2.1 Integral tropical manifolds

We first recall the notion of integral tropical manifolds from [Reference Gross and Siebert29, §1.1]. Given a lattice M of rank n, a rational convex polyhedron $\sigma $ is a convex subset in $M_{\mathbb {R}}$ given by a finite intersection of rational (i.e., defined over $M_{\mathbb {Q}}$ ) affine half-spaces. We usually drop the attributes ‘rational’ and ‘convex’ for polyhedra. A polyhedron $\sigma $ is said to be integral if all its vertices lie in M; a polytope is a compact polyhedron. The group $\mathbf {Aff}(M):= M \rtimes \mathrm {GL}(M)$ of integral affine transformations acts on the set of polyhedra in $M_{\mathbb {R}}$ . Given a polyhedron $\sigma \subset M_{\mathbb {R}}$ , let $\Lambda _{\sigma ,\mathbb {R}} \subset M_{\mathbb {R}}$ be the smallest affine subspace containing $\sigma $ , and denote by $\Lambda _{\sigma } := \Lambda _{\sigma ,\mathbb {R}} \cap M$ the corresponding lattice. The relative interior $\mathrm {int}_{\mathrm {re}}(\sigma )$ refers to taking the interior of $\sigma $ in $\Lambda _{\sigma ,\mathbb {R}}$ . There is an identification $T_{\sigma ,x} \cong \Lambda _{\sigma ,\mathbb {R}}$ for the tangent space at $x \in \mathrm {int}_{\mathrm {re}}(\sigma )$ . Write $\partial \sigma = \sigma \setminus \mathrm {int}_{\mathrm {re}}(\sigma )$ . Then a face of $\sigma $ is the intersection of $\partial \sigma $ with a supporting hyperplane. Codimension one faces are called facets.

Let $\underline {\mathrm {LPoly}}$ be the category whose objects are integral polyhedra and morphisms consist of the identity and integral affine isomorphisms onto faces (i.e., an integral affine morphism $\tau \rightarrow \sigma $ which is an isomorphism onto its image and identifies $\tau $ with a face of $\sigma $ ). An integral polyhedral complex is a functor $\mathtt {F}\colon \mathscr {P} \rightarrow \underline {\mathrm {LPoly}}$ from a finite category $\mathscr {P}$ to $\underline {\mathrm {LPoly}}$ such that every face of $\mathtt {F}(\sigma )$ still lies in the image of $\mathtt {F}$ , and there is at most one arrow $\tau \rightarrow \sigma $ for every pair $\tau ,\sigma \in \mathscr {P}$ . By abuse of notation, we usually drop the notation $\mathtt {F}$ and write $\sigma \in \mathscr {P}$ to represent an integral polyhedron in the image of the functor. From an integral polyhedral complex, we obtain a topological space $B := \varinjlim _{\sigma \in \mathscr {P}} \sigma $ via gluing of the polyhedra along faces. We further assume that:

1. 1. the natural map $\sigma \rightarrow B$ is injective for each $\sigma \in \mathscr {P}$ , so that $\sigma $ can be identified with a closed subset of B called a cell, and a morphism $\tau \rightarrow \sigma $ can be identified with an inclusion of subsets;

2. 2. a finite intersection of cells is a cell; and

3. 3. B is an orientable connected topological manifold of dimension n without boundary which in addition satisfies the condition that $H^1(B,\mathbb {Q}) = 0$ .

The set of k-dimensional cells is denoted by $\mathscr {P}^{[k]}$ and the k-skeleton by $\mathscr {P}^{[\leq k]}$ . For every $\tau \in \mathscr {P}$ , we define its open star by

$$ \begin{align*}U_{\tau}:= \bigcup_{\sigma \supset \tau} \mathrm{int}_{\mathrm{re}}(\sigma), \end{align*} $$

which is an open subset of B containing $\mathrm {int}_{\mathrm {re}}(\tau )$ . A fan structure along $\tau \in \mathscr {P}^{[n-k]}$ is a continuous map $S_{\tau } \colon U_{\tau } \rightarrow \mathbb {R}^{k}$ such that

• • $S^{-1}_{\tau }(0) = \mathrm {int}_{\mathrm {re}}(\tau )$ ,

• • for every $\sigma \supset \tau $ , the restriction $S_{\tau }|_{\mathrm {int}_{\mathrm {re}}(\sigma )}$ is an integral affine submersion onto its image (meaning that it is induced by some epimorphism $\Lambda _{\sigma } \rightarrow W \cap \mathbb {Z}^k$ for some vector subspace $W\subset \mathbb {R}^k$ ), and

• • the collection of cones $\{ K_{\tau }\sigma := \mathbb {R}_{\geq 0} S_{\tau }(\sigma \cap U_{\tau }) \}_{\sigma \supset \tau }$ forms a complete finite fan $\Sigma _{\tau }$ .

Two fan structures along $\tau $ are equivalent if they differ by composition with an integral affine transformation of $\mathbb {R}^{k}$ . If $S_\tau $ is a fan structure along $\tau $ and $\sigma \supset \tau $ , then $U_{\sigma } \subset U_{\tau }$ and there is a fan structure along $\sigma $ induced from $S_{\tau }$ via the composition:

$$ \begin{align*}U_{\sigma} \hookrightarrow U_{\tau} \rightarrow \mathbb{R}^{k} \twoheadrightarrow \mathbb{R}^{l}, \end{align*} $$

where $\mathbb {R}^{k} \rightarrow \mathbb {R}^{k}/ \mathbb {R} S_{\tau }(\sigma \cap U_{\tau }) \cong \mathbb {R}^{l}$ is the quotient map.

Taking sufficiently small and mutually disjoint open subsets $W_{v} \subset U_{v}$ for $v \in \mathscr {P}^{[0]}$ and $\mathrm {int}_{\mathrm {re}}(\sigma )$ for $\sigma \in \mathscr {P}^{[n]}$ , there is an integral affine structure on $\bigcup _{v \in \mathscr {P}^{[0]}} W_v \cup \bigcup _{\sigma \in \mathscr {P}^{[n]}} \mathrm {int}_{\mathrm {re}}(\sigma )$ . We will further choose the open subsets $W_v$ ’s and $\mathrm {int}_{\mathrm {re}}(\sigma )$ ’s so that the affine structure is defined outside a closed subset $\Gamma $ of codimension two in B, as in [Reference Gross and Siebert27, §1.3]. This affine structure allows us to use parallel transport to identify the tangent spaces $T_x B$ for different points x outside the closed subset. For every $\tau $ , we choose a maximal cell $\sigma \supset \tau $ and consider the lattice of normal vectors $\mathscr {Q}_{\tau }=\Lambda _{\sigma }/\Lambda _{\tau }$ (we suppress the dependence on $\sigma $ because we will see that $\Lambda _{\tau }$ is monodromy invariant under the monodromy transformation given by any two vertices of $\tau $ and any two maximal cells containing $\tau $ ). We can identify $\mathscr {Q}_{\tau }$ with $\mathbb {Z}^{k}$ via $S_{\tau }$ and write the fan structure as $S_{\tau } \colon U_{\tau } \rightarrow \mathscr {Q}_{\tau ,\mathbb {R}}$ .

Example 2.3. We take a two-dimensional example from [Reference Aspinwall, Bridgeland, Craw, Douglas, Gross, Kapustin, Moore, Segal, Szendrői and Wilson1, Ex. 6.74] to illustrate the above definitions. Let $\Xi $ be the convex hull of the points

$$ \begin{align*} p_0 = \begin{bmatrix} -1 \\ -1 \\ -1 \end{bmatrix}, \ p_1 = \begin{bmatrix} 3\\-1\\-1\end{bmatrix}, \ p_2 = \begin{bmatrix} -1 \\ 3 \\ -1 \end{bmatrix}, \ p_3 = \begin{bmatrix} -1 \\ -1 \\3 \end{bmatrix}, \end{align*} $$

so $\Xi $ is a $3$ -simplex. Take B (as a topological space) to be the boundary of $\Xi $ . The polyhedral decomposition $\mathscr {P}$ is defined so that the integral points are vertices as shown in Figure 1.

Figure 1

The polyhedral decomposition.

Then we define affine coordinate charts on $\bigcup _{\sigma \in \mathscr {P}^{[n]}} \mathrm {int}_{\mathrm {re}}(\sigma ) \cup \bigcup _{v\in \mathscr {P}^{[0]}} W_v$ as follows. On $\mathrm {int}_{\mathrm {re}}(\sigma )$ , we take $\psi _{\sigma } \colon \mathrm {int}_{\mathrm {re}}(\sigma ) \rightarrow \Lambda _{\sigma ,\mathbb {R}}$ which maps homeomorphically onto its image. At a vertex v treated as a vector in $\mathbb {R}^3$ , we let $\psi _v\colon W_v \subset \mathbb {R}^3 \rightarrow \mathbb {R}^3 / \mathbb {R} v$ , where $\mathbb {R}^3 \rightarrow \mathbb {R}^3/\mathbb {R} v$ is the natural projection onto the quotient. By [Reference Aspinwall, Bridgeland, Craw, Douglas, Gross, Kapustin, Moore, Segal, Szendrői and Wilson1, Prop. 6.81], this gives an integral affine manifold with singularities. The affine structure can be extended to the complement of a subset $\Gamma $ consisting of $24$ points lying on the six edges of $\Xi $ , with each edge containing $4$ points (colored in red in Figure 1). The fan structure $S_\tau $ can be defined similarly.

Locally near each singular point $p \in \Gamma $ contained in an edge $\rho $ , the affine structure is described as a gluing of two affine charts $U_{\mathrm {I}}\subset \mathbb {R}^2 \setminus \{0\} \times \mathbb {R}_{\geq 0}$ and $U_{\mathrm {II}}\subset \mathbb {R}^2 \setminus 0 \times \mathbb {R}_{\leq 0}$ as in [Reference Gross and Siebert30, §3.2]. The change of coordinates from $U_{\mathrm {I}}$ to $U_{\mathrm {II}}$ is given by the restriction of the map $\Upsilon $ from $(\mathbb {R} \setminus \{0\}) \times \mathbb {R}$ to itself defined by

$$ \begin{align*} (x,y) \mapsto \begin{cases} (x,y), & x<0\\ (x,x+y), & x>0. \end{cases} \end{align*} $$

The fan structure $S_\rho \colon U_{\rho } \rightarrow \mathbb {R}$ is given as $S_{\rho }(x,y) = x$ and the fan $\Sigma _{\rho }$ is the toric fan for $\mathbb {P}^1$ . Figure 2 below illustrates the situation.

Figure 2

Affine coordinate charts.

With the structure of an integral tropical manifold, the corners and edges in Figure 1 are flattened via the affine coordinate charts, and we can view $(B,\mathscr {P})$ as the 2-sphere equipped with a polyhedral decomposition and with $24$ affine singularities. Such an affine structure with singularities also appears in the base B of an SYZ fibration of a K3 surface.

Example 2.4. A three-dimensional example can be constructed as in [Reference Aspinwall, Bridgeland, Craw, Douglas, Gross, Kapustin, Moore, Segal, Szendrői and Wilson1, Ex. 6.74]. Take $\Xi $ to be the convex hull of the points

$$ \begin{align*} p_0 = \begin{bmatrix} -1 \\ -1 \\ -1 \\-1 \end{bmatrix}, \ p_1 = \begin{bmatrix} 4 \\-1\\-1 \\ -1 \end{bmatrix}, \ p_2 = \begin{bmatrix} -1 \\ 4 \\ -1 \\ -1 \end{bmatrix}, \ p_3 = \begin{bmatrix} -1 \\ -1 \\4 \\ -1 \end{bmatrix}, \ p_4 = \begin{bmatrix} -1 \\ -1 \\-1 \\ 4 \end{bmatrix}, \end{align*} $$

which gives a $4$ -simplex. Take B (as a topological space) to be the boundary of $\Xi $ . There are five three-dimensional maximal cells intersecting along 10 two-dimensional facets. The polyhedral decomposition $\mathscr {P}$ on each facet is as in Figure 3.

Figure 3

The polyhedral decomposition on a facet.

The affine structure can be extended to the complement of codimension 2 closed subset $\Gamma $ whose intersection with a triangle in Figure 3 is a Y-shaped locus. Locally near each of these triangles, it looks like Figure 4a.

Figure 4

Two types of Y-vertex.

$\Xi $ has 10 one-dimensional faces, each of which is an edge with affine length $5$ . The polyhedral decomposition $\mathscr {P}$ divides each edge into $5$ intervals as we can see in Figure 3. Locally near each of these length $1$ intervals, there are three $2$ -cells of $\mathscr {P}$ intersecting along it. The locus $\Gamma $ on each $2$ -cell intersects on the interval as shown in Figure 4b.

There is a natural inclusion $\mathcal {A}{ff}\hookrightarrow \mathcal {PL}_{\mathscr {P}}$ , and we let $\mathcal {MPL}_{\mathscr {P}}$ be the quotient:

$$ \begin{align*}0\to \mathcal{A}{ff}\to\mathcal{PL}_{\mathscr{P}}\to\mathcal{MPL}_{\mathscr{P}}\to 0.\end{align*} $$

Locally, an element $\varphi \in \Gamma (B,\mathcal {MPL}_{\mathscr {P}})$ is a collection of piecewise affine functions $\{\varphi _U\}$ such that on each overlap $U\cap V$ , the difference $\varphi _U|_{V}-\varphi _V|_{U}$ is an integral affine function on $U\cap V$ .

The set of all convex multivalued piecewise affine functions gives a submonoid of $H^0(B,\mathcal {MPL}_{\mathscr {P}})$ under addition, denoted as $H^0(B,\mathcal {MPL}_{\mathscr {P}},\mathbb {N})$ ; we let Q be the dual monoid.

We always assume that $\mathscr {P}$ is regular with a fixed strictly convex $\varphi \in H^0(B,\mathcal {MPL}_{\mathscr {P}})$ .

2.2 Monodromy, positivity and simplicity

To describe monodromy, we consider two maximal cells $\sigma _{\pm }$ and two of their common vertices $v_{\pm }$ . Taking a path $\gamma $ going from $v_+$ to $v_-$ through $\sigma _+$ , and then from $v_-$ back to $v_+$ through $\sigma _-$ , we obtain a monodromy transformation $T_{\gamma }$ . As in [Reference Gross and Siebert27, §1.5], we are interested in two cases. The first case is when $v_+$ is connected to $v_-$ via a bounded edge $\omega \in \mathscr {P}^{[1]}$ . Let $d_{\omega } \in \Lambda _{\omega }$ be the unique primitive vector pointing to $v_-$ along $\omega $ . For an integral tangent vector $m \in T_{v_+} := T_{v_+,\mathbb {Z}}B$ , the monodromy transformation $T_{\gamma }$ is given by

(2.1) $$ \begin{align} T_{\gamma}(m) = m + \langle m , n^{\sigma_+ \sigma_-}_{\omega} \rangle d_{\omega} \end{align} $$

for some $n^{\sigma _+ \sigma _-}_{\omega } \in \mathscr {Q}_{\sigma _+ \cap \sigma _-}^*\subset T_{v_+}^* $ , where $\langle \cdot , \cdot \rangle $ is the natural pairing between $T_{v_+}$ and $T_{v_+}^*$ . The second case is when $\sigma _+$ and $\sigma _-$ are separated by a codimension one cell $\rho \in \mathscr {P}^{[n-1]}$ . Let $\check {d}_{\rho }\in \mathscr {Q}_{\rho }^*$ be the unique primitive covector which is positive on $\sigma _+$ . The monodromy transformation is given by

(2.2) $$ \begin{align} T_{\gamma}(m) = m + \langle m , \check{d}_{\rho} \rangle m^{\rho}_{v_+v_-} \end{align} $$

for some $m^{\rho }_{v_+v_-} \in \Lambda _{\tau }$ , where $\tau \subset \rho $ is the smallest face of $\rho $ containing $v_{\pm }$ . In particular, if we fix both $v_{\pm } \in \omega \subset \rho \subset \sigma _{\pm }$ , one obtains the formula

(2.3) $$ \begin{align} T_{\gamma}(m) = m + \kappa_{\omega\rho}\langle m , \check{d}_{\rho} \rangle d_{\omega} \end{align} $$

for some integer $\kappa _{\omega \rho }$ .

