega_N & = \iota_{v-v'} ( \iota_{f^*_I} \Omega_N) = (-1)^I \iota_{v-v'} (f_1 \wedge f_{I'}) = - (-1)^If_1 \wedge (\iota_{v-v'} (f_{I'})) \end{aligned} \end{align*}$$

for some multi-index $I'$ such that $1 \notin I'$ . Now we have $f_1 \in \langle C(\sigma ) \rangle $ and $\iota _{v-v'} (f_{I'}) \in \bigwedge ^{n-p-1} (v-v')^\perp \subseteq \mathcal F_{n-p-1}(0, \tau )$ . Hence, $\iota _{v \wedge z}\Omega _N - \iota _{v' \wedge z} \Omega _N \in \langle C(\sigma ) \rangle \wedge \mathcal {F}_{n-p-1}(0,\tau )$ .

We now prove that (20) is invertible. In fact, consider the same map,but in the opposite direction $\mathcal M_{n-p}(\sigma _\infty , \hat {\tau }) \rightarrow \mathcal M_p(\tau ,\sigma )$ , using a vertex w of $\sigma _{\infty }$ (instead of v) and the $(n+1)$ -form $\Omega _M\in \bigwedge ^{n+1} N$ . This map (up to a sign) is the inverse of (20). Indeed, we prove that

$$\begin{align*}[\iota_{w \wedge (\iota_{v \wedge z} \Omega_{N})} \Omega_M] = \epsilon [z], \end{align*}$$

where $\epsilon = (-1)^{\tfrac {n(n+5)}{2}}$ .

It is enough to consider $z \in \bigwedge ^{p} f_1^{\perp }$ ; in particular, we can assume

$$\begin{align*}z = f_I^{*}, \end{align*}$$

where I is some multi-index with $1 \notin I$ and $|I| = p$ . Then

$$\begin{align*}\iota_{v \wedge z} \Omega_{N} = (-1)^{I} \iota_v(f_1 \wedge f_{I'}) = (-1)^I (f_{I'} - f_1 \wedge (\iota_v f_{I'})),\end{align*}$$

where $I'$ is the complement of I in $\{ 2, \ldots , n+1 \}$ . We can assume that the vertex w of $\sigma _{\infty }$ is $f_1$ . Hence,

$$\begin{align*}w \wedge (\iota_{v \wedge z} \Omega_{N}) = (-1)^I f_1 \wedge f_{I'}. \end{align*}$$

We have

$$\begin{align*}\Omega_M = f_1^* \wedge \ldots \wedge f_{n+1}^*. \end{align*}$$

Therefore,

$$\begin{align*}\iota_{w \wedge (\iota_{v \wedge z} \Omega_{N})} \Omega_M = (-1)^I \iota_{f_1 \wedge f_{I'}} \Omega_M = (-1)^I (-1)^{I'} f_{I}^* = (-1)^I (-1)^{I'} z. \end{align*}$$

Since $I \cup I' = \{2, \ldots , n+1 \}$ , it is easy to show that $ (-1)^I (-1)^{I'} = (-1)^{\tfrac {n(n+5)}{2}}$ . This concludes the proof that (20) is an isomorphism.

Let us prove the compatibility with cosheaf maps. We have an inclusion $(\tau _1, \sigma _1) \hookrightarrow (\tau , \sigma )$ if and only if $\tau \subset \tau _1$ and $\sigma \subset \sigma _1$ . In particular, in defining (20), the vertex v of $\tau $ is also a vertex of $\tau _1$ . Moreover, the coheaf maps $\mathcal F_p(0, \sigma ) \rightarrow \mathcal F_p(0, \sigma _1)$ and $\mathcal F_{n-p}(0, \hat \tau ) \rightarrow \mathcal F_{n-p}(0, \hat \tau _1)$ are just inclusions of sub-modules. So that we have a commuting diagram:

where the vertical arrows are given by the map $z \mapsto \iota _{v \wedge z} \Omega _N$ . When we pass to the quotients, we get the commuting diagram in the statement of the theorem.

