Tropical Lagrangian multi-sections
and
tropical locally free sheaves

The Gross-Siebert program [6, 7, 8] gives an algebro-geometric understanding of SYZ mirror symmetry [14]. In [4], together with Chan and Ma, the author of this paper attempted to understand homological mirror symmetry [10] in terms of the Gross-Siebert setup. We introduced there the notion of tropical Lagrangian multi-sections over any 2-dimensional integral affine manifold of singularities $B$ equipped with polyhedral decomposition $\mathscr{P}$ and constructed, by fixing certain local model, a locally free sheaf $\mathcal{E}_{0}$ over the associated scheme $X_{0}(B,\mathscr{P})$. We also provided a nice combinatorial condition for smoothability of the pair $(X_{0}(B,\mathscr{P}),\mathcal{E}_{0})$ under some extra assumptions.

In this article, we will generalize the notion of tropical Lagrangian multi-sections to any dimension. We begin by reviewing some preliminary of the Gross-Siebert program in Section 2. In Section 3, we introduce the notion of tropical Lagrangian multi-sections over any integral affine manifold with singularities $B$ equipped with a polyhedral decomposition $\mathscr{P}$. For this purpose, we need the notion of tropical spaces, which has been introduced in [12]. Roughly speaking, tropical spaces are spaces that allow us to talk about sheaf of affine and piecewise linear functions. A tropical Lagrangian multi-section $\mathbb{L}$ over $(B,\mathscr{P})$ is then a branched covering map $\pi:(L,\mathscr{P}^{\prime},\mu)\to(B,\mathscr{P})$ between tropical spaces that respect the polyhedral decomposition $\mathscr{P},\mathscr{P}^{\prime}$ together with a multi-valued piecewise linear function $\varphi$ on $L$. Here $\mu:\mathscr{P}\to\mathbb{Z}_{>0}$ is the multiplicity map. The multi-valued function $\varphi$ should be thought of as certain tropical limit of the local potential of a Lagrangian multi-section of the SYZ fibration.

Given a tropical Lagrangian multi-section $\mathbb{L}$ over $(B,\mathscr{P})$. Due to its discrete nature, one shouldn’t expect $\mathbb{L}$ can determine a sheaf on $X_{0}(B,\mathscr{P})$ uniquely. Therefore, we need to prescribe some continuous data on top of the discrete data determined by $\mathbb{L}$. We will introduce in 4 two continuous data $({\bf{g}},{\bf{h}})$ which guarantee the existence of a locally free sheave on $X_{0}(B,\mathscr{P})$. The idea is to apply the technique in [15] to construct a collection of toric vector bundles $\{\mathcal{E}({\bf{g}}(\tau^{\prime}))\}_{\pi(\tau^{\prime})=\tau}$ on each toric piece $X_{\tau}$ by using the multi-valued piecewise linear function $\varphi$ on $L$. The rank of each $\mathcal{E}({\bf{g}}(\tau^{\prime}))$ is given by the ramification degree of the cell $\tau^{\prime}\in\mathscr{P}^{\prime}$. Put

$$\mathcal{E}({\bf{g}}(\tau)):=\bigoplus_{\pi(\tau^{\prime})=\tau}\mathcal{E}({\bf{g}}(\tau^{\prime})),$$

which is a toric vector bundle on $X_{\tau}$, whose rank is exactly the degree of the branched covering map $\pi:L\to B$. Then we glue the vector bundles $\{\mathcal{E}({\bf{g}}(\tau))\}_{\tau\in\mathscr{P}}$ together by using the data ${\bf{h}}$. However, in general, we may encounter an extra twisting data $\overline{s}$ when we glue the toric strata $\{X_{\tau}\}_{\tau\in\mathscr{P}}$ together. In this case, there is an obstruction $o_{\mathbb{L}}([\overline{s}])\in H^{2}(L,\mathbb{C}^{\times})$ for gluing $\{\mathcal{E}({\bf{g}}(\tau))\}_{\tau\in\mathscr{P}}$. This obstruction generalizes the obstruction maps appear in [6], Theorem 2.34 (the ample line bundle case) and [4], Theorem 5.5 (the 2-dimensional case).

