Proof. We construct a cell complex \(B\) as follows. For each triple intersection point \(X_{ijk}\) we introduce a \(0\)-cell \(B_{ijk}\). We fix an order on the \(0\)-cells, and for any double intersection component \(X_{ij}\) we attach the interval \(B_{ij}=[0,\omega(X_{ij})]\) to the two triple intersection points \(X_{ijk},X_{ijk'}\) in an order preserving manner. Finally, for the 2-cells we take the base diagrams \(B_i\) and attach \(\mu_i(X_{ij})\) to \(B_{ij}\) by the unique affine map to the interval taking \(\mu_i(X_{ijk})\) and \(\mu_i(X_{ijk'})\) to \(B_{ijk}\) and \(B_{ijk'}\) respectively. It is clear that the construction is independent of the choice of order on the \(0\)-cells. Namely, if we change the order between a pair of successive vertices \(X_{ijk}\) and \(X_{ijk'}\) we get a map on the corresponding \(1\)-skeleta by mapping the interval \(B_{ij}\) to itself via \(t\mapsto \omega(X_{ij})-t\). This map extends to an isomorphism of the corresponding cell complexes.

We obtain a homeomorphism between \(B\) and the intersection complex \(\Delta_{\mathcal{X}}\) by mapping the 1-skeleton of \(B\) to the \(1\)-skeleton of \(\Delta_{\mathcal{X}}\) in the obvious way. For each \(i\) we then pick any homeomorphism from the \(2\)-cell associated with \(X_i\) to \(B_i\) in a way that commutes with the attaching maps. While this identification involves arbitrary choices, the piecewise nodal induced integral affine structures are all isomorphic. ◻

Proof. By assumption, there is an open neighborhood \(V\) of the origin in \(\mathbb{C}^2\) and a stratified equivariant symplectomorphism \(\psi:V\to U\) where the action on \(V\) is \(\rho_{st}\). Without loss of generality, we may take \(V\) to be the unit ball. We first observe that there exists a family \(\psi_t:V=B_1\to \mathbb{C}^2\) of symplectic embeddings, such that \(\psi_0\) is the inclusion, \(\psi_1=\psi\), and such that \(\psi_t\) preserves the stratification by the coordinate axes. Explicitly, we use Alexander’s trick and consider the path \[\psi_t(v):=\begin{cases} \frac{1}{t}\psi(tv),&\quad t\neq 0\\ d\psi(0,0),&\quad t=0. \end{cases}\] The path can be shown to be smooth using Hadamard’s Lemma. Note that as \(\psi\) preserves the stratification, \(d\psi(0,0)\) is a split symplectomorphism i.e a symplectomorphism for which the axes \(\mathbb{C}\times \{0\}\) and \(\{0\} \times \mathbb{C}\) are eigen directions for \(d\psi(0,0)\). We may further connect the linear split symplectomorphism \(d\psi(0,0)\) to the identity via a path of linear split symplectomorphisms. Let us still denote the concatenation by \(\psi_t\). We thus get a time dependent symplectic vector field \(X_t\) with domain \(\psi_s(V)\), by \(p\mapsto \frac{d}{dt}\psi_{s+t}(p)|_{t=0}\). By simply connectedness we can find a time dependent Hamiltonian \(H_t:\psi_t(V)\to \mathbb{R}\) so that \(X_t\) is its Hamiltonian vector field. Let \(\epsilon>0\) be such that \(B_{\epsilon}\subset \psi_t(V)\) for all time \(t\in[0,1]\).

Let \(f:\mathbb{R}^2\to[0,1]\) be a function which is identically \(0\) for \(x+y\geq\epsilon\), identically \(1\) close enough to the origin, and such that \(\frac{\partial f}{\partial x}\) is identically \(0\) in a sufficiently small neighborhood of the \(y\) axis, and \(\frac{\partial f}{\partial y}\) is identically \(0\) in a sufficiently small neighborhood of the \(x\) axis. Let \(g:\mathbb{C}^2\to\mathbb{R}\) be defined by \(g(z_1,z_2):=f(|z_1|^2,|z_2|^2)\). Then the Hamiltonian flow of \(gH_t\) is globally defined and is readily checked to preserve the axes. Let \(\rho\) be the action defined by pulling back the standard action via the inverse of the time \(1\) flow \(\phi:\mathbb{C}^2\to\mathbb{C}^2\) of \(gH_t\). Then \(\rho\) satisfies all the requirements. Indeed, since \(\psi\) fixes the origin, we verify that a small enough neighborhood of the origin is mapped for all \(t\in[0,1]\) under \(\psi_t\) into the region where \(f\equiv 1\). In that neighborhood we have \(\phi=\psi.\) Admissibility of \(\rho\) follows since \(\phi\) preserves the stratification. ◻