Following [Reference Gross and Siebert27, Definition 1.58], we package the monodromy data into polytopes associated to $\tau \in \mathscr {P}^{[k]}$ for $1\leq k \leq n-1$ . The simplest case is when $\rho \in \mathscr {P}^{[n-1]}$ , whose monodromy polytope is defined by fixing a vertex $v_0 \in \rho $ and setting

(2.4) $$ \begin{align} \Delta(\rho):= \mathrm{Conv}\{ m^{\rho}_{v_0 v} \ | \ v \in \rho, \ v \in \mathscr{P}^{[0]} \} \subset \Lambda_{\rho,\mathbb{R}}, \end{align} $$

where $\mathrm {Conv}$ refers to taking the convex hull. It is well defined up to translation and independent of the choice of $v_0$ . The normal fan of $\rho $ in $\Lambda _{\rho ,\mathbb {R}}^*$ is a refinement of the normal fan of $\Delta (\rho )$ . Similarly, when $\omega \in \mathscr {P}^{[1]}$ , one defines the dual monodromy polytope by fixing $\sigma _0 \supset \omega $ and setting

(2.5) $$ \begin{align} \check{\Delta}(\omega):= \mathrm{Conv}\{ n^{\sigma_0 \sigma}_{\omega} \ | \ \sigma\supset \omega, \ \sigma \in \mathscr{P}^{[n-1]} \} \subset \mathscr{Q}_{\omega,\mathbb{R}}^*. \end{align} $$

Again, this is well defined up to translation and independent of the choice of $\sigma _0$ . The fan $\Sigma _{\omega }$ in $\mathscr {Q}_{\omega ,\mathbb {R}}$ is a refinement of the normal fan of $\check {\Delta }(\omega )$ . For $1< \dim _{\mathbb {R}}(\tau ) <n-1$ , a combination of monodromy and dual monodromy polytopes is needed. We let $\mathscr {P}_1(\tau ) = \{ \omega \ | \ \omega \in \mathscr {P}^{[1]}, \ \omega \subset \tau \}$ and $\mathscr {P}_{n-1}(\tau ) = \{ \rho \ | \ \rho \in \mathscr {P}^{[n-1]}, \ \rho \supset \tau \}$ . For each $\rho \in \mathscr {P}_{n-1}(\tau )$ , we choose a vertex $v_0 \in \rho $ and let

$$ \begin{align*}\Delta_{\rho}(\tau):= \mathrm{Conv}\{ m^{\rho}_{v_0 v} \ | \ v \in \tau, \ v \in \mathscr{P}^{[0]} \} \subset \Lambda_{\tau,\mathbb{R}}.\end{align*} $$

Similarly, for each $\omega \in \mathscr {P}_1(\tau )$ , we choose $\sigma _0 \supset \tau $ and let

$$ \begin{align*}\check{\Delta}_{\omega}(\tau):= \mathrm{Conv} \{ n^{\sigma_0 \sigma}_{\omega} \ | \ \sigma\supset \tau, \ \sigma \in \mathscr{P}^{[n-1]} \} \subset \mathscr{Q}_{\tau,\mathbb{R}}^*.\end{align*} $$

These are well defined up to translation and independent of the choices of $v_0$ and $\sigma _0$ , respectively.

Definition 2.9 ([Reference Gross and Siebert27], Definition 1.60).

We say $(B,\mathscr {P})$ is simple if, for every $\tau \in \mathscr {P}$ , there are disjoint nonempty subsets

$$ \begin{align*} \Omega_1,\dots,\Omega_p \subset \mathscr{P}_1(\tau), \quad R_1,\dots, R_p \subset \mathscr{P}_{n-1}(\tau) \end{align*} $$

(where p depends on $\tau $ ) such that

1. 1. for $\omega \in \mathscr {P}_1(\tau )$ and $\rho \in \mathscr {P}_{n-1}(\tau )$ , $\kappa _{\omega \rho } \neq 0$ if and only if $\omega \in \Omega _i$ and $\rho \in R_i$ for some $1 \leq i \leq p$ ;

2. 2. $\Delta _{\rho }(\tau )$ is independent (up to translation) of $\rho \in R_i$ and will be denoted by $\Delta _i(\tau )$ ; similarly, $\check {\Delta }_{\omega }(\tau )$ is independent (up to translation) of $\omega \in \Omega _i$ and will be denoted by $\check {\Delta }_i(\tau )$ ;

3. 3. if $\{e_1,\dots ,e_p\}$ is the standard basis in $\mathbb {Z}^{p}$ , then

   $$ \begin{align*} \Delta(\tau):= \mathrm{Conv} \left\{ \bigcup_{i=1}^{p} \Delta_i(\tau) \times \{e_i\} \right\}, \quad \check{\Delta}(\tau):= \mathrm{Conv} \left\{ \bigcup_{i=1}^{p} \check{\Delta}_i(\tau) \times \{e_i\} \right\} \end{align*} $$
   are elementary simplices (i.e., a simplex whose only integral points are its vertices) in $\left (\Lambda _{\tau } \oplus \mathbb {Z}^{p}\right )_{\mathbb {R}}$ and $\left ( \mathscr {Q}_{\tau }^*\oplus \mathbb {Z}^{p} \right )_{\mathbb {R}}$ , respectively.

We need the following stronger condition in order to apply [Reference Gross and Siebert28, Thm. 3.21] in a later stage:

Example 2.11. Consider the two-dimensional example in Example 2.3. Following [Reference Aspinwall, Bridgeland, Craw, Douglas, Gross, Kapustin, Moore, Segal, Szendrői and Wilson1, Ex. 6.82(1)], we may choose the two adjacent vertices in Figure 1 to be $v_1 = \begin {bmatrix} -1 & -1 & -1 \end {bmatrix}^T$ and $v_2 = \begin {bmatrix} 0 & -1 & -1 \end {bmatrix}^T$ which bound a $1$ -cell $\rho $ . The two adjacent maximal cells are given by $\sigma _+ \subset \{ b \ | \ \langle w_+,b\rangle =1\}$ , where $w_+ = \begin {bmatrix} 0 & 0 & -1 \end {bmatrix}^T$ and $\sigma _- \subset \{ b \ | \ \langle w_-,b\rangle =1\}$ , where $w_- = \begin {bmatrix} 0 & -1 & 0 \end {bmatrix}^T$ . The tangent lattice $T_{v_1}$ can be identified with $\mathbb {Z}^3/\mathbb {Z} \cdot v_1$ equipped with the basis $e_1 = \begin {bmatrix} 1 & 0 &0 \end {bmatrix}^T$ , $e_2 = \begin {bmatrix} 0 & 1& 0 \end {bmatrix}^T$ . If we let $\gamma $ be a loop going from $v_1$ to $v_2$ through $\sigma _+$ and going back to $v_1$ through $\sigma _-$ , we have

$$ \begin{align*} T_{\gamma}(m) = m + \langle \begin{bmatrix} 0 & 1 & -1 \end{bmatrix}^T, m \rangle e_1 \end{align*} $$

for $m \in T_{v_1}$ . Therefore, we have $p=1$ , $\Delta _1(\rho ) = \mathrm {Conv} \{0,e_1\}$ and $\check {\Delta }_{1}(\rho ) = \mathrm {Conv} \{ 0 , w_+ - w_-\}$ . This is an example of a positive and strongly simple $(B,\mathscr {P})$ (Definitions 2.8 and 2.10).

Example 2.12. Next, we consider the two types of Y-vertex in Example 2.4.

We begin with Y-vertex of type I in Figure 4a. Following [Reference Aspinwall, Bridgeland, Craw, Douglas, Gross, Kapustin, Moore, Segal, Szendrői and Wilson1, Ex. 6.82(2)], the three vertices $v_1,v_2,v_3$ can be chosen to be

$$ \begin{align*} v_1 = \begin{bmatrix} -1 & -1& -1 & -1 \end{bmatrix}^T, \ v_2 = \begin{bmatrix} 0 & -1 & -1 & -1 \end{bmatrix}^T, \ v_3 = \begin{bmatrix} -1 & 0 & -1 & -1 \end{bmatrix}^T, \end{align*} $$

and $\sigma _+ \subset \{ b \in \mathbb {R}^4 \ | \ \langle w_+ , b \rangle = 1 \}$ , $\sigma _- \subset \{ b \in \mathbb {R}^4 \ | \ \langle w_- , b \rangle = 1 \}$ are $3$ -cells of B lying in the affine hyperplanes with dual vector $w_+ = \begin {bmatrix} 0 & 0 & -1 & 0 \end {bmatrix}^T$ and $w_- = \begin {bmatrix} 0 & 0 & 0 & -1 \end {bmatrix}^T$ , respectively. If we identify $T_v$ with $\Lambda _{\sigma _+}$ via parallel transport and choose the basis of $\Lambda _{\sigma _+}$ as

$$ \begin{align*} e_1 = \begin{bmatrix} 1 & 0 & 0 & 0 \end{bmatrix}^T, \ e_2 = \begin{bmatrix} 0 & -1 & 0 & 0 \end{bmatrix}^T, \ e_3 = \begin{bmatrix} 0 & 0 & 0 & 1 \end{bmatrix}^T, \end{align*} $$

then the monodromy transformations are given by

$$ \begin{align*} T_{\gamma_1}= \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \ T_{\gamma_2} = \begin{bmatrix} 1 & 0 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{bmatrix}, \ T_{\gamma_3} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}, \end{align*} $$

where $\gamma _i$ is the loop going from $v_i$ to $v_{i+1}$ through $\sigma _+$ and going back to $v_i$ through $\sigma _-$ , with indices of $v_i$ ’s taken modulo $3$ . In this case, we have $p=1$ , $\Delta _1(\rho ) = \mathrm {Conv}\{0, e_1,-e_2\}$ is a $2$ -simplex and $\check {\Delta }_{1}(\rho ) = \mathrm {Conv}\{0, w_+ - w_-\}$ is a $1$ -simplex.

For the Y-vertex of type II in Figure 4b, we can choose

$$ \begin{align*}v_1 = \begin{bmatrix} -1 & -1 & -1 & -1 \end{bmatrix}^T,\ v_2 = \begin{bmatrix} 0 & -1 & -1 & -1 \end{bmatrix}^T,\end{align*} $$

which are the end points of a $1$ -cell $\tau $ . We choose the three maximal cells $\sigma _1$ , $\sigma _2$ and $\sigma _3$ intersecting at $\tau $ to be the $3$ -cells lying in affine hyperplanes defined by $\{b \ | \ \langle w_i, b \rangle = 1\}$ , where

$$ \begin{align*} w_1 = \begin{bmatrix} 0 & 0& -1 & 0 \end{bmatrix}^T, \ w_2 = \begin{bmatrix} 0 & 0 & 0 & -1 \end{bmatrix}^T, \ w_3 = \begin{bmatrix} 0 & -1 & 0 & 0 \end{bmatrix}^T. \end{align*} $$

Let $\tilde {\gamma }_i$ be the loop going from $v_1$ to $v_2$ through $w_i$ and then going back to $v_1$ through $w_{i+1}$ , with indices taken to be modulo $3$ . Then the corresponding monodromy transformations are given by

$$ \begin{align*}T_{\gamma_1}= \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \ T_{\gamma_2} = \begin{bmatrix} 1 & 1& 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \ T_{\gamma_3} = \begin{bmatrix} 1 & -1& -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \end{align*} $$

with respect to the basis

$$ \begin{align*} e_1 = \begin{bmatrix} 1 & 0 & 0 & 0 \end{bmatrix}^T, \ e_2 = \begin{bmatrix} 0 & 1 & 0 & 0 \end{bmatrix}^T, \ e_3 = \begin{bmatrix} 0 & 0 & -1 & 0 \end{bmatrix}^T. \end{align*} $$

In this case, $p=1$ , $\Delta _1(\tau ) = \mathrm {Conv}\{0, v_2 - v_1\}$ is a $1$ -simplex and $\check {\Delta }_1(\tau )= \mathrm {Conv}\{0, w_1 - w_2, w_1-w_3\}$ is a $2$ -simplex.

Both examples are positive and strongly simple.

Throughout this paper, we always assume that $(B,\mathscr {P})$ is positive and strongly simple. In particular, both $\Delta _i(\tau )$ and $\check {\Delta }_i(\tau )$ are standard simplices of positive dimensions, and $\Lambda _{\Delta _1(\tau )}\oplus \cdots \oplus \Lambda _{\Delta _p(\tau )}$ (resp. $\Lambda _{\check {\Delta }_1(\tau )} \oplus \cdots \oplus \Lambda _{\check {\Delta }_p(\tau )}$ ) is an internal direct summand of $\Lambda _{\tau }$ (resp. $\mathscr {Q}_{\tau }^*$ ).

2.3 Cone construction by gluing open affine charts

In this subsection, we recall the cone construction of the maximally degenerate Calabi–Yau , following [Reference Gross and Siebert27] and [Reference Gross and Siebert29, §1.2]. For this purpose, we take $\mathbf {K} = \mathbb {Z}$ and Q to be the positive real axis in Notation 1.4. Throughout this paper, we will work in the category of analytic schemes.

We will construct as a gluing of affine analytic schemes $V(v)$ parametrized by the vertices of $\mathscr {P}$ . For each vertex v, we consider the fan $\Sigma _v$ and take the analytic affine toric variety

$$ \begin{align*}V(v):=\mathrm{Spec}_{\mathrm{an}}(\mathbb{C}[\Sigma_v]),\end{align*} $$

where $\mathrm {Spec}_{\mathrm {an}}$ means analytification of the algebraic affine scheme given by $\mathrm {Spec}$ . Here, the monoid structure for a general fan $\Sigma \subset M_{\mathbb {R}}$ is given by

$$ \begin{align*}p+q=\begin{cases} p+q &\text{ if }p,q \in M \text{ are in a common cone of } \Sigma, \\\infty & \text{ otherwise}, \end{cases}\end{align*} $$

and we set $z^{\infty } =0$ in taking $\mathrm {Spec}(\mathbb {C}[\Sigma ])$ (by abuse of notation, we use $\Sigma $ to stand for both the fan and the monoid associated to a fan if there is no confusion); in other words, the ring $\mathbb {C}[\Sigma ]$ is defined explicitly as

$$ \begin{align*}\mathbb{C}[\Sigma]:= \bigoplus_{p \in |\Sigma| \cap M} \mathbb{C} \cdot z^p, \quad z^p\cdot z^q=\begin{cases} z^{p+q} &\text{ if }p,q \in M \text{ are in a common cone of } \Sigma, \\0 & \text{ otherwise}, \end{cases}\end{align*} $$

where $|\Sigma |$ denotes the support of the fan $\Sigma $ .