In particular, this proves that the pairs $(\mathcal {J}^S(T^\circ ,T), \mathcal M_p \otimes A)$ and $( \mathcal {J}^S(T, T^\circ ) , \mathcal M_{n-p} \otimes A)$ are canonically isomorphic, and therefore, they have canonically isomorphic homologies. This concludes the proof.

4.1 Combinatorial patchworking

The patchworking method was invented by Viro in the 1980s, and it has proved to be a very powerful tool to construct real algebraic hypersurfaces (and complete intersections) in toric varieties with prescribed topology of the real part [Reference Viro22], [Reference Viro21]. Let us recall here the (primitive) combinatorial version. Let $\Delta $ be a lattice polytope in $M_{\mathbb {R}}$ and T a primitive lattice triangulation of $\Delta $ . Let $\varepsilon :\Delta \cap M\rightarrow \mathbb {F}_2$ be a sign distribution on the lattice points of $\Delta $ (since the triangulation is primitive, note that the lattice points of $\Delta $ coincide with the vertices of the triangulation). When we say signs in the context of Viro’s patchworking, we will identify $+$ with $0\in \mathbb {F}_2$ and $-$ with $1$ .

Recall that the real part of the toric variety associated to $\Sigma _\Delta $ is constructed by considering a certain quotient of copies $\Delta (\xi )$ of $\Delta $ for any group homomophism $\xi :M \rightarrow \mathbb {F}_2$ ; see Equation (3). Denote by $T(\xi )$ the triangulation of $\Delta (\xi )$ induced by T. Given a lattice point $v\in \Delta $ , the sign of its copy $v(\xi )$ in $\Delta (\xi )$ is defined by $\varepsilon (v(\xi )) = \xi (v)+\varepsilon (v) \in \mathbb {F}_2$ . For any $\xi $ and any $(n+1)$ -simplex $\sigma $ of $\Delta (\xi )$ , separate the vertices having different signs by taking the union of convex hulls of barycenters of simplices $\sigma _1\subset \cdots \subset \sigma $ such that $\sigma _1$ is an edge with different signs. This gives a simplicial complex defining a PL-hypersurface in the quotient $\mathbb {R}\Sigma _\Delta $ , denoted by $X_{T,\varepsilon }$ . See Figures 5 and 8 for examples (ignore the divisors for the moment). In the standard description of combinatorial patchworking [Reference Itenberg17], the hypersurface $X_{T,\varepsilon }$ is defined by considering convex hulls of midpoints of edges with different signs, but the two descriptions give the same result up to isotopy. In the case of a convex triangulation induced by a convex function $\mu : \Delta \rightarrow \mathbb {R}$ , Viro’s combinatorial patchworking theorem states that the PL-hypersurface $X_{T,\varepsilon }$ is ambient isotopic (in $\mathbb {R}\Sigma _\Delta $ ) to the closure (in $\mathbb {R}\Sigma _\Delta $ ) of the hypersurface in $(\mathbb {R}^*)^{n+1}$ defined by the polynomial

(23) $$ \begin{align} \sum_{v\in \Delta\cap M} (-1)^{\varepsilon(v)} t^{\mu(v)} x^v, \end{align} $$

for t small enough and positive. In this text, we consider also different compactifications. In fact, given primitive central triangulations $T^{\circ }$ and T of a pair of dual reflexive polytopes $\Delta ^{\circ }$ and $\Delta $ , we consider the toric variety coming from the fan $\Sigma _{T^\circ }$ . We denote by $X_{T^\circ ,T,\varepsilon }$ the compactification of $X_{T,\varepsilon }\cap (\mathbb {R}^*)^{n+1}$ in $\mathbb {R}\Sigma _{T^\circ }$ . Again, under convexity hypothesis, the PL-hypersurface $X_{T^\circ ,T,\varepsilon }$ is ambient isotopic in $\mathbb {R}\Sigma _{T^\circ }$ to the compactification in $\mathbb {R}\Sigma _{T^\circ }$ of the hypersurface given by (23).