The vanishing of $o_{\mathbb{L}}([\overline{s}])$ will give us another continuous data ${\bf{k}}_{s}$, which can be combined with ${\bf{h}}$ to form ${\bf{h}}_{s}$. We then write ${\bf{D}}_{s}$ for the data $({\bf{g}},{\bf{h}}_{s})$ and denote by $\mathscr{D}_{s}(\mathbb{L})$ the set of all such data. Theorem 1.1 says that a choice of data ${\bf{D}}_{s}\in\mathscr{D}_{s}(\mathbb{L})$ will provide us a locally free sheaf $\mathcal{E}_{0}(\mathbb{L},{\bf{D}}_{s})$ on the scheme $X_{0}(B,\mathscr{P},s)$.

In [5], the notion of unobstructed Lagrangian submanifolds ([1] for immersed Lagrangian submanifolds) was introduced. The main feature of an unobstructed Lagrangian submanifolds is that its Floer cohomology is well-defined and hence defines an object in the Fukaya category. In particular, unobstructed Lagrangian submanifolds should have the corresponding mirror objects. Therefore, we borrow this terminology here. Namely, for a fixed gluing data $s$, a tropical Lagrangian multi-section $\mathbb{L}$ over $(B,\mathscr{P})$ is said to be unobstructed if $\mathscr{D}_{s}(\mathbb{L})\neq\emptyset$ and a pair $(\mathbb{L},{\bf{D}}_{s})$ is called a tropical Lagrangian brane (Definition 4.8).

Section 5 will be devoted to the reverse construction. Based on the construction in Section 4, we introduce the notion of tropical locally free sheaves (Definition 5.1). To such a locally free sheaf $\mathcal{E}_{0}$, we are able to construct a canonical tropical Lagrangian multi-section $\mathbb{L}_{\mathcal{E}_{0}}$. Moreover, there is a natural data ${\bf{D}}_{s}(\mathcal{E}_{0})\in\mathscr{D}_{s}(\mathbb{L}_{\mathcal{E}_{0}})$ such that

$$\mathcal{E}_{0}\cong\mathcal{E}_{0}(\mathbb{L}_{\mathcal{E}_{0}},{\bf{D}}_{s}(\mathcal{E}_{0})).$$

We will end Section 5 by showing that we actually have an abundant sources of examples given by restricting toric vector bundles on the toric boundary of a toric variety (Theorem 5.9).

Given $(\mathbb{L},{\bf{D}}_{s})$, Section 4 has taught us how to construct a tropical locally free sheaf $\mathcal{E}_{0}:=\mathcal{E}_{0}(\mathbb{L},{\bf{D}}_{s})$ while Section 5 has provided a canonical tropical Lagrangian brane $(\mathbb{L}_{\mathcal{E}_{0}},{\bf{D}}_{s}(\mathcal{E}_{0}))$ out of $\mathcal{E}_{0}$. It is natural to compare $(\mathbb{L},{\bf{D}}_{s})$ and $\mathbb{L}_{\mathcal{E}_{0}}$. In general, we may not have $\mathbb{L}=\mathbb{L}_{\mathcal{E}_{0}}$. This indicates the phenomenon that non-Hamiltonian isotopic or topologically different Lagrangian submanifolds can still be equivalent to each other in the derived (immersed) Fukaya category. See, for example, [3]. Therefore, we would still like to regard them as the same object. This brings us to Section 6, where we will introduce the notion of combinatorial equivalence of tropical Lagrangian branes and prove the following

Theorem 1.2 provides a slightly more geometric understanding of homological mirror symmetry in the sense that we don’t need any derived objects on both sides to achieve the correspondence. Although a tropical Lagrangian multi-section is still not yet an honest Lagrangian multi-section, one should expect that an unobstructed Lagrangian multi-section can be constructed from the data $(\mathbb{L},{\bf{D}}_{s})$. We left this for future research.

We give a brief review of how the scheme $X_{0}(B,\mathscr{P},s)$ is constructed. We follow [6] and use the fan construction.