To glue these affine analytic schemes together, we need affine subschemes $\{V(\tau )\}$ associated to $\tau \in \mathscr {P}$ with $v\in \tau $ and natural open embeddings $V(\tau )\hookrightarrow V(\omega )$ for $v \in \omega \subset \tau $ . First, for $\tau \in \mathscr {P}$ such that $v\in \tau $ , we consider the localization of $\Sigma _v$ at $\tau $ defined by

$$ \begin{align*}\tau^{-1}\Sigma_v:=\{K_v\sigma + \Lambda_{\tau,\mathbb{R}}\,|\,K_v\sigma\text{ is a cone in } \Sigma_v \text{ such that }\sigma \supset \tau\};\end{align*} $$

here recall that $K_v\sigma = \mathbb {R}_{\geq 0} S_{v}(\sigma \cap U_v)$ is the cone in $\Sigma _v$ (see the definition of a fan structure before Definition 2.2). This defines a new complete fan in $T_{v,\mathbb {R}}$ consisting of convex, but not necessarily strictly convex, cones. If $\tau $ contains another vertex $v'$ , we can identify the fans $\tau ^{-1}\Sigma _v$ and $\tau ^{-1}\Sigma _{v'}$ as follows: For each maximal $\sigma \supset \tau $ , we identify the maximal cones $K_{v}\sigma + \Lambda _{\tau ,\mathbb {R}}$ and $K_{v'}\sigma + \Lambda _{\tau ,\mathbb {R}}$ by identifying the tangent spaces $T_v\cong T_{v'}$ using parallel transport through $\sigma \supset \tau $ . Patching these identifications for all $\sigma \supset \tau $ together, we get a piecewise linear transformation from $T_{v}$ to $T_{v'}$ , identifying the fans $\tau ^{-1}\Sigma _v$ and $\tau ^{-1}\Sigma _{v'}$ and hence the corresponding monoids. This defines the affine analytic scheme

$$ \begin{align*}V(\tau):=\mathrm{Spec}_{\mathrm{an}}(\mathbb{C}[\tau^{-1}\Sigma_v]),\end{align*} $$

up to a unique isomorphism. Notice that $\tau ^{-1}\Sigma _v$ can be identified (noncanonically) with the fan $\Sigma _{\tau } \times \Lambda _{\tau ,\mathbb {R}}$ in $\mathscr {Q}_{\tau ,\mathbb {R}}\times \Lambda _{\tau ,\mathbb {R}}$ , so actually

$$ \begin{align*}V(\tau) \cong \mathrm{Spec}_{\mathrm{an}}(\mathbb{C}[\Lambda_{\tau}]) \times \mathrm{Spec}_{\mathrm{an}}(\mathbb{C}[\Sigma_{\tau}]),\end{align*} $$

where $\mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\Lambda _{\tau }]) \cong \Lambda _{\tau }^* \otimes _{\mathbb {Z}} \mathbb {C}^* \cong (\mathbb {C}^{*})^l$ is a complex torus.

For any $v\in \omega \subset \tau $ , there is a map of monoids $ \omega ^{-1}\Sigma _v \to \tau ^{-1}\Sigma _v$ given by

$$ \begin{align*}p\mapsto\begin{cases} p & \text{ if }p\in K_{v}\sigma +\Lambda_{\omega,\mathbb{R}}\text{ for some }\sigma\supset\tau, \\\infty & \text{ otherwise} \end{cases}\end{align*} $$

(though there is no fan map from $\omega ^{-1}\Sigma _v$ to $\tau ^{-1}\Sigma _v$ in general), and hence a ring map

$$ \begin{align*}\iota_{\omega\tau}^*\colon \mathbb{C}[\omega^{-1}\Sigma_v]\to\mathbb{C}[\tau^{-1}\Sigma_v].\end{align*} $$

This gives an open inclusion of affine schemes

$$ \begin{align*}\iota_{\omega\tau}\colon V(\tau)\hookrightarrow V(\omega),\end{align*} $$

and hence a functor $F\colon \mathscr {P}\to \mathbf {{Sch}}_{\mathrm {an}}$ defined by

$$ \begin{align*}F(\tau):= V(\tau), \quad F(e):= \iota_{\omega\tau} \colon V(\tau)\to V(\omega) \end{align*} $$

for $\omega \subset \tau $ .

We can further introduce twistings of the gluing of the affine analytic schemes $\{V(\tau )\}_{\tau \in \mathscr {P}}$ . Toric automorphisms $\mu $ of $V(\tau )$ are in bijection with the set of $\mathbb {C}^{*}$ -valued piecewise multiplicative maps on $T_v\cap |\tau ^{-1}\Sigma _v|$ with respect to the fan $\tau ^{-1}\Sigma _v$ . Explicitly, for each maximal cone $\sigma \in \mathscr {P}^{[n]}$ with $\tau \subset \sigma $ , there is a monoid homomorphism $\mu _{\sigma }\colon \Lambda _{\sigma }\to \mathbb {C}^{*}$ such that if $\sigma '\in \mathscr {P}^{[n]}$ also contains $\tau $ , then $\mu _{\sigma }|_{\Lambda _{\sigma \cap \sigma '}}=\mu _{\sigma '}|_{\Lambda _{\sigma \cap \sigma '}}$ . Denote by $\mathrm {PM}(\tau )$ the multiplicative group of $\mathbb {C}^{*}$ -valued piecewise multiplicative maps on $T_v\cap |\tau ^{-1}\Sigma _v|$ . The group $\mathrm {PM}(\tau )$ a priori depends on the choice of $v \in \tau $ ; however, for different choices, say v and $v'$ , the groups can be identified via the identification $\tau ^{-1}\Sigma _v \cong \tau ^{-1}\Sigma _{v'}$ . For $\omega \subset \tau $ , there is a natural restriction map $|_{\tau } \colon \mathrm {PM}(\omega ) \rightarrow \mathrm {PM}(\tau )$ given by restricting to those maximal cells $\sigma \supset \omega $ with $\sigma \supset \tau $ .

The set of cohomology classes of choices of open gluing data is a group under multiplication, denoted as $H^1(\mathscr {P},\mathscr {Q}_{\mathscr {P}}\otimes \mathbb {C}^{\times })$ . For $s\in \mathrm {PM}(\tau )$ , we will denote also by s the corresponding toric automorphism on $V(\tau )$ which is explicitly given by $s^*(z^m) = s_{\sigma }(m) z^m$ for $m \in \sigma \supset \tau $ . If s is a choice of open gluing data, then we can define an s-twisted functor $F_s \colon \mathscr {P} \to \mathbf {{Sch}}_{\mathrm {an}}$ by setting $F_s(\tau ):=F(\tau )=V(\tau )$ on objects and $F_s(\omega \subset \tau ):=F(\omega \subset \tau )\circ s_{\omega \tau }^{-1} \colon V(\tau ) \to V(\omega )$ on morphisms. This defines the analytic scheme

Gross–Siebert [Reference Gross and Siebert27] showed that as schemes when $s,s'$ are cohomologous.

We recall the following definition from [Reference Gross and Siebert27], which serves as an alternative set of combinatorial data for encoding $\mu \in \mathrm {PM}(\tau )$ .

Definition 2.15 ([Reference Gross and Siebert27], Definition 3.25 and [Reference Gross and Siebert29], Definition 1.20).

Let $\mu \in \mathrm {PM}(\tau )$ and $\rho \in \mathscr {P}^{[n-1]}$ with $\tau \subset \rho $ . For a vertex $v\in \tau $ , we define

$$ \begin{align*}D(\mu,\rho,v):=\frac{\mu_{\sigma}(m)}{\mu_{\sigma'}(m')}\in\mathbb{C}^{\times},\end{align*} $$

where $\sigma ,\sigma '$ are the two unique maximal cells such that $\sigma \cap \sigma '=\rho $ , $m \in \Lambda _{\sigma }$ is an element projecting to the generator in $\mathscr {Q}_{\rho } \cong \Lambda _{\sigma }/\Lambda _{\rho }\cong \mathbb {Z}$ pointing to $\sigma '$ , and $m'$ is the parallel transport of $m\in \Lambda _{\sigma }$ to $\Lambda _{\sigma '}$ through v. $D(\mu ,\rho ,v)$ is independent of the choice of m.

Let $\rho \in \mathscr {P}^{[d-1]}$ and $\sigma _+,\sigma _-$ be the two unique maximal cells such that $\sigma _+\cap \sigma _-=\rho $ . Let $\check {d}_{\rho }\in \mathscr {Q}_{\rho }^*$ be the unique primitive generator pointing to $\sigma _+$ . For any two vertices $v,v'\in \tau $ , we have the formula

(2.6) $$ \begin{align} D(\mu,\rho,v)=\mu(m_{vv'}^{\rho})^{-1}\cdot D(\mu,\rho,v') \end{align} $$

relating monodromy data to the open gluing data, where $m_{vv'}^{\rho }\in \Lambda _{\rho }$ is as discussed in equation (2.2). The formula (2.6) describes the interaction between monodromy and a fixed $\mu \in \mathrm {PM}(\tau )$ . We shall further impose the following lifting condition from [Reference Gross and Siebert27, Prop. 4.25] relating $s_{v\tau }, s_{v'\tau } \in \mathrm {PM}(\tau )$ and monodromy data:

2.4 Log structures

We need to equip the analytic scheme with log structures. The main reference is [Reference Gross and Siebert27, §3 - 5].

Example 2.18. Let X be an analytic space and $D\subset X$ be a closed analytic subspace of pure codimension one. We denote by $j \colon X\setminus D \hookrightarrow X $ the inclusion. Then the sheaf of monoids

$$ \begin{align*}\mathcal{M}_{X}:=j_*(\mathcal{O}_{X\setminus D}^*) \cap \mathcal{O}_X, \end{align*} $$

together with the natural inclusion $\alpha _X \colon \mathcal {M}_{X} \rightarrow \mathcal {O}_X$ defines a log structure on X.

We write $X^{\dagger }$ if we want to emphasize the log structure on X. A general way to define a log structure is to take an arbitrary homomorphism of sheaves of monoids

$$ \begin{align*}\tilde{\alpha} \colon \mathcal{P} \rightarrow \mathcal{O}_X \end{align*} $$

and then define the associated log structure by

$$ \begin{align*} \mathcal{M}_X := (\mathcal{P}\oplus \mathcal{O}_X^*)/\{(p,\tilde{\alpha}(p)^{-1}) \ | \ p \in \tilde{\alpha}^{-1}(\mathcal{O}_X^*) \}. \end{align*} $$

In particular, this allows us to define log structures on an analytic space Y by pulling back those on another analytic space X via a morphism $f \colon Y \rightarrow X$ . More precisely, given a log structure on X, the pullback log structure on Y is defined to be the log structure associated to the composition $\tilde {\alpha }_Y \colon f^{-1}(\mathcal {M}_X) \rightarrow f^{-1}(\mathcal {O}_X) \rightarrow \mathcal {O}_Y$ . For more details of the theory of log structures, readers are referred to, for example, [Reference Gross and Siebert27, §3].

Before we describe the log structures on , let us first specify a ghost sheaf $\overline {\mathcal {M}}$ over . Recall that the polyhedral decomposition $\mathscr {P}$ is assumed to be regular, namely, there exists a strictly convex multivalued piecewise linear function $\varphi \in H^0(B,\mathcal {MPL}_{\mathscr {P}})$ . For any $\tau \in \mathscr {P}$ , we take a strictly convex representative $\bar {\varphi }_{\tau }$ of $\varphi $ on $\mathscr {Q}_{\tau ,\mathbb {R}}$ and define

$$ \begin{align*}\Gamma(V(\tau),\overline{\mathcal{M}}) := \bar{P}_{\tau} = C_{\tau} \cap (\mathscr{Q}_{\tau} \oplus \mathbb{Z}),\end{align*} $$

where $C_\tau :=\{(m,h)\in \mathscr {Q}_{\tau ,\mathbb {R}} \oplus \mathbb {R} \,|\, h\geq \bar {\varphi }_\tau (m)\}$ . For any $\omega \subset \tau $ , we take an integral affine function $\psi _{\omega \tau }$ on $U_{\omega }$ such that $\psi _{\omega \tau }+ S_{\omega }^*( \bar {\varphi }_{\omega })$ vanishes on $K_{\omega }\tau $ and agrees with $S_{\tau }^*(\bar {\varphi }_{\tau })$ on all of $\sigma \cap U_{\tau }$ for any $\sigma \supset \tau $ . This induces a map $C_{\omega } \rightarrow C_{\omega \tau } :=\{(m,h)\in \mathscr {Q}_{\omega ,\mathbb {R}} \oplus \mathbb {R} \,|\, h\geq \psi _{\omega \tau }(m)+\bar {\varphi }_\omega (m)\}$ by sending $(m,h)\mapsto (m,h+\psi _{\omega \tau }(m))$ , whose composition with the quotient map $\mathscr {Q}_{\omega ,\mathbb {R}}\oplus \mathbb {R} \rightarrow \mathscr {Q}_{\tau ,\mathbb {R}}\oplus \mathbb {R}$ gives a map $C_{\omega } \rightarrow C_{\tau }$ of cones that corresponds to the monoid homomorphism $\bar {P}_{\omega }\rightarrow \bar {P}_{\tau }$ . The $\bar {P}_{\tau }$ ’s glue together to give the ghost sheaf $\overline {\mathcal {M}}$ over . There is a well-defined section given by gluing $(0,1) \in C_{\tau }$ for each $\tau $ .

One may then hope to find a log structure on which is log smooth and with ghost sheaf given by $\overline {\mathcal {M}}$ . However, due to the presence of nontrivial monodromies of the affine structure, this can only be done away from a complex codimension $2$ subset not containing any toric strata. Such log structures can be described by sections of a coherent sheaf $\mathcal {LS}^+_{\mathrm {pre}}$ supported on the scheme-theoretic singular locus . We now describe the sheaf $\mathcal {LS}^+_{\mathrm {pre}}$ and some of its sections called slab functions; readers are referred to [Reference Gross and Siebert27, §3 and 4] for more details.