Proof of Proposition 16. For any collection of ATF’s \(\mu_i:X_i\to B_i\) on the components of the central fiber \(\mathcal{X}_0\), the construction in Proposition 14 gives a cell complex \(B\) endowed with a piecewise nodal integral affine structure, and which is homeomorphic to the intersection complex \(\Delta_{\mathcal{X}}\). The maps \(\mu_i\) do not a priori agree on the intersections \(X_{ij},X_{jk},X_{ik}\). However, we have for each triple intersections an equality of points \(\mu_i(X_{ijk})=\mu_j(X_{ijk})=\mu_k(X_{ijk})\) and for each double intersection an equality of sets \(\mu_i(X_{ij})=\mu_j(X_{ij})\). To obtain a continuous map on the central fiber we first modify the fibrations \(\mu_i\) to make them agree as maps to the resulting \(B\) on a neighborhood of each triple intersection \(X_{ijk}\). To do so we first note that by symplectic orthogonality the configuration of divisors \(X_{ij},X_{jk},X_{ik}\) is symplectomorphic near \(X_{ijk}\) to the configuration of coordinate hyperplanes near the origin by a symplectomorphism \(\psi_{ijk}\). We now apply Lemma 2 to modify \(\mu_i\) on a neighborhood of \(X_{ijk}\) in \(X_i\) as follows. On the one hand, the almost toric fibration \(\mu_i\) restricts to a toric action in an open neighborhood \(U\) of \(X_{ijk}\) inside \(X_i\). On the other hand, restricting to a smaller open neighborhood \(V\) of \(X_{ijk}\) inside \(X_i\), the symplectomorphism \(\psi_{ijk}\) gives rise to an admissible toric action \(\rho_1\) on \(V\) by considering the moment map \(\mu_0\circ\psi_{ijk}\). Applying Lemma 2 to this action \(\rho_1\), we obtain a new admissible toric action \(\rho\) on \(U\) which agrees with \(\rho_1\) on a smaller neighborhood \(V_1\) of the origin and agrees with the original action outside a larger neighborhood \(V_2\subset U\). After performing this construction for each of the three coordinate planes (corresponding to the three divisors meeting at \(X_{ijk}\)) and choosing appropriate affine coordinates on \(B_i\), the modified \(\mu_i\) satisfies \(\mu_i=\mu_0\circ\psi_{ijk}\) on a neighborhood \(V\) of \(X_{ijk}\) in \(X_i\). After this modification, the \(\mu_i\) glue to the continuous function \(\mu_0\circ\psi_{ijk}\) on \(V\cap\mathcal{X}_0\).

We now proceed to dealing with the edges. Note that \(\mu_i|_{X_i \cap X_j}\) and \(\mu_j|_{X_i \cap X_j}\) both induce circle actions on the sphere \(S^2\) with the same two fixed points each of weight \(1\). There is a Hamiltonian \(H_{ij}\) on \(X_{ij}\) whose time 1-flow \(\psi=\psi^{ij}\) satisfies \(\mu_j\circ\psi|_{X_i \cap X_j} = \mu_i|_{X_i \cap X_j}\). This symplectomorphism can be constructed using the Arnold-Liouville normal form. Let \(\Pi_i:N^{{\omega_i}}_{X_i}(X_i\cap X_j) \rightarrow X_i\cap X_j\) denote the symplectic normal bundle of \(X_i\cap X_j\) inside \(X_i\). Consider \(H_{ij}\circ\Pi\) on the normal bundle. By the symplectic neighbourhood theorem, this gives us an extension of \(H_{ij}\) to a symplectic normal neighbourhood \(V_i\) of \(X_i \cap X_j\) in \((X_i,\omega_i)\). Choose a bump function \(\rho_i\) supported on \(V_i\). Let \(\psi^{ij}_i\) denote the time 1 flow of this new Hamiltonian \(\tilde{H}^{ij}_{i}:=\rho_iH_{ij}\circ\Pi\) on \(X_i\). Then \(({\psi^{ij}_i})^*\mu_i\) is a new almost toric fibration on \(X_i\) which equals the original almost toric fibration \(\mu\) outside of some open set. Furthermore, \(({\psi^{ij}_i})^*\mu_i\) and \(({\psi^{ij}_j})^*\mu_j\) both agree on \(X_i \cap X_j\). Moreover, the new almost toric fibration agrees with the old one sufficiently close to the triple intersection points.