Let $B$ be an integral affine manifold with singularities equipped with a polyhedral decomposition $\mathscr{P}$. Elements in $\mathscr{P}$ are celled cells. Throughout the whole article, we assume $B$ is compact without boundary and all cells have no self-intersections. By taking the barycentric decomposition situation of $\mathscr{P}$, there is a canonical open cover $\mathcal{W}:=\{W_{\sigma}\}_{\sigma\in\mathscr{P}}$ of $B$ so that $W_{\tau}\cap W_{\sigma}\neq\emptyset$ if and only if $\tau\subset\sigma$. Denote by $\Delta\subset B$ the singular locus of $B$, which is a union of locally closed codimension 2 submanifolds inside the codimension 1 strata of $B$ and $\Lambda$ the lattice induced by the integral structure on $B\backslash\Delta$. For $\tau\in\mathscr{P}$, let

$$\Lambda_{\tau}:=\{v\in\Lambda_{y}:v\text{ is tangent to }\tau\text{ at }y\},$$

for any $y\in\mathrm{Int}(\tau)\backslash\Delta$. This lattice is independent of the choice of $y\in\mathrm{Int}(\tau)$. We also define

$$\mathcal{Q}_{\tau}:=\Lambda/\Lambda_{\tau}.$$

For $\tau\subset\sigma$, by parallel transport along a path in $B\backslash\Delta$ starting on $\mathrm{Int}(\tau)\backslash\Delta$ and ending on $\mathrm{Int}(\sigma)$, we get a projection $p:\mathcal{Q}_{\tau}\to\mathcal{Q}_{\sigma}$. We always assume $\mathscr{P}$ is toric, that is, for each $\tau\in\mathscr{P}$, there is a submersion $S_{\tau}:W_{\tau}\to\mathcal{Q}_{\tau}\otimes_{\mathbb{Z}}\mathbb{R}$. This gives a complete fan

$$\Sigma_{\tau}:=\{r\cdot S_{\tau}(\sigma)\,|\,\sigma\supset\tau,r\geq 0\}$$

on $\mathcal{Q}_{\tau}\otimes_{\mathbb{Z}}\mathbb{R}$ and hence a complete toric variety $X_{\tau}$. To glue them together, Gross-Siebert introduced the category ${\textbf{Cat}}(\mathscr{P})$, whose objects are elements in $\mathscr{P}$ and

$$\mathrm{Hom}_{{\textbf{Cat}}(\mathscr{P})}(\tau,\sigma):=\begin{cases}\emptyset&\text{ if }\tau\not\subset\sigma,\\ \{e\}&\text{ if }\tau\subset\sigma.\end{cases}$$

For $e:\tau\to\sigma$, the inclusion $p_{e}^{*}:\mathcal{Q}_{\sigma}^{*}\to\mathcal{Q}_{\tau}^{*}$ induces a natural inclusion $F(e):X_{\sigma}\to X_{\tau}$ that satisfies

$$F(e_{2}\circ e_{1})=F(e_{1})\circ F(e_{2}),$$

for all $e_{1}:\sigma_{1}\to\sigma_{2},e_{2}:\sigma_{2}\to\sigma_{3}$. Hence we can take the limit

$$\lim_{\longrightarrow}X_{\sigma}$$

to obtain an algebraic space. One can twist this construction by a cocycle $[\overline{s}]\in H^{1}(\mathcal{W},\mathcal{Q}_{\mathscr{P}}\otimes\mathbb{C}^{\times})$. Such an extra twisting is called a closed gluing data for the fan picture. Such cocycle give us for each $e:\tau\to\sigma$, an element $\overline{s}_{e}\in\mathcal{Q}_{\sigma}\otimes\mathbb{C}^{\times}$, which defines an automorphism $\overline{s}_{e}:X_{\sigma}\to X_{\sigma}$. Put

$$F_{\overline{s}}(e):=F(e)\circ\overline{s}_{e}.$$

Since $\overline{s}$ is a 1-cocycle, we have $\overline{s}_{e_{1}\circ e_{2}}=\overline{s}_{e_{1}}\circ\overline{s}_{e_{2}}$ and hence the $\overline{s}$-twisted limit

$$X_{0}(B,\mathscr{P},\overline{s}):=\lim_{\longrightarrow}X_{\sigma}$$

makes sense and exists in the category of algebraic spaces. In [6], Gross-Siebert also introduced an other more refined gluing data, called the open gluing data.