For every $\rho \in \mathscr {P}^{[n-1]}$ , we consider , where is the toric variety associated to the polytope $\rho \subset \Lambda _{\rho ,\mathbb {R}}$ . From the fact that the normal fan $\mathscr {N}_{\rho } \subset \Lambda _{\rho ,\mathbb {R}}^*$ of $\rho $ is a refinement of the normal fan $\mathscr {N}_{\Delta (\rho )}\subset \Lambda _{\rho ,\mathbb {R}}^*$ of the $r_{\rho }$ -dimensional simplex $\Delta (\rho )$ (as in §2.2), we have a toric morphism

(2.7)

Now, $\Delta (\rho )$ corresponds to $\mathcal {O}(1)$ on $\mathbb {P}^{r_\rho }$ . We let $\mathcal {N}_{\rho }:= \varkappa _{\rho }^*(\mathcal {O}(1))$ on and define

(2.8) $$ \begin{align} \mathcal{LS}^+_{\mathrm{pre}} := \bigoplus_{\rho \in \mathscr{P}^{[n-1]}} \iota_{\rho,*} (\mathcal{N}_{\rho}). \end{align} $$

Sections of $\mathcal {LS}^+_{\mathrm {pre}}$ can be described explicitly. For each $v \in \mathscr {P}^{[0]}$ , we consider the open subscheme $V(v)$ of and the local trivialization

$$ \begin{align*}\mathcal{LS}^+_{\mathrm{pre}}|_{V(v)} = \bigoplus_{\rho: v \in \rho} \mathcal{O}_{V_{\rho}(v)}, \end{align*} $$

whose sections over $V(v)$ are given by $(f_{v\rho })_{v \in \rho }$ . Given $v ,v' \in \tau $ where $\tau $ corresponding to $V(\tau )$ , these local sections obey the change of coordinates given by

(2.9) $$ \begin{align} D(s_{v'\tau},\rho, v')^{-1} s_{v'\tau}^{-1} (f_{v'\rho}) = z^{-m^{\rho}_{vv'}} D(s_{v\tau},\rho, v)^{-1} s_{v\tau}^{-1} (f_{v\rho}), \end{align} $$

where $\rho \supset \tau $ and $s_{v\tau }, s_{v'\tau }$ are part of the open gluing data s. The section $f := (f_{v\rho })_{v \in \rho }$ is said to be normalized if $f_{v\rho }$ takes the value $1$ at the zero-dimensional toric stratum corresponding to a vertex v, for all $\rho $ . We will restrict ourselves to normalized sections f of $\mathcal {LS}^+_{\mathrm {pre}}$ . The complex codimension $2$ subset is taken to be the zero locus of f on .

Only a subset of normalized sections of $\mathcal {LS}^+_{\mathrm {pre}}$ corresponds to log structures. For every vertex $v \in \mathscr {P}^{[0]}$ and $\tau \in \mathscr {P}^{[n-2]}$ containing v, we choose a cyclic ordering $\rho _1,\dots ,\rho _l$ of codimension one cells containing $\tau $ according to an orientation of $\mathscr {Q}_{\tau ,\mathbb {R}}$ . Let $\check {d}_{\rho _i} \in \mathscr {Q}_v^*$ be the positively oriented normal to $\rho _i$ . Then the condition for $f = (f_{v\rho })_{v\in \rho }\in \mathcal {LS}^+_{\mathrm {pre}}|_{V(v)}$ to define a log structure is given by

(2.10) $$ \begin{align} \prod_{i=1}^l \check{d}_{\rho_i} \otimes f_{v\rho_i}|_{V_{\tau}(v)} = 0 \otimes 1,\quad \text{in } \mathscr{Q}_v^* \otimes \Gamma(V_{\tau}(v)\setminus Z, \mathcal{O}_{V_{\tau}(v)}^*), \end{align} $$

where the group structure on $\mathscr {Q}_v^*$ is additive and that on $\Gamma (V_{\tau }(v)\setminus Z, \mathcal {O}_{V_{\tau }(v)}^*)$ is multiplicative. If $f = (f_{v\rho })_{v\in \rho }$ is a normalized section satisfying this condition, we call the $f_{v\rho }$ ’s slab functions.

From now on, we always assume that B is compact. To describe the log structure in Theorem 2.20, we first construct some local smoothing models: For each vertex $v \in \mathscr {P}^{[0]}$ , we represent the strictly convex piecewise linear function $\varphi $ in a small neighborhood U of v by a strictly convex piecewise linear $\varphi _v\colon \mathscr {Q}_{v,\mathbb {R}}\to \mathbb {R}$ (so that $\varphi = S_v^*(\varphi _v)$ ) and set

$$ \begin{align*} C_v := \{(m,h)\in \mathscr{Q}_{v,\mathbb{R}} \oplus\mathbb{R} \,|\, h\geq\varphi_v(m)\},\quad P_v := C_v\cap(\mathscr{Q}_{v}\oplus\mathbb{Z}). \end{align*} $$

The element $\varrho = (0,1)\in \mathscr {Q}_v\oplus \mathbb {Z}$ gives rise to a regular function $q:=z^{\varrho }$ on $\mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[P_v])$ . We have a natural identification

$$ \begin{align*}V(v):=\mathrm{Spec}_{\mathrm{an}}(\mathbb{C}[\Sigma_v]) \cong \mathrm{Spec}_{\mathrm{an}}(\mathbb{C}[P_v]/q),\end{align*} $$

through which we view $V(v)$ as the toric boundary divisor in $\mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[P_v])$ that corresponds to the holomorphic function q, and $\pi _v \colon \mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[P_v]) \rightarrow \mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[q])$ as a local model for smoothing $V(v)$ .

Using these local models, we can now describe the log structure around a point . On a neighborhood $V \subset V(v)\setminus Z$ of x, the local smoothing model is given by composing the two inclusions and $V(v) \hookrightarrow \mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[P_{v}])$ . The natural monoid homomorphism $P_v \rightarrow \mathbb {C}[P_v]$ defined by sending $m \mapsto z^m$ determines a log structure on $\mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[P_v])$ which restricts to one on the toric boundary divisor $V(v) = \mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\Sigma _v])$ . We further twist the inclusion as

(2.11) $$ \begin{align} z^{m} \mapsto h_m\cdot z^{m}\text{ for } m \in \Sigma_v; \end{align} $$

here, for each $m \in \Sigma _v$ , $h_m$ is chosen as an invertible holomorphic function on $V \cap \mathrm {Zero}(z^m;v)$ , where we denote $\mathrm {Zero}(z^m;v):=\overline { \{x \in V(v) \ | \ z^{m} \in \mathcal {O}_{x}^* \}}$ , and such that they satisfy the relations

(2.12) $$ \begin{align} h_m \cdot h_{m'} = h_{m+m'},\quad \text{on } V \cap \mathrm{Zero}(z^{m+m'};v). \end{align} $$

Then pulling back the log structure on $V(v)$ via produces a log structure on V which is log smooth.

These local choices of $h_m$ ’s are also required to be determined by the slab functions $f_{v\rho }$ ’s, up to equivalences. Here, we shall just give the formula relating them; see [Reference Gross and Siebert27, Thm. 3.22] for details. For any $\rho \in \mathscr {P}^{[n-1]}$ containing v and two maximal cells $\sigma _{\pm }$ such that $\sigma _{+} \cap \sigma _- = \rho $ , we take $m_+ \in \mathscr {Q}_{v} \cap K_{v} \sigma _+$ generating $\mathscr {Q}_{\rho }$ with some $m_0 \in \mathscr {Q}_v\cap K_v \rho $ such that $m_0 - m_+ \in \mathscr {Q}_v \cap K_v \sigma _-$ . Then the required relation is given by

(2.13) $$ \begin{align} f_{v\rho} = \frac{h_{m_0}^2}{h_{m_0-m_+} \cdot h_{m_0 + m_+}}\Big|_{V_\rho(v)\cap V} \in \mathcal{O}^*_{V_\rho(v)}( V_\rho(v) \cap V), \end{align} $$

which is independent of the choices of $m_0$ and $m_+$ .

By abuse of notation, we also let be the k-th order thickening of V over $\mathbb {C}[q]/q^{k+1}$ in the model $\mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[P_v])$ under the above embedding. Then there is a natural divisorial log structure on over coming from restriction of the log structure on $\mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[P_{v}])^{\dagger }$ over $S^{\dagger }$ (i.e., Example 2.18, which is the same as the one given by Example 2.19 in this case). Restricting to V reproduces the log structure we constructed above, which is the log structure of over the log point locally around x. We have a Cartesian diagram of log spaces

(2.14)

Next, we describe the log structure around a singular point for some $\tau $ . Viewing $f = \sum _{\rho \in \mathscr {P}^{[n-1]}} f_{\rho }$ where $f_{\rho }$ is a section of $\mathcal {N}_{\rho }$ , we let and write $Z = \bigcup _{\rho } Z_{\rho }$ . For every $\tau \in \mathscr {P}$ , we have the data $\Omega _i$ ’s, $R_i$ ’s, $\Delta _i(\tau )$ and $\check {\Delta }_i(\tau )$ described in Definition 2.9 because $(B,\mathscr {P})$ is simple. Since the normal fan $\mathscr {N}_\tau \subset \Lambda _{\tau ,\mathbb {R}}^*$ of $\tau $ is a refinement of $\mathscr {N}_{\Delta _i(\tau )} \subset \Lambda _{\tau ,\mathbb {R}}^*$ , we have a natural toric morphism

(2.15)

and the identification $\iota _{\tau \rho }^*(\mathcal {N}_{\rho }) \cong \varkappa _{\tau ,i}^*(\mathcal {O}(1))$ . By the proof of [Reference Gross and Siebert27, Thm. 5.2], $\iota _{\tau \rho }^*(f_{\rho })$ is completely determined by the gluing data s and the associated monodromy polytope $\Delta _i(\tau )$ , where $\rho \in R_i$ . In particular, we have $\iota _{\tau \rho }^*(f_{\rho }) = \iota _{\tau \rho '}^*(f_{\rho '})$ and for $\rho ,\rho ' \in R_i$ . Locally, if we write $V(\tau ) = \mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\tau ^{-1}\Sigma _v])$ by choosing some $v \in \tau $ , then, for each $ 1 \leq i \leq p$ , there exists an analytic function $f_{v,i}$ on $V(\tau )$ such that $f_{v,i}|_{V_{\rho }(\tau )} =s_{v\tau }^{-1}( f_{v\rho })$ for $\rho \in R_i$ .

According to [Reference Gross and Siebert28, §2.1], for each $ 1 \leq i \leq p$ , we have $\check {\Delta }_i(\tau ) \subset \mathscr {Q}_{\tau ,\mathbb {R}}^*$ , which gives

(2.16) $$ \begin{align} \psi_i(m) = - \inf \{ \langle m,n \rangle \ | \ n \in \check{\Delta}_i(\tau)\}. \end{align} $$

By convention, we write $\psi _0:= \bar {\varphi }_{\tau }$ . By rearranging the indices i’s, we can assume that $x \in Z^\tau _{1} \cap \cdots \cap Z^\tau _{r}$ and $x \notin Z^{\tau }_{r+1} \cup \cdots \cup Z^{\tau }_{p} $ . We introduce the convention that $\psi _{x,i} = \psi _{i}$ for $0\leq i \leq r$ and $\psi _{x,i} \equiv 0$ for $r<i\leq \dim _{\mathbb {R}}(\tau )$ . Then the local smoothing model near x is constructed as $\mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[P_{\tau ,x}])$ , where

(2.17) $$ \begin{align} P_{\tau,x}:= \{ (m,a_0,\dots,a_{l})\in \mathscr{Q}_{\tau} \times \mathbb{Z}^{l+1} \ | \ a_i \geq \psi_{x,i}(m) \}, \end{align} $$

$l = \dim _{\mathbb {R}}(\tau )$ , and the distinguished element $\varrho = (0,1,0,\dots ,0)$ defines a family

$$ \begin{align*}\mathrm{Spec}_{\mathrm{an}}(\mathbb{C}[P_{\tau,x}]) \rightarrow \mathrm{Spec}_{\mathrm{an}}(\mathbb{C}[q])\end{align*} $$

by sending $q \mapsto z^{\varrho }$ . The central fiber is given by $\mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[Q_{\tau ,x}])$ , where

(2.18) $$ \begin{align} Q_{\tau,x} = \{ (m,a_0,\dots,a_l) \ | \ a_0 = \psi_{x,0}(m) \} \cong P_{\tau,x}/(\varrho+P_{\tau,x}) \end{align} $$

is equipped with the monoid structure

$$ \begin{align*}m + m' = \begin{cases} m+m' & \text{if } m+m' \in Q_{\tau,x},\\ \infty & \text{otherwise.} \end{cases} \end{align*} $$

We have the ring isomorphism $\mathbb {C}[Q_{\tau ,x}] \cong \mathbb {C}[\Sigma _{\tau } \oplus \mathbb {N}^{l}]$ induced by the monoid isomorphism defined by sending $(m,a_0,a_1,\dots ,a_l) \mapsto (m,a_1 - \psi _1(m),\dots ,a_l - \psi _l(m))$ .

We also fix some isomorphism $\mathbb {C}[\tau ^{-1}\Sigma _v] \cong \mathbb {C}[\Sigma _{\tau }\oplus \mathbb {Z}^{l}]$ coming from the identification of $\tau ^{-1}\Sigma _v $ with the fan $\Sigma _{\tau } \oplus \mathbb {R}^{l} = \{ \omega \oplus \mathbb {R}^{l} \ | \ \omega \text { is a cone of } \tau \} $ in $\mathscr {Q}_{\tau ,\mathbb {R}} \oplus \mathbb {R}^{l}$ . Taking a sufficiently small neighborhood V of x such that $Z_\rho \cap V = \emptyset $ if $x \notin Z_{\rho }$ , we define a map $V \rightarrow \mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[Q_{\tau ,x}])$ by composing the inclusion $V \hookrightarrow \mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\tau ^{-1}\Sigma _v]) \cong \mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\Sigma _{\tau }\oplus \mathbb {Z}^{l}])$ with the map

$$ \begin{align*}\mathrm{Spec}_{\mathrm{an}}(\mathbb{C}[\Sigma_{\tau}\oplus \mathbb{Z}^{l}]) \rightarrow \mathrm{Spec}_{\mathrm{an}}(\mathbb{C}[\Sigma_{\tau}\oplus \mathbb{N}^{l}])\end{align*} $$

described on generators by

(2.19) $$ \begin{align} \begin{cases} z^m \mapsto h_m \cdot z^m & \text{if } m\in \Sigma_{\tau} ;\\ u_i \mapsto f_{v,i} & \text{if } 1\leq i \leq r;\\ u_i \mapsto z_i-z_i(x) & \text{if } r<i\leq l; \end{cases} \end{align} $$

here, $u_i$ is the i-th coordinate function of $\mathbb {C}[\mathbb {N}^{l}]$ , $z_i$ is the i-th coordinate function of $\mathbb {C}[\mathbb {Z}^{l}]$ chosen so that $\left (\frac {\partial f_{v,i}}{\partial {z_j}}\right )_{1\leq i \leq r, 1\leq j \leq r}$ is nondegenerate on V; also, each $h_m$ is an invertible holomorphic functions on $V \cap \mathrm {Zero}(z^m;v)$ , and they satisfy the equations (2.12) and (2.13) where we replace $f_{v\rho }$ by

$$ \begin{align*}\tilde{f}_{v\rho} = \begin{cases} s_{v\tau}^{-1}( f_{v\rho}) & \text{if } x \notin Z_{\rho},\\ 1 & \text{if } x \in Z_{\rho}. \end{cases} \end{align*} $$

Letting be the k-th order thickening of V over $\mathbb {C}[q]/q^{k+1}$ in the model $\mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[P_{\tau ,x}])$ under the above embedding, we have a natural divisorial log structure on over induced from the inclusion $\mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[Q_{\tau ,x}]) \hookrightarrow \mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[P_{\tau ,x}])$ (i.e., Example 2.18). Restricting it to V gives the log structure of over the log point locally around x.