 ◻

Proof of Theorem 20. We construct a chart in a neighborhood of each edge and show it agrees with the canonical structure near the vertices. Fix \(e_{ij}:=\mu(X_{ij})\) an edge of the \(1\) skeleton. Let \(U\subset B_i\cup B_j\subset B\) be an open neighborhood of the interior of \(e_{ij}\) containing no nodes. We consider continuous functions on \(U\) whose restriction to each of \(U\cap B_i\), \(U\cap B_j\) is affine. To reduce the amount of freedom, we fix a basepoint in the interior of \(e_{ij}\) and we consider only the piecewise integral linear functions. Namely, piecewise integral affine functions which vanish at the basepoint and whose restriction to each side of the edge is affine. The space \(\mathcal{A}\) of all these is a free Abelian lattice of rank \(3\). Indeed, a linear function is determined by two coefficients. \(\mathcal{A}\) consists of pairs of linear functions that agree and the edge \(e_{ij}\), and so there are \(3\) degrees of freedom. Singling out an integral affine structure on \(U\) which agrees with the piecewise nodal integral affine structure already given means declaring which of these piecewise linear functions are considered to be linear functions. This amounts to specifying a rank 2 sublattice of \(\mathcal{A}\) whose restriction to each side is the lattice of linear functions on that side near the edge. We now define a homomorphism \(wt:\mathcal{A}\to \mathbb{Z}\), the weight, which will single out the desired sublattice.

To define the weight homomorphism, we observe that each element \(f\) of \(\mathcal{A}\) gives rise to a circle action \(\rho_f\) on the restriction of the tangent bundle of the total space \(T\mathcal{X}|_{X_{ij}}\). That is, an action on \(X_{ij}\) which lifts to a linear action on the fibers. To see this, let \(f_k=f_k|_{B_k}\) for \(k=i,j\). Then \(f_k\circ\mu\) is the Hamiltonian of a circle action on \(X_k\) near the \(1\)-dimensional stratum. The linearization of this action gives circle actions on the vector bundles \(TX_i|_{X_{ij}}\) and \(TX_j|_{X_{ij}}\). The continuity equation \(f_i|_{X_{ij}}=f_j|_{X_{ij}}\) implies that these two linearized actions agree on the common subbundle \(TX_{ij}\), because on \(TX_{ij}\) both are the differential of the same circle action on \(X_{ij}\). Since \(X_i\) and \(X_j\) meet transversely along \(X_{ij}\), we have a canonical identification \[T\mathcal{X}|_{X_{ij}}\cong \left(TX_i|_{X_{ij}}\oplus TX_j|_{X_{ij}}\right)\Big/\{(v,-v):v\in TX_{ij}\}.\] The map \([u_i,u_j]\mapsto [\rho_i(\theta)\,u_i,\rho_j(\theta)\,u_j]\) for each \(\theta\in S^1\) is well defined on the quotient and defines the required circle action \(\rho_f\) on \(T\mathcal{X}|_{X_{ij}}\).

Pick any point \(p\in X_{ij}\) and any basis \(v_1,v_2,v_3\) to \(T_p\mathcal{X}\) over the complex numbers. Let \(\Omega\) be the Calabi-Yau form on \(\mathcal{X}\). Then we get map \(\lambda_f:S^1\to S^1\) by \(\theta\mapsto\arg\left(\Omega(\rho_f(\theta)\cdot (v_1\wedge v_2\wedge v_3))\right)\). Define \(wt(f):= deg(\lambda_f)\). The following are easy to verify:

• The definition of \(wt\) is independent of the choice of \(p\) or the basis \(\{v_1,v_2,v_3\}\) since any for any two choices the corresponding maps \(\lambda_f\) are connected by a homotopy. It is also independent of the choice of \(U\)

• \(wt:\mathcal{A}\to\mathbb{Z}\) is a homomorphism.

• \(wt\) is surjective.