4.2 Patchworking and divisors in the mirror

Consider the Picard group of a toric variety $\mathbb {C}\Sigma $ with $\mathbb {F}_{2}$ coefficients, denoted by $\operatorname {\mathrm {Pic}}_{\mathbb {F}_2}(\mathbb {C}\Sigma )$ . For a smooth toric variety $\mathbb {C}\Sigma $ with $\Sigma $ a complete fan in M, this group is computed as follows. Let $\operatorname {\mathrm {Div}}_{\text {toric}} \mathbb {C}\Sigma $ be the free module generated by toric divisors; that is,

$$\begin{align*}\operatorname{\mathrm{Div}}_{\text{toric}} \mathbb{C}\Sigma = \sum_{\rho \in \Sigma(1)} \mathbb{F}_{2} D_{\rho}, \end{align*}$$

where the sum runs over all rays of the fan (i.e., one-dimensional cones) and $D_{\rho }$ is the toric divisor associated to the ray. Then we have the following short exact sequence:

(24) $$ \begin{align} 0 \rightarrow N \otimes \mathbb{F}_2 \rightarrow \operatorname{\mathrm{Div}}_{\text{toric}} \mathbb{C}\Sigma \rightarrow \operatorname{\mathrm{Pic}}_{\mathbb{F}_2}(\mathbb{C}\Sigma) \rightarrow 0. \end{align} $$

The second arrow is the map $n \mapsto \sum _{\rho } \langle n,v_{\rho } \rangle D_{\rho }$ , where $v_{\rho } \in M \otimes \mathbb {F}_{2}$ is the $\mod 2$ reduction of the primitive generator of the ray $\rho $ , see [Reference Fulton13] and [Reference Cox, Little and Schenck11].

We now go back to the dual reflexive polytopes $\Delta $ and $\Delta ^{\circ }$ with primitive central triangulations T and $T^{\circ }$ . Suppose we do patchworking with T and some sign distribution $\varepsilon $ , thus obtaining a PL-hypersurface $X_{T^{\circ }, T,\varepsilon }$ in $\mathbb {R}\Sigma _{T^\circ }$ , which, in the convex case, is isotopic to a real Calabi-Yau variety.

The sign distribution $\varepsilon $ also determines a toric divisor $D^{\varepsilon } \in \operatorname {\mathrm {Div}}_{\text {toric}} \mathbb {C}\Sigma _T $ as follows. Let $\mathbf {o} \in T$ be the unique interior lattice point. Then the other lattice points $v \in \partial T$ are the primitive generators of the rays of the fan $\Sigma _T$ . Therefore, we can define

$$\begin{align*}D^{\varepsilon} = \sum_{\varepsilon(v) = \varepsilon (\mathbf o)} D_{v}. \end{align*}$$

It is clear that, up to inverting the sign of every lattice point in T, there is a one to one correspondence between toric divisors with $\mathbb {F}_2$ coefficients on $\mathbb {C}\Sigma _T$ and a choice of patchworking signs on T. Therefore, we denote by $X_{T^{\circ }, T, D}$ the PL-hypersurface in $\mathbb {R}\Sigma _{T^{\circ }}$ constructed by the patchworking signs determined by $D \in \operatorname {\mathrm {Div}}_{\text {toric}} \mathbb {C}\Sigma _T$ . We have then the following:

Proof. We will prove that D and $D'$ are equivalent $\mathbb {F}_2$ divisors if and only if they correspond to sign distributions $\varepsilon $ and $\varepsilon '$ satisfying $\varepsilon '=\varepsilon +\xi $ for some $\xi \in \operatorname {\mathrm {Hom}}(M,\mathbb {F}_2)$ . This obviously implies the statement. Using (24), D and $D'$ are equivalent divisors if and only if

$$\begin{align*}D' = D + \sum_{v \in \partial T} \langle n, v - \mathbf o \rangle D_v \end{align*}$$

for some $n \in N \otimes \mathbb {F}_2$ . Note that $N\otimes \mathbb {F}_2=\operatorname {\mathrm {Hom}}(M,\mathbb {F}_2)$ .

Let

$$\begin{align*}D = \sum_{v \in \partial T} a_v D_v \end{align*}$$

with $a_v \in \mathbb {F}_2$ .