3.1 A generalized moment map

From this point onward, we will assume the vanishing of an obstruction class associated to the open gluing data s, namely, $o(s) = 1$ , where the obstruction class $o(s)$ is written multiplicatively (see [Reference Gross and Siebert27, Thm. 2.34]). Under this assumption, one can construct an ample line bundle $\mathcal {L}$ on as follows: For each polytope $\tau \subset \Lambda _{\tau ,\mathbb {R}}$ , by identifying (a closed stratum of described in Remark 2.14) with the projective toric variety associated to $\tau $ , we obtain an ample line bundle $\mathcal {L}_\tau $ on . When the assumption holds, then there exists an isomorphism $\mathbf {h}_{\omega \tau }\colon \iota _{\omega \tau }^*(\mathcal {L}_{\tau }) \cong \mathcal {L}_{\omega }$ , for every pair $\omega \subset \tau $ such that the isomorphisms $\mathbf {h}_{\omega \tau }$ ’s satisfy the cocycle condition, that is, $\mathbf {h}_{\omega \tau }\circ \iota _{\omega \tau }^*(\mathbf {h}_{\tau \sigma }) = \mathbf {h}_{\omega \sigma }$ for every triple $\omega \subset \tau \subset \sigma $ .Footnote ³ In particular, the degenerate Calabi–Yau is projective.

Sections of $\mathcal {L}$ correspond to the lattice points $B_{\mathbb {Z}} \subset B$ . More precisely, given $m\in B_{\mathbb {Z}}$ , there is a unique $\tau \in \mathscr {P}$ such that $m \in \mathrm {int}_{\mathrm {re}}(\tau )$ , and this determines a section $\vartheta _{m,\tau }$ of $\mathcal {L}_{\tau }$ by toric geometry. This section extends uniquely as $\vartheta _{m}$ to $\sigma \supset \tau $ such that $\mathbf {h}_{\tau \sigma }(\vartheta _m) = \vartheta _{m,\tau }$ . Further extending $\vartheta _m$ by $0$ to other cells gives a section of $\mathcal {L}$ corresponding to m, called a ( $0^{\text {th}}$ -order) theta function. Now, for a vertex $v \in \mathscr {P}^{[0]}$ , we can trivialize $\mathcal {L}$ over $V(v)$ using $\vartheta _{v}$ as the holomorphic frame. Then, for m lying in a cell $\sigma $ that contains v, $\vartheta _{m}$ is of the form $g \vartheta _v$ , where g is a constant multiple of $z^{m}$ .

Under the above projectivity assumption, one can define a generalized moment map

(3.1)

following [Reference Ruddat and Siebert43, Prop. 2.1]: First of all, the theta functions $\{\vartheta _m\}_{m \in B_{\mathbb {Z}}}$ defines an embedding of , denoted by . Restricting to each closed toric stratum , the only nonzero theta functions are those corresponding to $m \in B_{\mathbb {Z}} \cap \tau $ . Also, there is an embedding $\mathfrak {j}_{\tau }\colon \mathtt {T}_{\tau } := \Lambda _{\tau ,\mathbb {R}}^*/\Lambda _{\tau ,\mathbb {Z}}^* \hookrightarrow \mathtt {U}(1)^{N}$ of real tori such that the composition of $\Phi $ with the inclusion is equivariant. The map $\mu $ is then defined by setting

(3.2)

which can be understood as a composition of maps

where $\mu _{\mathbb {P}}$ is the standard moment map for $\mathbb {P}^{N}$ and $d\mathfrak {j}_{\tau }\colon \Lambda _{\tau ,\mathbb {R}}^* \rightarrow \mathbb {R}^{N}$ is the Lie algebra homomorphism induced by $\mathfrak {j}_{\tau }\colon \mathtt {T}_{\tau } \rightarrow \mathtt {U}(1)^N$ .

Fixing a vertex $v \in \mathscr {P}^{[0]}$ , we can naturally embed $\Lambda _{\tau ,\mathbb {R}} \hookrightarrow T_{v,\mathbb {R}}$ for all $\tau $ containing v. Furthermore, we can patch the $d\mathfrak {j}^*_{\tau }$ ’s into a linear map $d\mathfrak {j}^* \colon (\mathbb {R}^{N})^* \rightarrow T_{v,\mathbb {R}}$ so that $\mu _{\tau } = d\mathfrak {j}^* \circ \mu _{\mathbb {P}} \circ \Phi _{\tau }$ for each $\tau $ which contains v. In particular, on the local chart $V(\tau )=\mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\tau ^{-1}\Sigma _v])$ associated with $v \in \tau $ , we have the local description $\mu |_{V(\tau )} = d \mathfrak {j}^*\circ \mu _{\mathbb {P}}\circ \Phi |_{V(\tau )}$ of the generalized moment map $\mu $ .

We consider the amoeba $\mathcal {A}:= \mu (Z)$ . As , where $Z^{\tau }_i$ is the zero set of a section of $\varkappa _{\tau ,i}^*(\mathcal {O}(1))$ (see the discussion right after equation (2.15)), we can see that $\mathcal {A} \cap \tau = \bigcup _{i=1}^{p} \mu _{\tau }(Z^{\tau }_i)$ is a union of amoebas $\mathcal {A}^{\tau }_i:= \mu _{\tau }(Z^{\tau }_i)$ . It was shown in [Reference Ruddat and Siebert43] that the affine structure defined right after Definition 2.2 extends to $B \setminus \mathcal {A}$ .

3.2 Construction of charts on B

For any $\tau \in \mathscr {P}$ , we have

$$ \begin{align*}\mu(V(\tau)) = \bigcup_{\tau \subset \omega } \mathrm{int}_{\mathrm{re}}(\omega) =: W(\tau).\end{align*} $$

For later purposes, we would like to relate sufficiently small open convex subsets $W \subset W(\tau )$ with Stein (or strongly $1$ -completed, as defined in [Reference Demailly13]) open subsets $U \subset V(\tau )$ . To do so, we need to introduce a specific collection of (nonaffine) charts on B.

Recall that there are natural maps $\Lambda _{\tau } \hookrightarrow \tau ^{-1} \Sigma _v $ and $\tau ^{-1} \Sigma _v \twoheadrightarrow \Sigma _{\tau }$ . By choosing a piecewise linear splitting $\mathsf {split}_\tau \colon \Sigma _{\tau } \rightarrow \tau ^{-1} \Sigma _v$ , we have an identification of monoids $\tau ^{-1} \Sigma _v \cong \Sigma _{\tau } \times \Lambda _{\tau }$ , which induces the biholomorphism

$$ \begin{align*}V(\tau) = \mathrm{Spec}_{\mathrm{an}}(\mathbb{C}[\tau^{-1} \Sigma_v]) \cong \mathrm{Spec}_{\mathrm{an}}(\mathbb{C}[\Lambda_{\tau}]) \times \mathrm{Spec}_{\mathrm{an}}(\mathbb{C}[\Sigma_{\tau}]),\end{align*} $$

where $\Lambda _{\tau ,\mathbb {C}^*}^*:=\mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\Lambda _{\tau }]) \cong \Lambda _{\tau }^* \otimes _{\mathbb {Z}} \mathbb {C}^* \cong (\mathbb {C}^{*})^l$ is a complex torus. Fixing a set of generators $\{m_i \}_{i \in \mathtt {B}_{\tau }}$ of the monoid $\Sigma _{\tau }$ , which is not necessarily a minimal set, we can define an embedding $\mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\Sigma _{\tau }])\hookrightarrow \mathbb {C}^{|\mathtt {B}_{\tau }|}$ as an analytic subset using the functions $z^{m_i}$ ’s. We consider the real torus $\mathtt {T}_{\tau ,\perp } := \mathscr {Q}_{\tau ,\mathbb {R}}^*/\mathscr {Q}_{\tau }^* \cong \mathtt {U}(1)^{n - l}$ and its action on $\mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\Sigma _{\tau }])$ defined by $t\cdot z^{m} = e^{2\pi i (t,m)} z^m$ , together with an embedding $\mathtt {T}_{\tau ,\perp } \hookrightarrow \mathtt {U}(1)^{|\mathtt {B}_{\tau }|}$ of real tori via $t\mapsto (e^{2\pi i (t,m_i)})_{i \in \mathtt {B}_\tau }$ so that the inclusion $\mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\Sigma _{\tau }])\hookrightarrow \mathbb {C}^{|\mathtt {B}_{\tau }|}$ is $\mathtt {T}_{\tau ,\perp }$ -equivariant.

We consider the moment map $\hat {\mu }_{\tau } \colon \mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\Sigma _{\tau }]) \rightarrow \mathscr {Q}_{\tau ,\mathbb {R}}$ defined by

(3.3) $$ \begin{align} \hat{\mu}_{\tau} := \sum_{i \in \mathtt{B}_{\tau}} \frac{1}{2} |z^{m_i}|^2 \cdot m_i, \end{align} $$

which is obtained by composing the standard moment map $\mathbb {C}^{|\mathtt {B}_{\tau }|} \rightarrow \mathbb {R}^{|\mathtt {B}_{\tau }|}_{\geq 0}$ , $(z_i)_{i\in \mathtt {B}_{\tau }} \mapsto (\frac {1}{2} |z_i|^2)_{i\in \mathtt {B}_{\tau }}$ with the projection $\mathbb {R}^{|\mathtt {B}_{\tau }|} \rightarrow \mathscr {Q}_{\tau ,\mathbb {R}}$ , $e_i \mapsto m_i$ . By [Reference Fulton and William21, §4.2], $\hat {\mu }_{\tau }$ induces a homeomorphism between the quotient $\mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\Sigma _{\tau }])/\mathtt {T}_{\tau ,\perp }$ and $\mathscr {Q}_{\tau ,\mathbb {R}}$ . Taking product with the log map $\log \colon \Lambda _{\tau ,\mathbb {C}^*}^* \rightarrow \Lambda _{\tau ,\mathbb {R}}^*$ (which is induced from the standard log map $\log \colon \mathbb {C}^* \rightarrow \mathbb {R}$ defined by $\log (e^{2\pi (x+i\theta )}) = x$ ), we obtain a map $\mu _{\tau } := (\log , \hat {\mu }_{\tau }) \colon V(\tau ) \rightarrow \Lambda _{\tau ,\mathbb {R}}^* \times \mathscr {Q}_{\tau ,\mathbb {R}}$ ,Footnote ⁴ and the following diagram

(3.4)

where $\Upsilon _{\tau }$ is a homeomorphism which serves as a chart.

The homeomorphism $\Upsilon _{\tau }$ exists because if we fix a vertex $v \in \tau $ , then we can equip $V(\tau )$ with an action by the real torus $\mathtt {T}^{n} := T^*_{v, \mathbb {R}} / T^*_v$ such that both $\mu $ and $\mu _\tau $ induce homeomorphisms from the quotient $V(\tau )/\mathtt {T}^{n}$ onto the images. The restriction of $\Upsilon _{\tau }$ to $\Lambda _{\tau ,\mathbb {R}}^* \times \{o\}$ , where $\{o\}$ is the zero cone, is a homeomorphism onto $\mathrm {int}_{\mathrm {re}}(\tau ) \subset W(\tau )$ , which is nothing but (a generalized version of) the Legendre transform (see [Reference Fulton and William21, §4.2] for the explicit formula); also, this homeomorphism is independent of the choices of the splitting $\mathsf {split}_{\tau }$ and the generators $\{m_i \}_{i \in \mathtt {B}_{\tau}}$ .

The dependencies of the chart $\Upsilon _{\tau }$ on the choices of the splitting $\mathsf {split}_{\tau }\colon \Sigma _{\tau } \rightarrow \tau ^{-1} \Sigma _v$ and the generators $\{m_i \}_i$ can be described as follows. First, if we choose another piecewise linear splitting $\widetilde {\mathsf {split}}_{\tau }\colon \Sigma _{\tau } \rightarrow \tau ^{-1} \Sigma _v$ , then there is a piecewise linear map $b \colon \Sigma _{\tau } \rightarrow \Lambda _{\tau ,\mathbb {R}}$ recording the difference between $\mathsf {split}_\tau $ and $\widetilde {\mathsf {split}}_{\tau }$ . The two corresponding coordinate charts $\Upsilon _{\tau }$ and $\tilde {\Upsilon }_{\tau }$ are then related by a homeomorphism $\gimel $ such that

$$ \begin{align*}\gimel\left(x,\sum_{i} y_i m_i\right) = \left(x , \sum_{i}y_i e^{4\pi \langle b(m_i),x\rangle } m_i \right), \end{align*} $$

where $y_i= \frac {1}{2} |z^{m_i}|^2$ for some point $z\in \mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\Sigma _{\tau }])$ and i runs through $m_i \in \sigma $ , via the formula $\tilde {\Upsilon }_{\tau } = \Upsilon _{\tau } \circ \gimel $ . Second, if we choose another set of generators $\tilde {m}_j$ ’s, then the corresponding maps $\hat {\mu }_{\tau }, \tilde {\mu }_{\tau } \colon \mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\Sigma _{\tau }]) \rightarrow \mathscr {Q}_{\tau ,\mathbb {R}}$ are related by a continuous map $\hat {\gimel } \colon \mathscr {Q}_{\tau ,\mathbb {R}} \rightarrow \mathscr {Q}_{\tau ,\mathbb {R}}$ which maps each cone $\sigma \in \Sigma _{\tau }$ back to itself. This is because both $\hat {\mu }_{\tau }, \tilde {\mu }_{\tau } $ induce a homeomorphism between $\mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\Sigma _{\tau }])/\mathtt {T}_{\tau ,\perp }$ and $\mathscr {Q}_{\tau ,\mathbb {R}}$ .

Now, suppose that $\omega \subset \tau $ . We want to see how the charts $\Upsilon _{\omega }$ , $\Upsilon _{\tau }$ can be glued together in a compatible manner. We first make a compatible choice of splittings. So we fix a vertex $v \in \omega $ and a piecewise linear splitting $\mathsf {split}_{\omega }\colon \Sigma _{\omega } \rightarrow \omega ^{-1} \Sigma _{v}$ . We then choose a piecewise linear splitting $\mathsf {split}_{\omega \tau }\colon \Sigma _{\tau } \rightarrow \Sigma _{\omega }$ such that $K_\tau \sigma $ is mapped into $K_\omega \sigma $ for any $\sigma \supset \tau $ . Together with the natural maps $\Lambda _{\tau }/\Lambda _{\omega } \hookrightarrow \tau ^{-1}\Sigma _{\omega }$ and $\tau ^{-1}\Sigma _{\omega } \twoheadrightarrow \Sigma _{\tau }$ , we obtain an isomorphism $\tau ^{-1}\Sigma _{\omega } \cong (\Lambda _{\tau }/\Lambda _{\omega }) \times \Sigma _{\tau }$ . By composing together $\mathsf {split}_{\omega \tau }\colon \Sigma _{\tau } \rightarrow \Sigma _{\omega }$ , $\mathsf {split}_{\omega }\colon \Sigma _{\omega } \rightarrow \omega ^{-1} \Sigma _{v}$ and the natural monoid homomorphism $ \omega ^{-1} \Sigma _{v} \rightarrow \tau ^{-1}\Sigma _{v}$ , we get a splitting $\mathsf {split}_{\tau }\colon \Sigma _{\tau } \rightarrow \tau ^{-1} \Sigma _{v}$ .