We now show that \(\ker(wt)\), when restricted to each side, gives exactly the lattice of linear functions on that side. Fix the side \(U\cap B_i\) (the argument for \(U\cap B_j\) is identical). Let \(\mathcal{A}_i:=\{g\in\mathcal{A}: g|_{U\cap B_i}=0\}\); this is rank \(1\). For nonzero \(g\in\mathcal{A}_i\), the induced action is trivial on \(TX_i|_{X_{ij}}\) but nontrivial on the normal line of \(X_{ij}\) inside \(X_j\), so \(wt(g)\neq 0\) (for a primitive generator, \(wt(g)=\pm 1\)). Now let \(\ell\) be any linear function on \(U\cap B_i\), and choose \(f\in\mathcal{A}\) with \(f|_{U\cap B_i}=\ell\). Pick a primitive \(g\in\mathcal{A}_i\). Then \[f':=f-wt(f)\,g\] still satisfies \(f'|_{U\cap B_i}=\ell\), and \(wt(f')=0\), so \(f'\in\ker(wt)\). Hence every linear function on \(U\cap B_i\) is the restriction of an element of \(\ker(wt)\). Since \(wt\) is surjective, \(\ker(wt)\) is rank \(2\). We therefore declare \(\ker(wt)\) to be the lattice of linear functions, and this defines the integral affine structure on \(U\).

It remains to verify compatibility of this chart with the canonical integral affine structure near each vertex. This follows from the following observations:

• The integral affine functions for the canonical structure near a vertex can equivalently be characterized as being those whose induced action preserves the standard volume form on \(\mathbb{C}^3\).

• For \(p=X_{ijk}\), the volume form \(\Omega_p\) equals the one induced from \(\mathbb{C}^3\) by any local chart up to multiplication by a scalar, which does not affect the computation of the weight.

 ◻

Proof. We establish well-definedness and continuity in the following three steps.

Step 1: Trapping neighborhoods. We show that for each \(p\in\mathcal{X}_0\setminus\bigcup_{i,j}C_{ij}\) and any open neighborhood \(V\subset \mathcal{X}\) of \(p\) there exist an open neighborhood \(U\ni p\) in \(\mathcal{X}\) with the following trapping property: any parallel transport path \(y(t)=P_{\pi,\gamma,t}(z)\) satisfying \(y(t_0)\in U\) for some \(t_0\) sufficiently close to \(0\) remains in \(V\) for all \(t\in[t_0,0)\).

Case (a): \(p\) is a smooth point of \(\mathcal{X}_0\). By shrinking \(V\) we may assume that \(\pi|_V\) is a submersion. The symplectic parallel transport along the reversed path \(\bar\gamma(s)=\gamma(-s)\) is well defined on \(\mathcal{X}_0\cap V\). By smooth ODE theory (continuous dependence on initial conditions), paths starting in a sufficiently small neighborhood \(W\subset \mathcal{X}_0\cap V\) of \(p\) remain in \(V\) for sufficiently small time \(t_0\). We may take \(U\) to be the sweepout of \(W\) by the flow parallel transport along \(\bar\gamma|_{[t_0,0)}\).

Case (b): \(p\in X_{ij}\setminus C_{ij}\) for some \(i,j\). By definition of \(C_{ij}\), there exists a neighborhood \(V\ni p\) in \(\mathcal{X}\) on which \(\omega\) is compatible with \(J\). Write \(\pi=\pi_1+i\pi_2\) and let \(\nabla\pi_1,\nabla\pi_2\) be the gradients with respect to the Kähler metric determined by \(J\) and \(\omega|_V\). Since \(\pi\) is holomorphic and \(\omega|_V\) is compatible, these gradients span the symplectic complement of the fiber, giving the horizontal lift formula \[\label{eqHolHorLift} y'(t)=\left(\gamma'_1(t)\nabla \pi_1(y(t))+\gamma'_2(t)\nabla \pi_2(y(t))\right)/|\nabla \pi|^2.\tag{1}\] By a direct calculation of the gradient of \(z_1\dots z_n\), for a fibration \(\pi\) with normal crossings singularities, \[\label{eqNormalCrossing} |\nabla \pi|>C|\pi|^{1-1/n}\tag{2}\] for some constant \(C=C(\pi)\) and \(n\) the complex dimension, yielding \[\label{eqParrallelTransportBound} |y'|\leq|\nabla\pi||\gamma'(t)|/|\nabla\pi|^2<\frac{1}{C}|\gamma(t)|^{\left(\frac{1}{n}-1\right)}|\gamma'(t)|.\tag{3}\] This bound holds as long as \(y(t)\in V\). Reparametrizing \(\gamma\) as \(t\mapsto t^ne^{i\theta(t)}\) makes the right-hand side bounded by a constant \(C'\), so the total arc length of \(y\) on \([t_0,0)\) is at most \(C'|t_0|\) while the path stays in \(V\). Take \(V\) to contain the open ball of radius \(2\epsilon\) around \(p\) and \(U\) the concentric ball of radius \(\epsilon\). If \(y(t_0)\in U\) and \(t_0\) is close enough to \(0\) that \(C'|t_0|<\epsilon\), then \(y\) cannot exit \(V\) during \([t_0,0)\).