Fix the sign of $\mathbf o$ to be $\varepsilon (\mathbf o) = 1\in \mathbb {F}_2$ . Then the signs corresponding to D are $\varepsilon (v) = a_v$ . Let $\xi \in \operatorname {\mathrm {Hom}}(M,\mathbb {F}_2)$ . We see that for $v\in \partial T$ , one has

$$ \begin{align*}\varepsilon(v)+\xi (v) = \varepsilon(\mathbf o) + \xi (\mathbf o) \end{align*} $$

iff

$$ \begin{align*}a_v+\xi (v-\mathbf o)=1. \end{align*} $$

This means that if $\xi = n$ , the divisor corresponding to $\varepsilon +\xi $ is precisely

$$ \begin{align*}D + \sum_{v \in \partial T} \langle n, v - \mathbf o \rangle D_v, \end{align*} $$

which is $D'$ . This concludes the proof of the lemma.

Example 4.2. The real cubic curves in $\mathbb {P}^2$ obtained from patchworking with the sign distribution corresponding to $D= D_7 + D_8$ and to $D=D_8$ are depicted in Figure 5. Notice that the first one is connected, while the second one is not.

Figure 5

The signs corresponding to $D = D_7 + D_8$ and to $D=D_8$ and the corresponding real cubics.

4.3 Real structures and the patchworking spectral sequence

In this section, we describe in our context the spectral sequence from [Reference Renaudineau and Shaw20], allowing a better understanding of the Betti numbers of a combinatorial patchworking.

This definition appeared first in [Reference Renaudineau and Shaw20] for tropical hypersurfaces. Notice that $\mathcal E_{\sigma }$ can also be interpreted as a choice of an affine hyperplane of $N \otimes \mathbb {F}_2$ parallel to $\sigma ^{\perp }$ . The condition on the two-dimensional simplex $\tau $ is that the three affine hyperplanes, corresponding to the edges of $\tau $ , pairwise intersect in three distinct codimension $2$ subspaces parallel to $\tau ^{\perp }$ . A real phase structure is equivalent to a choice of patchworking signs on the triangulation T, up to inversion of all signs. Indeed, this follows by establishing the rule that the vertices of an edge $\sigma $ have the same sign if and only if $\mathcal E_{\sigma }$ is the generator of $\frac {N \otimes \mathbb {F}_2}{\sigma ^{\perp }}$ . Then a sign distribution determines a real phase structure, and vice versa, given a real phase structure $\mathcal E$ , the choice of the sign of one vertex determines the signs of all other vertices.

Definition 4.4. Let $\mathcal E$ be a real phase structure on $(\Delta , T)$ . For $(\tau ,\sigma )\in \mathcal {P}^1(T^\circ ,T)$ , define

$$ \begin{align*}\mathcal{E}(\tau,\sigma)=\left\lbrace s\in N\otimes \mathbb{F}_2 / \langle C(\tau) \rangle \:\: \vert \:\: \pi_{\gamma}(s)=\mathcal{E}_\gamma \text{ for some edge } \gamma\subset\sigma \right\rbrace, \end{align*} $$

where $\pi _{\gamma }: N\otimes \mathbb {F}_2 / \langle C(\tau ) \rangle \rightarrow N\otimes \mathbb {F}_2 / \gamma ^\perp $ .

Define also

$$ \begin{align*}\mathcal{P}_{\mathbb{R}}(T^\circ,T)= \left\lbrace (\tau, \sigma, s) \:\:\vert \:\: (\tau,\sigma)\in \mathcal{P}(T^\circ,T), s\in N\otimes \mathbb{F}_2 / \langle C(\tau) \rangle \right\rbrace. \end{align*} $$