Using these choices of splittings, we have a biholomorphism

$$ \begin{align*}\mathrm{Spec}_{\mathrm{an}}(\mathbb{C}[\tau^{-1}\Sigma_{\omega}]) \cong (\Lambda_{\tau}/\Lambda_{\omega})^*\otimes_{\mathbb{Z}}\mathbb{C}^* \times \mathrm{Spec}_{\mathrm{an}}(\mathbb{C}[\Sigma_{\tau}])\end{align*} $$

which fits into the following diagram

(3.5)

Here, the bottom left horizontal map is induced from a splitting $(\Lambda _{\tau }/\Lambda _{\omega }) \rightarrow \Lambda _{\tau }$ obtained by composing $\Lambda _{\tau }/\Lambda _{\omega } \rightarrow \tau ^{-1}\Sigma _{\omega }$ with the splitting $\tau ^{-1}\Sigma _{\omega } \rightarrow \tau ^{-1}(\omega ^{-1} \Sigma _{v})$ and then identifying with the image lattice $\Lambda _{\tau }$ . The appearance of $s_{\omega \tau }$ in the diagram is due to the twisting of $V(\tau )$ by the open gluing data $(s_{\omega \tau })_{\omega \subset \tau }$ when it is glued to $V(\omega )$ .

We also have to make a compatible choice of the generators $\{m_i\}_{i \in \mathtt {B}_{\omega }}$ and $\{m_i\}_{i \in \mathtt {B}_{\tau }}$ . First, note that the restriction of $\hat {\mu }_{\omega }$ to the open subset $ \mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\tau ^{-1}\Sigma _{\omega }]) \subset \mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\Sigma _{\omega }])$ depends only on the subcollection $\{m_i\}_{i \in \mathtt {B}_{\omega \subset \tau }}$ of $\{m_i\}_{i \in \mathtt {B}_{\omega }}$ which contains those $m_i$ ’s that belong to some cone $\sigma \supset \tau $ . We choose the set of generators $\{\tilde {m}_i\}_{i \in \mathtt {B}_{\tau }}$ for $\Sigma _{\tau }$ , with $\mathtt {B}_{\tau }=\mathtt {B}_{\omega \subset \tau }$ , to be the projection of $\{m_i\}_{i \in \mathtt {B}_{\omega \subset \tau }}$ through the natural map $\tau ^{-1} \Sigma _{\omega } \rightarrow \Sigma _{\tau }$ . Each $m_i$ can be expressed as $m_i = \mathsf {split}_{\omega \tau }(\tilde {m}_i) + b_i$ for some $b_i \in \Lambda _{\tau }/\Lambda _{\omega }$ , through the splitting $\mathsf {split}_{\omega \tau }\colon \Sigma _{\tau } \rightarrow \Sigma _{\omega }$ . Notice that if $m_i \in K_{\omega } \tau $ , then we have $\tilde {m}_i= o$ and hence $b_i \in K_{\omega }\tau $ . By tracing through the biholomorphism in equation (3.5) and taking either the modulus or the log map, we have a map

$$ \begin{align*}\gimel \colon \Lambda_{\omega,\mathbb{R}}^* \times (\Lambda_{\tau,\mathbb{R}}/\Lambda_{\omega,\mathbb{R}})^* \times \mathscr{Q}_{\tau,\mathbb{R}} \rightarrow \Lambda_{\omega,\mathbb{R}}^* \times \mathscr{Q}_{\omega,\mathbb{R}},\end{align*} $$

satisfying

(3.6) $$ \begin{align} \gimel\left(x_1 - c_{\omega\tau,1},x_2-c_{\omega\tau,2},\sum_i y_i |s_{\omega\tau}(\mathsf{split}_{\omega\tau}(\tilde{m}_i))|^{-2} \tilde{m}_i\right) = \left(x_1,\sum_{i} y_i e^{4\pi \langle b_i,x_2 \rangle} m_i\right), \end{align} $$

where $y_i = \frac {1}{2} | z^{\tilde {m}_i}|^2$ . Here, $s_{\omega \tau } \in \mathrm {PM}(\tau )$ is the part of the open gluing data associated to $\omega \subset \tau $ , and $c_{\omega \tau }=c_{\omega \tau ,1}+c_{\omega \tau ,2} \in \Lambda _{\tau ,\mathbb {R}}^*$ is the unique element representing the linear map $\log |s_{\omega \tau }| \colon \Lambda _{\tau ,\mathbb {R}} \rightarrow \mathbb {R}$ defined by $\log |s_{\omega \tau }|(b) = \log |s_{\omega \tau }(b)|$ . For instance, the holomorphic function $z^{m_i} \in \mathbb {C}[\tau ^{-1}\Sigma _{\omega }]$ is identified with $z^{b_i}\cdot z^{\tilde {m}_i}$ in $(\Lambda _{\tau }/\Lambda _{\omega })^*\otimes _{\mathbb {Z}}\mathbb {C}^* \times \mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\Sigma _{\tau }])$ , resulting in the expression $\sum _{i} y_i e^{4\pi \langle b_i,x_2 \rangle } m_i$ on the right-hand side. We have $\Upsilon _{\tau } = \Upsilon _{\omega } \circ \gimel $ , where we use the splitting $(\Lambda _{\tau }/\Lambda _{\omega }) \rightarrow \Lambda _{\tau }$ to obtain an isomorphism $\Lambda _{\omega ,\mathbb {R}}^* \times (\Lambda _{\tau ,\mathbb {R}}/\Lambda _{\omega ,\mathbb {R}})^* \cong \Lambda _{\tau ,\mathbb {R}}^*$ and an identification of the domains of the two maps $\Upsilon _{\tau }$ and $\Upsilon _{\omega } \circ \gimel $ .

Proof. First of all, it is well known that analytic spaces associated to affine varieties are Stein. So $V(\tau )$ is Stein for any $\tau $ . Now, we fix a point $x \in \mathrm {int}_{\mathrm {re}}(\tau ) \subset B$ . It suffices to show that there is a local base $\mathscr {B}_x$ of x such that the preimage $\mu ^{-1}(W)$ is Stein for each $W \in \mathscr {B}_x$ . We work locally on $\mu |_{V(\tau )} \colon V(\tau ) \rightarrow W(\tau )$ . Consider the diagram (3.4) and write $\Upsilon ^{-1}(x) = (\underline {x},o)$ , where $o \in \mathscr {Q}_{\tau ,\mathbb {R}}$ is the origin. By [Reference Demailly13, Ch. 1, Ex. 7.4], the preimage $\log ^{-1}(W)$ under the log map $\log \colon (\mathbb {C}^*)^l \rightarrow \Lambda _{\tau ,\mathbb {R}}^*$ is Stein for any convex $W \subset \Lambda _{\tau ,\mathbb {R}}^*$ which contains $\underline {x}$ . Again by [Reference Demailly13, Ch. 1, Ex. 7.4], any subset

$$ \begin{align*}\bigcap_{j=1}^N\{z \in \mathrm{Spec}_{\mathrm{an}}(\mathbb{C}[\Sigma_{\tau}]) \ | \ |f_j(z)|<\epsilon \},\end{align*} $$

where $f_j$ ’s are holomorphic functions, is Stein. By taking $f_j$ ’s to be the functions $z^{m_j}$ ’s associated to the set of all nonzero generators in $\{m_j\}_{j\in \mathtt {B}_{\tau }}$ and $\epsilon $ sufficiently small, we have a subset

$$ \begin{align*}W = \left\{ y \ \Big| \ y = \sum_{j} y_j m_j \text{ with } |y_j|<\frac{\epsilon^2}{2}, \ \text{where } y_j = \frac{1}{2} |z^{m_j}|^2 \text{ at some point } z \in \mathrm{Spec}_{\mathrm{an}}(\mathbb{C}[\Sigma_{\tau}])\right\}\end{align*} $$

of $\mathscr {Q}_{\tau ,\mathbb {R}}$ such that the preimage $\hat {\mu }_{\tau }^{-1}(W)$ is Stein. Therefore, we can construct a local base $\mathscr {B}_{o}$ of o such that the preimage $\hat {\mu }_{\tau }^{-1}(W)$ is Stein for any $W \in \mathscr {B}_o$ . Finally, since a product of Stein open subsets is Stein, we obtain our desired local base $\mathscr {B}_x$ by taking the products of these subsets.

3.3 The tropical singular locus $\mathscr {S}$ of B

We now specify a codimension $2$ singular locus $\mathscr {S} \subset B$ of the affine structure using the charts $\Upsilon _{\tau }$ introduced in (3.4) for $\tau $ such that $\dim _{\mathbb {R}}(\tau )<n$ . Given the chart $\Upsilon _{\tau }$ that maps $\Lambda _{\tau ,\mathbb {R}}^*$ to $\mathrm {int}_{\mathrm {re}}(\tau )$ , we define the tropical singular locus $\mathscr {S}$ by requiring that

(3.7) $$ \begin{align} \Upsilon_{\tau}^{-1}(\mathscr{S} \cap \mathrm{int}_{\mathrm{re}}(\tau)) = \bigcup_{\substack{\rho \in \mathscr{N}_{\tau};\\ \dim_{\mathbb{R}}(\rho) < \dim_{\mathbb{R}}(\tau)}} \big( (\mathrm{int}_{\mathrm{re}}(\rho) +c_{\tau}) \times \{o\} \big) , \end{align} $$

where $\mathscr {N}_{\tau } \subset \Lambda _{\tau ,\mathbb {R}}^*$ is the normal fan of the polytope $\tau $ , and $\{o\}$ is the zero cone in $\Sigma _{\tau } \subset \mathscr {Q}_{\tau ,\mathbb {R}}$ ; here, $c_{\tau } = \log |s_{v\tau }|$ is the element in $\Lambda _{\tau ,\mathbb {R}}^*$ representing the linear map $\log |s_{v\tau }|\colon \Lambda _{\tau ,\mathbb {R}} \rightarrow \mathbb {R}$ , which is independent of the vertex $v\in \tau $ . A subset of the form $\mathscr {S}_{\tau ,\rho } := (\mathrm {int}_{\mathrm {re}}(\rho )+c_{\tau }) \times \{o\}$ in (3.7) is called a stratum of $\mathscr {S}$ in $\mathrm {int}_{\mathrm {re}}(\tau )$ . The locus $\mathscr {S}$ is independent of the choices of the splittings $\mathsf {split}_{\tau }$ ’s and generators $\{m_i\}_{i \in \mathtt {B}_{\tau }}$ used to construct the charts $\Upsilon _{\tau }$ ’s.

Proof. We consider the map $\gimel $ described in equation (3.6) and take a neighborhood $W = W_1 \times \mathscr {Q}_{\omega ,\mathbb {R}}$ of a point $(\underline {x},o)$ in $ \Lambda _{\omega ,\mathbb {R}}^* \times \mathscr {Q}_{\omega ,\mathbb {R}}$ , where $W_1$ is some sufficiently small neighborhood of $\underline {x}$ in $\Lambda _{\omega ,\mathbb {R}}^*$ . By shrinking W if necessary, we may assume that $\gimel ^{-1}(W) = W_1 \times (a -\mathrm {int}_{\mathrm {re}}(K_{\omega }\tau ^{\vee })) \times \mathscr {Q}_{\tau ,\mathbb {R}}$ , where a is some element in $-\mathrm {int}_{\mathrm {re}}(K_{\omega }\tau ^{\vee }) \subset (\Lambda _{\tau ,\mathbb {R}}/\Lambda _{\omega ,\mathbb {R}})^*$ . Writing $c_{\tau } = c_{\tau ,1}+c_{\tau ,2}$ , where $c_{\tau ,1},c_{\tau ,2}$ are the components of $c_{\tau }$ according to the chosen decomposition $\Lambda _{\tau ,\mathbb {R}}^* \cong \Lambda _{\omega ,\mathbb {R}}^* \times (\Lambda _{\tau ,\mathbb {R}}/\Lambda _{\omega ,\mathbb {R}})^*$ , the equality $c_{\tau ,1} + c_{\omega \tau ,1} = c_{\omega }$ follows from the compatibility of the open gluing data in Definition 2.13. If $\mathscr {S}_{\tau ,\rho }$ intersects the open subset $\gimel ^{-1}(W)$ , then $\rho \subset \Lambda _{\tau ,\mathbb {R}}^*$ must be the dual cone of some face $\rho ^{\vee } \subset \omega \subset \tau $ in $\Lambda _{\tau ,\mathbb {R}}^*$ . The intersection is of the form

$$ \begin{align*}(\mathrm{int}_{\mathrm{re}}(\underline{\rho})+c_{\tau,1}) \times (a-\mathrm{int}_{\mathrm{re}}(K_{\omega}\tau^{\vee})) \times \{o\}\end{align*} $$

for some $\underline {\rho } \in \mathscr {N}_{\omega }$ ( $c_{\tau ,2}$ is absorbed by a), where $\underline {\rho } \subset \Lambda _{\omega ,\mathbb {R}}^*$ is the dual cone of $\rho ^{\vee }$ in $\Lambda _{\omega ,\mathbb {R}}^*$ , and hence we have $W \cap \mathscr {S}_{\tau ,\rho } =\gimel ((\mathrm {int}_{\mathrm {re}}(\underline {\rho })+c_{\tau ,1}) \times (a-\mathrm {int}_{\mathrm {re}}(K_{\omega }\tau ^{\vee })) \times \{o\})$ . Therefore, the intersection of $\overline {\mathscr {S}_{\tau ,\rho }}$ with $\Lambda _{\omega ,\mathbb {R}}^*$ in the open subset $W \subset \Lambda _{\omega ,\mathbb {R}}^* \times \mathscr {Q}_{\omega ,\mathbb {R}}$ is given by $(\underline {\rho }+c_{\omega }) \times \{o\}$ , which is a union of strata.

The tropical singular locus $\mathscr {S}$ is naturally equipped with a stratification, where a stratum is given by $\mathscr {S}_{\tau ,\rho }$ for some cone $\rho \subset \mathscr {N}_{\tau }$ of $\dim _{\mathbb {R}}(\rho ) < \dim _{\mathbb {R}}(\tau )$ for some $\tau \in \mathscr {P}^{[<n]}$ . We use the notation $\mathscr {S}^{[k]}$ to denote the set of k-dimensional strata of $\mathscr {S}$ . The affine structure on $\bigcup _{v \in \mathscr {P}^{[0]}} W_v \cup \bigcup _{\sigma \in \mathscr {P}^{[n]}} \mathrm {int}_{\mathrm {re}}(\sigma )$ introduced right after Definition 2.2 in §2.1 can be naturally extended to $B \setminus \mathscr {S}$ as in [Reference Gross and Siebert29].