The set of open gluing data modulo equivalence is parametrized by the group

$$H^{1}(\mathscr{P},\mathcal{Q}_{\mathscr{P}}\otimes\mathbb{C}^{\times}):=\frac{Z^{1}(\mathscr{P},\mathcal{Q}_{\mathscr{P}}\otimes\mathbb{C}^{\times})}{B^{1}(\mathscr{P},\mathcal{Q}_{\mathscr{P}}\otimes\mathbb{C}^{\times})}.$$

For an open gluing data $s$, one associates the closed gluing data $\overline{s}$ as follows. For $e:\tau\to\sigma$, define $\overline{s}_{e}$ to be the image of $s$ under the composition

$$\Gamma(\tau,\mathcal{Q}_{\mathscr{P}}\otimes\mathbb{C}^{\times})\to\mathcal{Q}_{\tau}\otimes\mathbb{C}^{\times}\xrightarrow{\sim}\Gamma(W_{\tau},\mathcal{Q}_{\mathscr{P}}\otimes\mathbb{C}^{\times})\to\Gamma(W_{e},\mathcal{Q}_{\mathscr{P}}\otimes\mathbb{C}^{\times}),$$

where the last map is given by restriction. This induces the open-to-closed map

$$H^{1}(\mathscr{P},\mathcal{Q}_{\mathscr{P}}\otimes\mathbb{C}^{\times})\to H^{1}(\mathcal{W},\mathcal{Q}_{\mathscr{P}}\otimes\mathbb{C}^{\times}),$$

which is injective by Proposition 2.32 in [6]. An important consequence of open gluing data is that the algebraic space $X_{0}(B,\mathscr{P},\overline{s})$ can be built from some standard affine charts defined as follows. The open gluing data and the associated closed gluing data give the following commutative diagram

The colimit of the left hand side

$$V(\sigma):=\lim_{\longrightarrow}V_{\tau\to\sigma}$$

is actually an affine scheme and hence one can construct an algebraic space $X_{0}(B,\mathscr{P},s)$ by gluing $\{V(\sigma)\}_{\sigma\in\mathscr{P}_{max}}$ and it was shown by using universal property of colimit that

$$X_{0}(B,\mathscr{P},s)\cong X_{0}(B,\mathscr{P},\overline{s}).$$

as algebraic spaces (Proposition 2.30 in [6]). The fact that all cells have no self-intersections implies $X_{0}(B,\mathscr{P},s)$ is actually a scheme. Moreover, Proposition 2.32 in [6] also showed that there is an isomorphism $X_{0}(B,\mathscr{P},s)\cong X_{0}(B,\mathscr{P},s^{\prime})$ preserving toric strata if and only if $[s]=[s^{\prime}]$.

Assume $\mathscr{P}$ is simple and positive (see [6] for their definitions). With a suitable choice of open gluing data $s$, Gross-Siebert have shown in [6] that $X_{0}(B,\mathscr{P},s)$ carries a log structure that is log smooth off a codimension 2 locus $Z$, not containing any toric strata. Moreover, they proved in [8] that $X_{0}(B,\mathscr{P},s)$ is smoothable to a formal family over $\mathrm{Spec}({\mathbb{C}[[t]]})$. In [13], Ruddat and Siebert proved that this formal family is in fact an analytic family.

We introduce the notion of tropical Lagrangian multi-sections over any integral affine manifold with singularities equipped with a polyhedral decomposition, generalizing the definition of tropical Lagrangian multi-sections in [4]. We use the notion of tropical space introduced in [12].

There is a natural notion of morphisms between tropical spaces.

The following lemma is evident.