The set $\mathcal {P}_{\mathbb {R}}(T^\circ ,T)$ is also a poset, where the order is given by $(\tau , \sigma , s)\leq (\tau ', \sigma ', s')$ if $(\tau , \sigma )\leq _{\mathcal {P}} (\tau ', \sigma ')$ and $\pi _{\tau ',\tau }(s')=s$ , where $\pi _{\tau ',\tau }:N\otimes \mathbb {F}_2 / \langle C(\tau ') \rangle \rightarrow N\otimes \mathbb {F}_2 / \langle C(\tau ) \rangle $ is the $\mathbb {F}_2$ reduction of the projection maps (8). Consider the geometric realization of the poset $\mathcal {P}_{\mathbb {R}}(T^\circ ,T)$ and its subdivision given by

$$ \begin{align*}\vert \mathcal{P}_{\mathbb{R}}(T^\circ,T)\vert = \bigcup_{(\tau,\sigma, s)\in \mathcal{P}_{\mathbb{R}}(T^\circ,T)} F(\tau,\sigma,s), \end{align*} $$

where

$$ \begin{align*}F(\tau,\sigma,s)= \bigcup_{(\tau,\sigma,s)_\bullet \in (\tau,\sigma,s)_{\leq}}\vert (\tau,\sigma,s)_\bullet \vert. \end{align*} $$

Proof. This is similar to Proposition 2.7. Here, the face poset of the real toric variety associated to a fan $\Sigma $ is the poset

$$ \begin{align*}\Sigma_{\mathbb{R}}:=\left\lbrace (\rho,s) \: \vert \: \rho\in\Sigma \text{ and } s\in N\otimes\mathbb{F}_2 / \langle \rho \rangle \right\rbrace. \end{align*} $$

Again, the poset $\mathcal {P}_{\mathbb {R}}(T^{\circ },T)$ defines a subdivision of $\Sigma _{T^\circ ,\mathbb {R}}$ , so their geometric realizations are homeomorphic.

Given a real phase structure $\mathcal {E}$ on $(\Delta ,T)$ , consider the following subposet of $\mathcal {P}_{\mathbb {R}}(T^{\circ }, T)$ :

$$ \begin{align*}\mathcal{P}^1_{\mathcal{E}}(T^\circ,T)= \left\lbrace (\tau, \sigma, s) \in \mathcal{P}_{\mathbb{R}}(T^\circ,T) \:\: \vert \:\: \dim \sigma \geq 1, \ \ s\in\mathcal{E}(\tau,\sigma) \right\rbrace. \end{align*} $$

Proof. Recall that the hypersurface $X_{T,\varepsilon }$ is obtained by considering for any $\xi \in N\otimes \mathbb {F}_2$ and any $(n+1)$ -simplex $\sigma $ of $\Delta (\xi )$ the union of convex hulls of barycenters of simplices $\sigma _1\subset \cdots \subset \sigma $ such that $\sigma _1$ is an edge with different signs. So after taking the quotient in $\mathbb {R}\Sigma _\Delta $ , the hypersurface $X_{T,\varepsilon }$ is realized by the poset

$$ \begin{align*}\left\lbrace ((\sigma_1\subset\cdots\subset\sigma_{k}),\xi) \:\: \vert \:\: \sigma_i\in T, \xi\in N\otimes\mathbb{F}_2/(F_k)^\perp, \pi_{\sigma_1}(\xi)=\mathcal{E}_{\sigma_1} \right\rbrace. \end{align*} $$

Here, $F_k$ denotes the minimal face of $\Delta $ containing $\sigma _k$ . Recall also that the poset $\mathcal {P}(T^\circ ,T)$ was defined as a small modification of the poset

$$ \begin{align*}\Phi:=\left\lbrace (\rho,\sigma)\in\Sigma_\Delta\times T \mid \sigma\subset \rho^\vee \right\rbrace. \end{align*} $$

One can consider

$$ \begin{align*}\Phi_{\mathcal{E}}:=\left\lbrace (\rho,\sigma,s) \: \vert \: (\rho,\sigma) \in \Phi \text{ and } s\in \mathcal{E}(\rho,\sigma) \right\rbrace. \end{align*} $$

The surjective map of posets

$$ \begin{align*}((\sigma_1\subset\cdots\subset\sigma_{k}),\xi)\rightarrow (F_k^\vee,\sigma_1,\xi) \end{align*} $$

shows that $\Phi _{\mathcal {E}}$ induces a regular CW-structure on $X_{T,\varepsilon }$ . The result is obtained by taking the preimages by the map $\operatorname {Bl}: \mathbb {R}\Sigma _{T^\circ } \rightarrow \mathbb {R}\Sigma _{\Delta }$ as in the proof of Proposition 2.7.