If we consider $\omega \subset \tau \subset \rho $ for some $\omega \in \mathscr {P}^{[1]}$ and $\rho \in \mathscr {P}^{[n-1]}$ , the corresponding monodromy transformation $T_{\gamma }$ is nontrivial if and only if $\omega \in \Omega _p$ and $\rho \in R_p$ , where p is as in Definition 2.9. Therefore, the part of the singular locus $\mathscr {S}$ lying in $\Upsilon _{\tau }^{-1}(\mathrm {int}_{\mathrm {re}}(\tau )) = \Lambda _{\tau ,\mathbb {R}}^* \times \{o\}$ is determined by the subsets $\Omega _p$ ’s. We may further define the essential singular locus $\mathscr {S}_e$ to include only those strata contained in $\mathscr {S}^{[n-2]}$ with nontrivial monodromy around them. We observe that the affine structure can be further extended to $B \setminus \mathscr {S}_e$ .

More explicitly, we have a projection

$$ \begin{align*}\mathtt{i}_{\tau} = \mathtt{i}_{\tau,1} \oplus \cdots \oplus \mathtt{i}_{\tau,p} \colon \Lambda_{\tau}^* \rightarrow \Lambda_{\Delta_1(\tau)}^* \oplus \cdots \oplus \Lambda_{\Delta_p(\tau)}^*,\end{align*} $$

in which $\Lambda _{\Delta _1(\tau )}^* \oplus \cdots \oplus \Lambda _{\Delta _p(\tau )}^*$ can be treated as a direct summand as in §2.2. So we can consider the pullback of the fan $\mathscr {N}_{\Delta _1(\tau )} \times \cdots \times \mathscr {N}_{\Delta _p(\tau )}$ via the map $\mathtt {i}_{\tau }$ and realize $\mathscr {N}_{\tau } \subset \Lambda _{\tau ,\mathbb {R}}^*$ as a refinement of this fan. Similarly, we have $\check {\mathtt {i}}_{\tau } =\check {\mathtt {i}}_{\tau ,1} \oplus \cdots \oplus \check {\mathtt {i}}_{\tau ,p} \colon \mathscr {Q}_{\tau }^* \rightarrow \Lambda ^*_{\check {\Delta }_1(\tau )} \oplus \cdots \oplus \Lambda ^*_{\check {\Delta }_p(\tau )}$ and the fan $\mathscr {N}_{\check {\Delta }_1(\tau )} \times \cdots \times \mathscr {N}_{\check {\Delta }_p(\tau )}$ in $\mathscr {Q}_{\tau ,\mathbb {R}}^*$ under pullback via $\check {\mathtt {i}}_{\tau }$ . The intersection $\mathscr {S}_e \cap \mathrm {int}_{\mathrm {re}}(\tau ) $ can be described by replacing $\rho \in \mathscr {N}_{\tau }$ with the condition $\rho \in \mathtt {i}_{\tau }^{-1}(\mathscr {N}_{\Delta _1(\tau )} \times \cdots \times \mathscr {N}_{\Delta _p(\tau )})$ , with a stratum denoted by $\mathscr {S}_{e,\tau ,\rho }$ . This gives a stratification on $\mathscr {S}_e$ .

Proof. Given $\omega \subset \tau $ , we take a change of coordinate map $\gimel $ together with a neighborhood W as in the proof of Lemma 3.3. We need to show that $W \cap \mathscr {S}_{\tau ,\rho } =\gimel ((\mathrm {int}_{\mathrm {re}}(\rho )+c_{\tau ,1}) \times (a-\mathrm {int}_{\mathrm {re}}(K_{\omega }\tau ^{\vee })) \times \{o\})$ for some cone $\rho \in \mathtt {i}_{\tau }^{-1}(\prod _{i=1}^p \mathscr {N}_{\Delta _i(\tau )})$ . Let $\Delta _1(\tau ),\dots ,\Delta _r(\tau ),\dots ,\Delta _p(\tau )$ be the monodromy polytopes of $\tau $ , and $\Delta _1(\omega ), \dots , \Delta _r(\omega ),\dots ,\Delta _{p'}(\omega )$ be those of $\omega $ such that $\Delta _j(\omega )$ is the face of $\Delta _j(\tau )$ parallel to $\Lambda _{\omega }$ for $j=1,\dots ,r$ . Then we have direct sum decompositions $\Lambda _{\Delta _{1}(\tau )}\oplus \cdots \oplus \Lambda _{\Delta _{p}(\tau )} \oplus A_{\tau } = \Lambda _{\tau }$ and $ \Lambda _{\Delta _{1}(\omega )}\oplus \cdots \oplus \Lambda _{\Delta _{p'}(\omega )} \oplus A_{\omega } = \Lambda _{\omega }$ . We can further choose an inclusion

$$ \begin{align*}\mathsf{a}_{\omega\tau}\colon \Lambda_{\Delta_{r+1}(\omega)}\oplus \cdots \oplus \Lambda_{\Delta_{p'}(\omega)} \oplus A_{\omega} \hookrightarrow A_{\tau}; \end{align*} $$

in other words, for every $j=r+1,\dots ,p'$ , any $f \in R_j \subset \mathscr {P}_{n-1}(\omega )$ in Definition 2.9 is not containing $\tau $ . For every $j=r+1,\dots ,p$ and any $f \in R_j \subset \mathscr {P}_{n-1}(\tau )$ , the element $m^{f}_{v_1v_2}$ is zero for any two vertices $v_1,v_2$ of $\omega $ . We have the identification

$$ \begin{align*}\Lambda_{\tau}/\Lambda_{\omega} = \bigoplus_{j=1}^r (\Lambda_{\Delta_j(\tau)}/\Lambda_{\Delta_j(\omega)}) \oplus \bigoplus_{l=r+1}^{p} \Lambda_{\Delta_l(\tau)} \oplus \mathrm{coker}(\mathsf{a}_{\omega\tau}). \end{align*} $$

As a result, any cone $\mathtt {i}^{-1}_{\tau }(\prod _{j=1}^{p} \rho _j) \in \mathtt {i}^{-1}_{\tau }\big (\prod _{i=1}^p \mathscr {N}_{\Delta _i(\tau )} \big )$ of codimension greater than $0$ intersecting $\gimel ^{-1}(W)$ will be a pullback of a cone under the projection to $\Lambda _{\Delta _1(\tau ),\mathbb {R}}^* \oplus \cdots \oplus \Lambda _{\Delta _r(\tau ),\mathbb {R}}^*$ . Consider the commutative diagram of projection maps

(3.8)

We see that, in the open subset $\gimel ^{-1}(W)$ , every cone of codimension greater than $0$ coming from pullback via $\mathtt {p}_\tau $ is a further pullback via $\Pi _{\omega \subset \tau } \circ \mathtt {p}_{\tau }$ . As a consequence, it must be of the form $\gimel ((\mathrm {int}_{\mathrm {re}}(\rho )+c_{\tau ,1}) \times (a-\mathrm {int}_{\mathrm {re}}(K_{\omega }\tau ^{\vee })) \times \{o\})$ in W.

3.3.1 Contraction of $\mathcal {A}$ to $\mathscr {S}$

We would like to relate the amoeba $\mathcal {A} = \mu (Z)$ with the tropical singular locus $\mathscr {S}$ introduced above.

Assumption 3.5. We assume the existence of a surjective contraction map $\mathscr {C} \colon B \rightarrow B$ which is isotopic to the identity and satisfies the following conditions:

1. 1. $\mathscr {C}^{-1}(B \setminus \mathscr {S}) \subset (B \setminus \mathscr {S})$ and the restriction $\mathscr {C}|_{\mathscr {C}^{-1}(B \setminus \mathscr {S})}\colon \mathscr {C}^{-1}(B \setminus \mathscr {S}) \to B \setminus \mathscr {S}$ is a homeomorphism.

2. 2. $\mathscr {C}$ maps $\mathcal {A}$ into the essential singular locus $\mathscr {S}_e$ .

3. 3. For each $\tau \in \mathscr {P}$ , we have $\mathscr {C}^{-1}(\mathrm {int}_{\mathrm {re}}(\tau )) \subset \mathrm {int}_{\mathrm {re}}(\tau )$ .

4. 4. For each $\tau \in \mathscr {P}$ with $0<\dim _{\mathbb {R}}(\tau )<n$ , we have a decomposition

   $$ \begin{align*}\tau \cap \mathscr{C}^{-1}(B\setminus \mathscr{S}) = \bigcup_{v \in \tau^{[0]}} \tau_{v}\end{align*} $$
   of the intersection $\tau \cap \mathscr {C}^{-1}(B\setminus \mathscr {S})$ into connected components $\tau _{v}$ ’s, where each $\tau _{v}$ is contractible and is the unique component containing the vertex $v \in \tau $ .

5. 5. For each $\tau \in \mathscr {P}$ and each point $x \in \mathrm {int}_{\mathrm {re}}(\tau ) \cap \mathscr {S}$ , $\mathscr {C}^{-1}(x) \subset \mathrm {int}_{\mathrm {re}}(\tau )$ is a connected compact subset.

6. 6. For each $\tau \in \mathscr {P}$ and each point $x \in \mathrm {int}_{\mathrm {re}}(\tau ) \cap \mathscr {S}$ , there exists a local base $\mathscr {B}_x$ around x such that $(\mathscr {C} \circ \mu )^{-1}(W) \subset V(\tau )$ is Stein for every $W \in \mathscr {B}_x$ , and for any $U \supset \mathscr {C}^{-1}(x)$ , we have $\mathscr {C}^{-1}(W) \subset U$ for sufficiently small $W \in \mathscr {B}_x$ .

Similar contraction maps appear in [Reference Ruddat and Siebert43, Rem. 2.4] (see also [Reference Ruddat and Zharkov45, 44]).

When $\dim _{\mathbb {R}}(B) = 2$ , we can take $\mathscr {C} = \mathrm {id}$ because from [Reference Gross and Siebert27, Ex. 1.62], we see that Z is a finite collection of points, with at most one point lying in each closed stratum , and the amoeba $\mathcal {A}$ is exactly the image of Z under the generalized moment map $\mu $ .

When $\dim _{\mathbb {R}}(B) = 3$ , the amoeba $\mathcal {A}$ can possibly be of codimension one and we need to construct a contraction map as shown in Figure 5.

Figure 5

Contraction map $\mathscr {C}$ when $\dim _{\mathbb {R}}(B) = 3$ .

For $\dim _{\mathbb {R}}(\tau ) = 1$ , again from [Reference Gross and Siebert27, Ex. 1.62], we see that if $\mathcal {A} \cap \mathrm {int}_{\mathrm {re}}(\tau ) \neq \emptyset $ , then there is exactly one $\Omega _1$ and $R_1$ , and $\Delta _{1}(\tau )$ is a line segment of affine length $1$ . In this case, consists of only one point, given by the intersection of the zero locus $s_{v\tau }^{-1}(f_{v\rho })$ with $\mathbb {C}^* \cong V_{\tau }(\tau ) \subset V(\tau )$ . Taking m to be the primitive vector in $\Lambda _{\tau }$ starting at v that points into $\tau $ , we can write $s_{v\tau }^{-1}(f_{v\rho }) = 1 + s_{v\tau }^{-1}(m) z^{m}$ . Applying the log map $\log \colon \mathbb {C}^* \rightarrow \mathbb {R}$ , we see that $\mathcal {A} \cap \mathrm {int}_{\mathrm {re}}(\tau ) = c_{\tau }$ . Therefore, for an edge $\tau \in \mathscr {P}^{[1]}$ , we can define $\mathscr {C}$ to be the identity on $\tau $ .

On a codimension one cell $\rho $ such that $\mathrm {int}_{\mathrm {re}}(\rho ) \cap \mathcal {A} \neq \emptyset $ (see Figure 6), we consider the log map $\log \colon \mathrm {Spec}_{\mathrm {an}}(\mathbb {C}[\Lambda _{\rho }]) \cong (\mathbb {C}^*)^2 \rightarrow \Lambda _{\rho ,\mathbb {R}}^* \cong \mathbb {R}^2$ and take a sufficiently large polytope $\mathtt {P}$ (colored purple in Figure 6) so that $\mathcal {A} \setminus \mathrm {int}_{\mathrm {re}}(\mathtt {P})$ is a disjoint union of legs. We first contract each leg to the tropical singular locus (colored blue in Figure 6) along the normal direction to the tropical singular locus. Next, we contract the polytope $\mathtt {P}$ to the zero-dimensional stratum of $\mathscr {S}_e$ . Notice that the restriction of $\mathscr {C}$ to the tropical singular locus $\mathscr {S}$ is not the identity but rather a contraction onto itself. Once the contraction map is constructed for all codimension one cells $\rho $ , we can then extend it continuously to the whole of B so that it is a diffeomorphism on $\mathrm {int}_{\mathrm {re}}(\sigma )$ for every maximal cell $\sigma $ . The map is chosen such that the preimage $\mathscr {C}^{-1}(x)$ for every point $x \in \mathrm {int}_{\mathrm {re}}(\rho )$ is a convex polytope in $\mathbb {R}^2$ . Therefore, given any open subset $U\subset \mathbb {R}^2$ which contains $\mathscr {C}^{-1}(x)$ , we can find some convex open neighborhood $W_1 \subset U$ of $\mathscr {C}^{-1}(x)$ giving the Stein open subset $\log ^{-1}(W_1) \subset (\mathbb {C}^*)^2$ . By taking $W = W_1 \times W_2$ in the chart $\Lambda _{\rho ,\mathbb {R}}^* \times \mathscr {Q}_{\rho ,\mathbb {R}}$ as in the proof of Lemma 3.1, we have the open subset W that satisfies condition (5) in Assumption 3.5.

Figure 6

Contraction at $\rho $ .

In general, we need to construct $\mathscr {C}|_{\mathrm {int}_{\mathrm {re}}(\tau )}$ inductively for each $\tau \in \mathscr {P}$ such that $\mathscr {C}^{-1}(x) \subset \mathrm {int}_{\mathrm {re}}(\tau )$ is convex in the chart $\Lambda _{\tau ,\mathbb {R}}^*\cong \mathrm {int}_{\mathrm {re}}(\tau )$ and the codimension one amoeba $\mathcal {A}$ is contracted to the codimension 2 tropical singular locus $\mathscr {S}_e$ . The reason for introducing such a contraction map is that we can modify the generalized moment map $\mu $ to one which is more closely related with tropical geometry:

Definition 3.6. We call the composition the modified moment map.

One immediate consequence of property $(6)$ in Assumption 3.5 is that we have $R\nu _* (\mathcal {F}) = \nu _*(\mathcal {F})$ for any coherent sheaf $\mathcal {F}$ on , thanks to Lemma 3.1 and Cartan’s Theorem B:

Theorem 3.7 (Cartan’s Theorem B [Reference Cartan6]; see e.g. Ch. IX, Cor. 4.11 in [Reference Demailly13]).