Given two topological spaces $L,B$, a continuous map $\pi:L\to B$ and a function $\mu:L\to\mathbb{Z}_{>0}$. If for any $x\in B$, the preimage set $\pi^{-1}(x)$ is finite, then we can define a function $Tr_{\pi}(\mu):B\to\mathbb{Z}_{>0}$ by

$$Tr_{\pi}(\mu)(x):=\sum_{x^{\prime}\in\pi^{-1}(x)}\mu(x^{\prime}).$$

Now we can define branched covering map between tropical spaces.

Let $B$ be an integral affine manifold with singularities equipped with a polyhedral decomposition $\mathscr{P}$. It carries a natural tropical space structure $\mathcal{A}ff_{B},\mathcal{PL}_{\mathscr{P}}$. See [6], Section 1. Unless specified, we use this tropical space structure for $(B,\mathscr{P})$ without further notice.

Given $\pi:(L,\mathscr{P}^{\prime},\mu)\to(B,\mathscr{P})$. For $\tau^{\prime}\in\mathscr{P}^{\prime}$ and $\tau:=\pi(\tau^{\prime})$, define $W_{\tau^{\prime}}$ to be the connected component of $\pi^{-1}(W_{\tau})$ that contains $\mathrm{Int}(\tau^{\prime})$.

Suppose $\pi:(L,\mathscr{P}^{\prime},\mu)\to(B,\mathscr{P})$ is toric and $\varphi:W_{\tau^{\prime}}\to\mathbb{R}$ is a piecewise linear function. Then there is an affine function $f^{\prime}:W_{\tau^{\prime}}\to\mathbb{R}$ and a piecewise linear function $\varphi_{\tau^{\prime}}$ on $|\Sigma_{\tau^{\prime}}|$ such that

$$\varphi-f^{\prime}=S_{\tau^{\prime}}^{*}\varphi_{\tau^{\prime}},$$

on $W_{\tau^{\prime}}$. For $\sigma^{\prime}\in\mathscr{P}_{max}^{\prime}$ contains $\tau^{\prime}$, we denote by $m_{\tau}(\sigma^{\prime})\in\mathcal{Q}_{\tau}^{*}$ the slope of $\varphi_{\tau^{\prime}}$ on $S_{\tau^{\prime}}(\sigma^{\prime})$. For $g:\tau_{1}\to\tau_{2}$, choose a path $\gamma\subset U_{\tau_{1}}$, which goes from a point in $\mathrm{Int}(\tau_{1})\backslash\Delta$ to a point in $\mathrm{Int}(\tau_{2})\backslash\Delta$. Parallel transport along $\gamma$ gives a surjection $p_{g}:\mathcal{Q}_{\tau_{1}}\to\mathcal{Q}_{\tau_{2}}$. Given $\varphi\in H^{0}(\mathcal{W}^{\prime},\mathcal{MPL}_{\mathscr{P}^{\prime}})$ and representatives $\{\varphi_{\tau^{\prime}}\}$, for $\tau_{1}^{\prime}\subset\tau_{2}^{\prime}$, there exists an affine function $f_{\tau_{1}^{\prime}\tau_{2}^{\prime}}:W_{\tau_{1}^{\prime}}\cap W_{\tau_{2}^{\prime}}\to\mathbb{R}$ such that

$$S_{\tau_{2}^{\prime}}^{*}\varphi_{\tau_{2}^{\prime}}=S_{\tau_{1}^{\prime}}^{*}\varphi_{\tau_{1}^{\prime}}+f_{\tau_{1}^{\prime}\tau_{2}^{\prime}},$$

whenever defined. Therefore, via the inclusion $p_{g}^{*}:\mathcal{Q}_{\tau_{2}}^{*}\to\mathcal{Q}_{\tau_{1}}^{*}$, for any $\sigma^{\prime}\supset\tau_{2}^{\prime}$,

$$p_{g}^{*}m_{\tau_{2}}(\sigma^{\prime})=m_{\tau_{1}}(\sigma^{\prime})+m_{\tau_{1}^{\prime}\tau_{2}^{\prime}},$$

for some $m_{\tau_{1}^{\prime}\tau_{2}^{\prime}}\in\mathcal{Q}_{\tau_{1}}^{*}$ only depends on $\tau_{1}^{\prime},\tau_{2}^{\prime}$. We simply write

$$m_{\tau_{2}}(\sigma^{\prime})=m_{\tau_{1}}(\sigma^{\prime})+m_{\tau_{1}^{\prime}\tau_{2}^{\prime}}$$

if there is no confusion.