There is an obvious surjective map from $\mathcal {P}_{\mathbb {R}}(T^\circ ,T)$ to $\mathcal {P}(T^\circ ,T)$ by forgetting the last coordinate, and results from [Reference Renaudineau and Shaw20] extend to this case. We recall them for reader convenience. The proofs are similar as in [Reference Renaudineau and Shaw20].

Definition 4.7. The sign cosheaf on $\mathcal {P}(T^\circ ,T)$ is given by

$$ \begin{align*}\mathcal{S}(\tau,\sigma):=\mathbb{F}_2^{\mathcal{E}(\tau,\sigma)}. \end{align*} $$

For $\varepsilon \in \mathcal {E}(\tau ,\sigma )$ , we will denote by $w_\varepsilon $ the corresponding generator of $\mathcal {S}(\tau ,\sigma )$ .

Proposition 4.9. Let $\mathcal {E}$ be a real phase structure on $(\Delta ,T)$ . There exists a filtration of cosheaves on $\mathcal {P}(T^\circ ,T)$

$$ \begin{align*}0=\mathcal{K}_{n+1}\subset\mathcal{K}_{n}\subset\mathcal{K}_{n-1}\subset\cdots\subset \mathcal{K}_0=\mathcal{S} \end{align*} $$

such that

$$ \begin{align*}\mathcal{K}_{p}/\mathcal{K}_{p+1}\simeq \mathcal{F}_p. \end{align*} $$

For the reader’s convenience, we recall briefly the definition of the filtration $\mathcal {K}_p$ and of the isomorphisms

$$ \begin{align*}\mathcal{K}_{p}/\mathcal{K}_{p+1}\simeq \mathcal{F}_p. \end{align*} $$

If $\dim \sigma =1$ , then $\mathcal {E}(\tau ,\sigma )$ is an affine space of direction $\sigma ^\perp / \langle C(\tau ) \rangle $ . Define in this case

$$ \begin{align*}\mathcal{K}_p(\tau,\sigma)=\mathbb{F}_2 \langle \sum_{\varepsilon\in H}w_\varepsilon \:\vert \: H\in \operatorname{\mathrm{Aff}}_p(\mathcal{E}(\tau,\sigma))\rangle,\end{align*} $$

where $\operatorname {\mathrm {Aff}}_p(\mathcal {E}(\tau ,\sigma ))$ is the space of affine subspaces of $\mathcal {E}(\tau ,\sigma )$ of dimension p. In the general case, consider the sum

$$ \begin{align*}\mathcal{K}_p(\tau,\sigma):=\sum_{\gamma\subset\sigma \: \dim\gamma=1}\mathcal{K}_p(\tau,\gamma). \end{align*} $$

The isomorphism

$$ \begin{align*}\mathcal{K}_{p}/\mathcal{K}_{p+1}\simeq \mathcal{F}_p \end{align*} $$

is induced by mapping $w_{G}$ to $v_1 \wedge \dots \wedge v_p$ , for $G \in \operatorname {\mathrm {Aff}}_p(\mathcal {E}(\tau ,\sigma ))$ where $\dim \sigma =1$ . Here, $w_{G}$ denotes $\sum _{\varepsilon \in G}w_\varepsilon $ and $v_1, \dots , v_p$ form a basis of the tangent space (over $\mathbb {F}_2$ ) of G. The spectral sequence $(E_{p,q}^{k}, \delta ^{[k]})$ associated to this filtration has first page

$$ \begin{align*}E^1_{p,q}=H_q(X_{T^\circ,T};\mathcal{F}_p)\end{align*} $$

and converges to $H_\bullet (X_{T^\circ ,T};\mathcal {S})$ , which by Proposition 4.8 is the homology of $X_{T^\circ ,T,\varepsilon }$ .