Step 2: Well-definedness. Fix \(z\in\mathcal{X}_x\) and let \(y(t)=P_{\pi,\gamma,t}(z)\). Since \(\pi^{-1}(\gamma)\) is compact, the \(\omega\)-limit set \[\Lambda:=\bigcap_{s<0}\,\overline{\{y(t):t\in[s,0)\}}\] is non-empty and connected. We claim \(\Lambda\) maps under the quotient map to a single point of \(\hat{\mathcal{X}}_0\).

Suppose \(p\in\Lambda\setminus\bigcup_{i,j}C_{ij}\), so \(y(t_k)\to p\) for some \(t_k\to 0^-\). By the trapping property, for \(k\) large enough \(y\) remains in \(V\) for all \(t\in[t_k,0)\) where \(V\) is any open neighborhood of \(p\). So, \(\lim_{t\to 0} y(t)=p\).

If instead \(\Lambda\cap(\mathcal{X}_0\setminus\bigcup_{i,j}C_{ij})=\emptyset\), then \(\Lambda\subset\bigcup_{i,j}C_{ij}\). Since the \(C_{ij}\) are pairwise disjoint and \(\Lambda\) is connected, \(\Lambda\) lies in a single component \(C_{ij}\). In either case, \(\Lambda\) maps to a single point in \(\hat{\mathcal{X}}_0\). Hence \(P_{\pi,\gamma}(z):=[\Lambda]\in\hat{\mathcal{X}}_0\) is well-defined.

Step 3: Continuity. Fix \(z\in\mathcal{X}_x\) and let \(p=P_{\pi,\gamma}(z)\in\hat{\mathcal{X}}_0\). If \(p\) lies in the top stratum of \(\mathcal{X}_0\), parallel transport is smooth near \(p\) and continuity follows from standard ODE theory.

For the lower strata, we distinguish two cases. If \(p\in X_{ij}\setminus C_{ij}\) for some \(i,j\), then the limit \(\lim_{t\to 0}y(t)=p\) exists in \(\mathcal{X}_0\). Let \(V\) be a neighborhood of \(p\) in \(\mathcal{X}_0\). Let \(V'\) be a neighborhood of \(p\) in \(\mathcal{X}\) such that \(V=V'\cap \mathcal{X}_0\). Let \(U\) and \(t_0\) be as in the trapping property for \(V'\). By continuous dependence on initial conditions on \([x,t_0]\), for \(z'\) sufficiently close to \(z\) we have \(P_{\pi,\gamma,t_0}(z')\) lands in \(U\). Hence by the trapping property \(\lim_{t\to 0} P_{\pi,\gamma,t}(z')\in V'\cap\mathcal{X}_0=V\).

The remaining case is when \(p=\hat{p}_{ij}\), meaning the \(\omega\)-limit set \(\Lambda\) of \(y(t)=P_{\pi,\gamma,t}(z)\) lies entirely in the compact cylinder \(C_{ij}\). We argue by contradiction. Suppose \(P_{\pi,\gamma}\) is not continuous at \(z\). Then there exists an open neighborhood \(\hat{A}\) of \(\hat{p}_{ij}\) in \(\hat{\mathcal{X}}\) and a sequence \(z_k \to z\) in \(\mathcal{X}_x\) such that for all \(k\) we have \(P_{\pi,\gamma}(z_k) \notin \hat{A}\). Let \(A \subset \mathcal{X}\) be the preimage of \(\hat{A}\) under the quotient map. Since \(C_{ij}\) is compact and \(C_{ij} \subset A\), we can choose an open neighborhood \(W\) of \(C_{ij}\) such that \(C_{ij} \subset W \subset \overline{W} \subset A\). Moreover, we can assume that \(\partial W\) is disjoint from each \(C_{kl}\).