Now we can define the main object that we are going to study in this paper.

There is a special type of morphisms between tropical Lagrangian multi-sections over the same base $(B,\mathscr{P})$.

Let $\mathbb{L}$ be a tropical Lagrangian multi-section over $(B,\mathscr{P})$ of degree $r$ and $s$ an open gluing data. By thinking $\mathbb{L}$ as a Lagrangian multi-section of a Lagrangian torus fibration over $B$, the SYZ philosophy suggests the mirror of $\mathbb{L}$ should be a holomorphic vector bundle, whose rank is same as the degree of the covering $\mathbb{L}\to B$. Therefore, in this section, we would like to construct a rank $r$ locally free sheaf on $X_{0}(B,\mathscr{P},s)$. However, as mentioned in the introduction, one shouldn’t expect $\mathbb{L}$ itself can determine a locally free sheaf due to its discrete nature. We need some extra continuous data in analogous to the linear algebra data defined in [9].

To begin, let $\tau\in\mathscr{P}$ and $\tau^{\prime}\in\mathscr{P}^{\prime}$ be a lift, we would like to construct a rank $\mu(\tau^{\prime})$ locally free sheaf on the strata $X_{\tau}$. Let $V_{\tau\to\sigma}\subset X_{\tau}$ be the affine chart corresponds to the cone $K_{\tau\to\sigma}:=\mathbb{R}_{\geq 0}\cdot S_{\tau}(\sigma)\in\Sigma_{\tau}$. Define

$$\mathcal{E}_{\sigma}(\tau^{\prime}):=\mathcal{O}_{V_{\tau\to\sigma}}^{\oplus\mu(\tau^{\prime})}.$$

For $\sigma\in\mathscr{P}_{max}$ contains $\tau$, $\mu(\tau^{\prime})$ equals to the number (count with multiplicity) of lifts of $\sigma$ that contain $\tau^{\prime}$. We then obtain a frame $\{1_{\sigma^{(\alpha)}}(\tau)\}_{\sigma^{(\alpha)}\supset\tau^{\prime}}$ for $\mathcal{E}_{\sigma}(\tau^{\prime})$, parametrized by lifts of $\sigma$ contains $\tau^{\prime}$, counting with multiplicity. To define transition maps, we use the function $\varphi$. By the toric assumption, there is a connected cone complex $\Sigma_{\tau^{\prime}}$ over $\Sigma_{\tau}$ and a piecewise linear function $\varphi_{\tau^{\prime}}$ such that $S_{\tau^{\prime}}^{*}\varphi_{\tau^{\prime}}$ represents $\varphi|_{W_{\tau^{\prime}}}$. Let $\sigma^{\prime}\in\mathscr{P}_{max}^{\prime}$ be a lift of $\sigma$ contains $\tau^{\prime}$ and $m_{\tau}(\sigma^{\prime})\in\mathcal{Q}_{\tau}^{*}$ be the slope of $\varphi_{\tau^{\prime}}$ on the cone $S_{\tau^{\prime}}(\sigma^{\prime})$. For $\sigma_{1},\sigma_{2}\supset\tau$, define $G_{\sigma_{1}\sigma_{2}}(\tau^{\prime}):\mathcal{E}_{\sigma_{1}}(\tau^{\prime})|_{V_{\tau\to\sigma_{1}\cap\sigma_{2}}}\to\mathcal{E}_{\sigma_{2}}(\tau^{\prime})|_{V_{\tau\to\sigma_{1}\cap\sigma_{2}}}$ by