For any coherent sheaf $\mathcal {F}$ over a Stein space U, we have $ H^{>0}(U,\mathcal {F}) = 0. $

3.3.2 Monodromy invariant differential forms on B

Outside of the essential singular locus $\mathscr {S}_e$ , we have a nice integral affine manifold $B \setminus \mathscr {S}_e$ , on which we can talk about the sheaf $\Omega ^*$ of ( $\mathbb {R}$ -valued) de Rham differential forms. But in fact, we can extend its definition to $\mathscr {S}_e$ as well using monodromy invariant differential forms.

We consider the inclusion $\iota \colon B_0 :=B \setminus \mathscr {S}_e \rightarrow B$ and the natural exact sequence

(3.9) $$ \begin{align} 0 \rightarrow \underline{\mathbb{Z}} \rightarrow \mathcal{A}{ff} \rightarrow \iota_* \Lambda_{B_0}^* \rightarrow 0, \end{align} $$

where $\Lambda _{B_0}^*$ denotes the sheaf of integral cotangent vectors on $B_0$ . For any $\tau \in \mathscr {P}$ , the stalk $(\iota _*\Lambda _{B_0}^*)_x$ at a point $x \in \mathrm {int}_{\mathrm {re}}(\tau ) \cap \mathscr {S}_e$ can be described using the chart $\Upsilon _{\tau }$ in equation (3.4). Using the description in §3.3, we have $x \in \mathscr {S}_{e,\tau ,\rho } = \mathrm {int}_{\mathrm {re}}(\rho ) \times \{o\}$ for some $\rho \in \mathtt {i}_{\tau }^{-1}(\mathscr {N}_{\Delta _1(\tau )} \times \cdots \times \mathscr {N}_{\Delta _p(\tau )})$ . Taking a vertex $v \in \tau $ , we can consider the monodromy transformations $T_{\gamma }$ ’s around the strata $\mathscr {S}_{e,\eta ,\rho }$ ’s that contain x in their closures. We can identify the stalk $\iota _*(\Lambda _{B_0}^*)_{x}$ as the subset of invariant elements of $T_{v}^*$ under all such monodromy transformations. Since $\rho \subset \Lambda _{\tau ,\mathbb {R}}^*$ is a cone, we have $\Lambda _{\rho } \subset \Lambda _{\tau }^*$ . Using the natural projection map $\pi _{v\tau }\colon T_{v}^* \rightarrow \Lambda _{\tau }^*$ , we have the identification $\iota _*(\Lambda _{B_0}^*)_{x} \cong \pi _{v\tau }^{-1}(\Lambda _{\rho })$ . There is a direct sum decomposition $\iota _*(\Lambda _{B_0}^*)_{x} = \Lambda _{\rho } \oplus \mathscr {Q}_{\tau }^*$ , depending on a decomposition $T_{v} = \Lambda _{\tau } \oplus \mathscr {Q}_{\tau }$ . This gives the map

(3.10) $$ \begin{align} \mathtt{x} \colon U_{x} \rightarrow \pi_{v\tau}^{-1}(\Lambda_{\rho})_{\mathbb{R}}^* \end{align} $$

in a sufficiently small neighborhood $U_{x}$ , locally defined up to a translation in $\pi _{v\tau }^{-1}(\Lambda _{\rho })^*_{\mathbb {R}}$ . We need to describe the compatibility between the map associated to a point $x \in \mathscr {S}_{e,\omega ,\rho }$ and that to a point $\tilde {x} \in \mathscr {S}_{e,\tau ,\tilde {\rho }}$ such that $\mathscr {S}_{e,\omega ,\rho } \subset \overline {\mathscr {S}_{e,\tau ,\tilde {\rho }}}$ .

The first case is when $\omega =\tau $ . We let $\tilde {x} \in \mathrm {int}_{\mathrm {re}}(\tilde {\rho }) \times \{o\} \cap U_{x}$ for some $\rho \subset \tilde {\rho }$ . Then, after choosing suitable translations in $\pi _{v\tau }^{-1}(\Lambda _{\rho })^*_{\mathbb {R}}$ for the maps $\mathtt {x}$ and $\tilde {\mathtt {x}}$ , we have the following commutative diagram:

(3.11)

The second case is when $\omega \subsetneq \tau $ . Making use of the change of charts $\gimel $ in equation (3.6), and the description in the proof of Lemma 3.4, we write

$$ \begin{align*}\tilde{x} \in \mathrm{int}_{\mathrm{re}}(\tilde{\rho}) \times \{o\}\end{align*} $$

for some cone $\tilde {\rho } = \mathtt {i}_{\tau }^{-1}(\prod _{j=1}^p \tilde {\rho }_j) \in \mathtt {i}_{\tau }^{-1}\big (\prod _{j=1}^p \Lambda _{\Delta _{j}(\tau )}^* \big )$ of positive codimension. In $\gimel ^{-1}(W)$ , we may assume $\tilde {\rho }$ is the pullback of a cone $\breve {\rho }$ via $\Pi _{\omega \subset \tau } \circ \mathtt {p}_{\tau }$ as in equation (3.8). Since $\mathscr {S}_{e,\omega ,\rho } \subset \overline {\mathscr {S}_{e,\tau ,\tilde {\rho }}}$ , we have $\rho \subset \mathtt {p}_{\omega }^{-1}(\breve {\rho })$ and hence $\mathtt {p}_{\omega \subset \tau }^{-1}(\Lambda _{\rho }) \subset \Lambda _{\tilde {\rho }}$ . Therefore, from $\mathtt {p}_{\omega \subset \tau } \circ \pi _{v\tau } = \pi _{v\omega }$ , we obtain $\pi _{v\omega }^{-1}(\Lambda _{\rho }) \subset \pi _{v\tau }^{-1}(\Lambda _{\tilde {\rho }})$ , inducing the map $\mathtt {p}\colon \pi _{v\tau }^{-1}(\Lambda _{\tilde {\rho }})_{\mathbb {R}}^* \rightarrow \pi _{v\omega }^{-1}(\Lambda _{\rho })_{\mathbb {R}}^*$ . As a result, we still have the commutative diagram (3.11) for a point $\tilde {x}$ sufficiently close to x.

Definition 3.8. Given $x \in \mathscr {S}_e$ as above, the stalk of $\Omega ^*$ at x is defined as the stalk $\Omega ^*_{x}:= (\mathtt {x}^{-1}\Omega ^*)_{x}$ of the pullback of the sheaf of smooth de Rham forms on $\pi _{v\tau }^{-1}(\Lambda _{\rho })^*_{\mathbb {R}}$ , which is equipped with the de Rham differential d. This defines the complex $(\Omega ^*, d)$ of monodromy invariant smooth differential forms on B. A section $\alpha \in \Omega ^*(W)$ is a collection of elements $\alpha _{x} \in \Omega ^*_{x}$ , $x \in W$ such that each $\alpha _{x}$ can be represented by $\mathtt {x}^{-1}\beta _{x}$ in a small neighborhood $U_{x} \subset \mathtt {p}^{-1}(\mathtt {U}_{x})$ for some smooth form $\beta _{x}$ on $\mathtt {U}_{x}$ , and satisfies the relation $\alpha _{\tilde {x}} = \tilde {\mathtt {x}}^{-1}(\mathtt {p}^* \beta _{x})$ in $\Omega ^*_{\tilde {x}}$ for every $\tilde {x} \in U_{x}$ .

Example 3.9. In the two-dimensional case in Example 2.11, we consider a singular point

$$ \begin{align*}\{ x \} = \mathscr{S}_e \cap \mathrm{int}_{\mathrm{re}}(\tau)\end{align*} $$

for some $\tau \in \mathscr {P}^{[1]}$ . In this case, we can take $\rho $ to be the zero-dimenisonal stratum in $\mathscr {N}_{\tau }=\mathtt {i}_{\tau }^{-1}(\mathscr {N}_{\Delta _1(\tau )})$ and we have $\iota _*(\Lambda _{B_0}^*)_{x} = \mathscr {Q}_{\tau }^*$ . Taking a generator of $\mathscr {Q}_{\tau }^*$ , we get an invariant affine coordinate $\mathtt {x}\colon U_{x} \rightarrow \mathbb {R}$ which is the normal affine coordinate of $\tau $ . The stalk $\Omega ^*_{x}$ is then identified with the pullback of the space of germs of smooth differential forms from $(\mathbb {R},0)$ via $\mathtt {x}$ . In particular, $\Omega ^2_{x} =0$ .

For the Y-vertex of type II in Example 2.12, the situation is similar to the $2$ -dimensional case. For $\{ x \} = \mathscr {S}_e \cap \mathrm {int}_{\mathrm {re}}(\tau )$ , we still have $\iota _*(\Lambda _{B_0}^*)_{x} = \mathscr {Q}_{\tau }^*$ , and in this case, $\mathtt {x} \colon U_{x} \rightarrow \mathbb {R}^2$ are the two invariant affine coordinates. We can identify $\Omega ^*_{x}$ as the pullback of the space of germs of smooth differential forms from $(\mathbb {R}^2,0)$ via $\mathtt {x}$ .

For the Y-vertex of type I in Example 2.12, we use the identification $ \Lambda _{\tau ,\mathbb {R}}^* \cong \mathrm {int}_{\mathrm {re}}(\tau )$ via $\Upsilon _{\tau }$ for the $2$ -dimensional cell $\tau $ separating two maximal cells $\sigma _+$ and $\sigma _-$ . In this case, $\mathscr {S}_e$ is as shown (in blue color) in Figure 6 and $\mathscr {N}=\mathtt {i}_{\tau }^{-1}(\mathscr {N}_{\Delta _1(\tau )})$ is the fan of $\mathbb {P}^2$ . If x is the zero-dimensional stratum of $\mathscr {S}_e \cap \mathrm {int}_{\mathrm {re}}(\tau )$ , we have $\iota _*(\Lambda _{B_0}^*)_{x} = \mathscr {Q}_{\tau }^*$ and $\mathtt {x}\colon U_{x} \rightarrow \mathbb {R}$ as an invariant affine coordinate. If x is a point on a leg of the Y-vertex, we have $\mathtt {x} =(\mathtt {x}_1,\mathtt {x}_2)\colon U_{x} \rightarrow \mathbb {R}^2$ with $\mathtt {x}_1$ coming from a generator of $\Lambda _{\rho }$ and $\mathtt {x}_2$ coming from a generator of $\mathscr {Q}_{\tau }^*$ .

It follows from the definition that $\underline {\mathbb {R}} \rightarrow \Omega ^*$ is a resolution. We shall also prove the existence of a partition of unity.

Lemma 3.10. Given any $x \in B$ and a sufficiently small neighborhood U, there exists $\varrho \in \Omega ^{0}(U)$ with compact support in U such that $0 \leq \varrho \leq 1$ and $\varrho \equiv 1$ near x. (Since $\Omega ^0$ is a subsheaf of the sheaf $\mathcal {C}^{0}$ of continuous functions on B, we can talk about the value $f(x)$ for $f \in \Omega ^{0}(W)$ and $x \in W$ .)

Proof. If $x \notin \mathscr {S}_e$ , the statement is a standard fact. So we assume that $x \in \mathrm {int}_{\mathrm {re}}(\tau ) \cap \mathscr {S}_e$ for some $\tau \in \mathscr {P}$ . As above, we can write $x \in \mathrm {int}_{\mathrm {re}}(\rho ) \times \{o\}$ . Since $\rho $ is a cone in the fan $\mathtt {i}_{\tau }^{-1}(\mathscr {N}_{\Delta _1(\tau )} \times \cdots \times \mathscr {N}_{\Delta _p(\tau )})$ , $\Lambda _{\tau }^*$ has $\Lambda _{\Delta _1(\tau )}^* \oplus \cdots \oplus \Lambda _{\Delta _p(\tau )}^*$ as a direct summand, and the description of $\iota _*(\Lambda _{B_0}^*)_x$ is compatible with the direct sum decomposition of $\Lambda _{\tau }^*$ . We may further assume that $p=1$ and $\tau = \Delta _1(\tau )$ is a simplex.

If $\rho $ is not the smallest cone (i.e., the one consisting of just the origin in $\mathscr {N}_{\tau }$ ), we have a decomposition $\Lambda _{\tau }^* = \Lambda _{\rho } \oplus \mathscr {Q}_{\rho }$ and the natural projection $\mathtt {p}\colon \Lambda _{\tau }^* \rightarrow \mathscr {Q}_{\rho }$ . Then, locally near $x_0$ , we can write the normal fan $\mathscr {N}_{\tau }$ as $\mathtt {p}^{-1}(\Sigma _{\rho })$ for some normal fan $\Sigma _{\rho } \subset \mathscr {Q}_{\rho }$ of a lower-dimensional simplex. For any vector v tangent to $\rho $ at $x_0$ and the corresponding affine function $l_{v}$ locally near $x_0$ , we always have $\frac {\partial l_{v}}{\partial {v}}>0$ . This allows us to construct a bump function $\varrho = \sum _{v_i} (l_{v_i}(x) - l_{v_i}(x_0))^2$ along the $\Lambda _{\rho ,\mathbb {R}}$ -direction. So we are reduced to the case when $\rho = \{o\}$ is the smallest cone in the fan $\mathscr {N}_{\tau }$ .

Now, we construct the function $\varrho $ near the origin $o \in \mathscr {N}_{\tau }$ by induction on the dimension of the fan $\mathscr {N}_{\tau }$ . When $\dim _{\mathbb {R}}(\mathscr {N}_{\tau }) = 1$ , it is the fan of $\mathbb {P}^1$ consisting of three cones $\mathbb {R}_-$ , $\{o\}$ and $\mathbb {R}_+$ . One can construct the bump function which is equal to $1$ near o and supported in a sufficiently small neighborhood of o. For the induction step, we consider an n-dimensional fan $\mathscr {N}_\tau $ . For any point x near but not equal to o, we have $x \in \mathrm {int}_{\mathrm {re}}(\rho )$ for some $\rho \neq \{o\}$ . Then we can decompose $\mathscr {N}_\tau $ locally as $\Lambda _{\rho } \oplus \mathscr {Q}_{\rho }$ . Applying the induction hypothesis to $\mathscr {Q}_{\rho }$ gives a bump function $\varrho _x$ compactly supported in any sufficiently small neighborhood of x (for the $\Lambda _{\rho }$ directions, we do not need the induction hypothesis to get the bump function). This produces a partition of unity $\{\varrho _i\}$ outside o. Finally, letting $\varrho := 1 - \sum _i \varrho _i$ and extending it continuously to the origin o gives the desired function.

Lemma 3.10 produces a partition of unity for the complex $(\Omega ^*, d)$ of monodromy invariant differential forms on B, which satisfies the requirement in Condition 4.7 below. In particular, the cohomology of $(\Omega ^*(B),d)$ computes $R\Gamma (B,\underline {\mathbb {R}})$ . Given a point $x \in B \setminus \mathscr {S}_e$ , we can take an element $\varrho _x \in \Omega ^n(B)$ , compactly supported in an arbitrarily small neighborhood $U_x \subset B \setminus \mathscr {S}_e$ , to represent a nonzero element in the cohomology $H^n(\Omega ^*,d) = H^n(B,\mathbb {C}) \cong \mathbb {C}$ .