$$G_{\sigma_{1}\sigma_{2}}(\tau^{\prime}):1_{\sigma_{1}^{(\alpha)}}(\tau^{\prime})\mapsto\sum_{\beta:\sigma_{2}^{(\beta)}\supset\tau^{\prime}}g_{\sigma_{1}^{(\alpha)}\sigma_{2}^{(\beta)}}(\tau^{\prime})z^{m_{\tau}(\sigma_{1}^{(\alpha)})-m_{\tau}(\sigma_{2}^{(\beta)})}1_{\sigma_{2}^{(\beta)}}(\tau^{\prime}),$$

where $\{1_{\sigma^{(\alpha)}}(\tau^{\prime})\}$ is a frame of $\mathcal{E}_{\sigma}(\tau^{\prime})$. Put

$$G_{\sigma_{1}\sigma_{2}}(\tau):=\sum_{\tau^{\prime}:\pi(\tau^{\prime})=\tau}G_{\sigma_{1}\sigma_{2}}(\tau^{\prime}).$$

The coefficient of each monomial entry of $G_{\sigma_{1}\sigma_{2}}(\tau)$ will be denoted by $g_{\sigma_{1}^{(\alpha)}\sigma_{2}^{(\beta)}}(\tau)\in\mathbb{C}$. We require them to satisfy the following

Condition (G2) implies entries of $G_{\sigma_{1}\sigma_{2}}(\tau^{\prime})$ are regular functions. Condition (G1) and the cocycle condition (G3) immediately implies the existence of a rank $\mu(\tau^{\prime})$ locally free sheaf $\mathcal{E}({\bf{g}}(\tau^{\prime}))$ on the closed toric strata $X_{\tau}$. Define

$$\mathcal{E}({\bf{g}}(\tau)):=\bigoplus_{\tau^{\prime}:\pi(\tau^{\prime})=\tau}\mathcal{E}({\bf{g}}(\tau^{\prime})),$$

which is a rank $r$ locally sheaf on $X_{\tau}$.

We would like to glue $\{\mathcal{E}({\bf{g}}(\tau))\}_{\tau\in\mathscr{P}}$ together. The idea is to embed $\mathcal{E}(\tau_{2}^{\prime})$ to $\mathcal{E}(\tau_{1}^{\prime})$ when $\tau_{1}^{\prime}\subset\tau_{2}^{\prime}$. To do this, we first use the data $\{(h_{\sigma^{(\alpha)}\sigma^{(\beta)}}(g))\}$ to construct an isomorphism $H(g):\mathcal{E}({\bf{g}}(\tau_{2}))\to F(g)^{*}\mathcal{E}({\bf{g}}(\tau_{1}))$. Define

$$H_{\sigma}(g):1_{\sigma^{(\alpha)}}(\tau_{2})\mapsto\sum_{\beta=1}^{r}h_{\sigma^{(\alpha)}\sigma^{(\beta)}}(g)z^{m_{\tau_{1}}(\sigma^{(\alpha)})-m_{\tau_{1}}(\sigma^{(\beta)})}|_{V_{\tau_{2}\to\sigma}}F(g)^{*}1_{\sigma^{(\beta)}}(\tau_{1}).$$

By Condition (H1), the entries of $H_{\sigma}(g)$ are regular functions on $V_{\tau_{2}\to\sigma}$. Given maximal $\sigma_{1},\sigma_{2}\supset\tau_{2}$, the composition $F(g)^{*}G_{\sigma_{1}\sigma_{2}}(\tau_{1})\circ H_{\sigma_{1}}(g)|_{V_{\tau_{2}\to\tau}}$ is given by

$$1_{\sigma_{1}^{(\alpha)}}(\tau_{1})\mapsto\sum_{\beta,\gamma=1}^{r}h_{\sigma_{1}^{(\alpha)}\sigma_{1}^{(\beta)}}(g)g_{\sigma_{1}\sigma_{2}}^{(\beta\gamma)}(\tau_{1})z^{m_{\tau_{1}}(\sigma_{1}^{(\alpha)})-m_{\tau_{1}}(\sigma_{2}^{(\gamma)})}|_{V_{\tau_{2}\to\tau}}F(g)^{*}1_{\sigma_{2}^{(\gamma)}}(\tau_{1}).$$

On the other hand, the composition $H_{\sigma_{2}}(g)|_{V_{\tau_{2}\to\tau}}\circ G_{\sigma_{1}\sigma_{2}}(\tau_{2})$ is given by

$$1_{\sigma_{1}^{(\alpha)}}(\tau_{2})\mapsto\sum_{\beta,\gamma=1}^{r}g_{\sigma_{1}^{(\alpha)}\sigma_{2}^{(\beta)}}(\tau_{2})h_{\sigma_{2}^{(\beta)}\sigma_{2}^{(\gamma)}}(g)z^{m_{\tau_{2}}(\sigma_{1}^{(\alpha)})-m_{\tau_{2}}(\sigma_{2}^{(\beta)})+m_{\tau_{1}}(\sigma_{2}^{(\beta)})-m_{\tau_{1}}(\sigma_{2}^{(\gamma)})}|_{V_{\tau_{2}\to\tau}}F(g)^{*}1_{\sigma_{2}^{(\gamma)}}(\tau_{1}).$$

Now, we introduce a frequently used trick, called the slope cancellation trick. By definition of $G_{\sigma_{1}\sigma_{2}}(\tau_{2})$, the constant $g_{\sigma_{1}^{(\alpha)}\sigma_{2}^{(\beta)}}(\tau_{2})$ is non-zero only if $\sigma_{1}^{(\alpha)},\sigma_{2}^{(\beta)}$ contain a common lift of $\tau_{2}$, say $\tau_{2}^{\prime}$, so in particular, they contains $\tau_{1}^{\prime}\subset\tau_{2}^{\prime}$. On the other hand, by the construction of $H_{\sigma}(g)$, the constant $h_{\sigma_{2}^{(\beta)}\sigma_{2}^{(\gamma)}}(g)$ is non-zero only if $\sigma_{2}^{(\beta)},\sigma_{2}^{(\gamma)}$ contains a common lift of $\tau_{1}$ and it must be $\tau_{1}^{\prime}$ as $\sigma_{2}^{(\beta)}\supset\tau_{1}^{\prime}$. As a whole, we conclude that $\sigma_{1}^{(\alpha)},\sigma_{2}^{(\beta)},\sigma_{2}^{(\gamma)}$ all contain the lift $\tau_{1}^{\prime}$ of $\tau_{1}$. Moreover, via the inclusion $p_{g}^{*}:\mathcal{Q}_{\tau_{2}}^{*}\to\mathcal{Q}_{\tau_{1}}^{*}$, the piecewise linear function

$$f:=p_{g}^{*}m_{\tau_{2}^{\prime}}(\sigma^{\prime})-m_{\tau_{1}^{\prime}}(\sigma^{\prime})$$

is independent of $\sigma^{\prime}$ as long as $\tau_{1}^{\prime}\subset\tau_{2}^{\prime}\subset\sigma^{\prime}$, which means $f$ is actually an affine function. This implies

	$\displaystyle m_{\tau_{2}^{\prime}}(\sigma_{1}^{(\alpha)})-m_{\tau_{2}^{\prime}}(\sigma_{2}^{(\beta)})+m_{\tau_{1}^{\prime}}(\sigma_{2}^{(\beta)})-m_{\tau_{1}^{\prime}}(\sigma_{2}^{(\gamma)})=$	$\displaystyle\,m_{\tau_{2}^{\prime}}(\sigma_{1}^{(\alpha)})-f-m_{\tau_{1}^{\prime}}(\sigma_{2}^{(\gamma)})$
	$\displaystyle=$	$\displaystyle\,m_{\tau_{1}^{\prime}}(\sigma_{1}^{(\alpha)})-m_{\tau_{1}^{\prime}}(\sigma_{2}^{(\gamma)}).$

It is worth mentioning that we are not allowed to absorb $f$ by $m_{\tau_{1}^{\prime}}(\sigma_{2}^{(\gamma)})$ as $\sigma_{2}^{(\gamma)}$ may not contain $\tau_{2}^{\prime}$.