Since the \(\omega\)-limit set \(\Lambda\) of \(y(t)\) is contained in \(C_{ij}\), there exists a time \(t_1 < 0\) such that \(y(t) \in W\) for all \(t \in [t_1, 0)\). Let \(y_k(t) = P_{\pi,\gamma,t}(z_k)\). By continuous dependence on initial conditions of the parallel transport equation on the compact interval \([x, t_1]\), we have \(y_k(t_1) \to y(t_1) \in W\). Thus, for all sufficiently large \(k\), \(y_k(t_1) \in W\).

However, since \(P_{\pi,\gamma}(z_k) \notin \hat{A}\), the \(\omega\)-limit set of \(y_k\) is not contained in \(A\), and consequently not contained in \(W\). Therefore, each path \(y_k\) must eventually exit \(W\). Let \(s_k \in (t_1, 0)\) be the first time \(y_k(t)\) exits \(W\), meaning \(y_k(s_k) \in \partial W\). Because the sequence \(s_k\) is bounded, we can pass to a subsequence such that \(s_k \to s^* \in [t_1, 0]\). If \(s^* < 0\), then \(y_k(s_k) \to y(s^*)\) by continuous dependence on initial conditions for closed intervals in \([-1,0)\). Since \(\partial W\) is closed, this would imply \(y(s^*) \in \partial W\), which directly contradicts that \(y(t) \in W\) for all \(t \in [t_1, 0)\). Therefore, we must have \(s^* = 0\).

Now consider the sequence of boundary points \(y_k(s_k) \in \partial W\). Since \(\pi(y_k(s_k)) = \gamma(s_k) \to \gamma(0) = 0\), properness of \(\pi\) near \(\mathcal{X}_0\) implies that, passing to a further subsequence, \(y_k(s_k)\) converges to some point \(q \in \partial W \cap \mathcal{X}_0\). Because \(W\) is a neighborhood of \(C_{ij}\), the point \(q\) is disjoint from \(C_{ij}\), so \(q \in \mathcal{X}_0 \setminus \bigcup_{k,l} C_{kl}\). Furthermore, since \(\overline{W} \subset A\), we have \(q \in A\).

Since \(A\) is an open neighborhood of \(q\), choose an open neighborhood \(V_q\subset A\) of \(q\) satisfying the trapping property from Step 1 for some time \(t_q < 0\). Since \(y_k(s_k) \to q\) and \(s_k \to 0\), for all sufficiently large \(k\) we have \(y_k(s_k) \in V_q\). The trapping property then guarantees that \(y_k(t) \in V_q\) for all \(t \in [s_k, 0)\), so the \(\omega\)-limit set of \(y_k\) lies in \(A\) and therefore \(P_{\pi,\gamma}(z_k)\in\hat{A}\), contradicting our initial assumption. This completes the proof of continuity of \(P_{\pi,\gamma}\).

Joint continuity. To prove continuity of \(\Phi:\mathcal{X}_{\gamma(-1)}\times[-1,0]\to\hat{\mathcal{X}}_0\), it suffices to verify continuity at points of the form \((w,0)\), since \(\Phi\) is continuous on \(\mathcal{X}_{\gamma(-1)}\times[-1,0)\) by smooth dependence of parallel transport on initial conditions. Fix \((w,0)\) and let \(\hat{A}\) be an open neighborhood of \(\Phi(w,0)=P_{\pi,\gamma}(w)\) in \(\hat{\mathcal{X}}_0\). The continuity of \(P_{\pi,\gamma}\) established above provides an open neighborhood \(U\subset\mathcal{X}_{\gamma(-1)}\) of \(w\) such that \(P_{\pi,\gamma}(U)\subset\hat{A}\). By the same trapping arguments used in Step 3, there exists \(\delta>0\) such that for all \(w'\in U\) and \(t\in[-\delta,0]\), we have \(\Phi(w',t)\in\hat{A}\). This establishes continuity of \(\Phi\) at \((w,0)\). ◻

Proof. Set \(W:=P_{\pi,\gamma}^{-1}(\overline{\hat{U}})\subset\mathcal{X}_{\gamma(-1)}\). By continuity of the map \(\Phi:\mathcal{X}_{\gamma(-1)}\times[-1,0]\to\mathcal{X}\) and compactness of \(W\), there is a \(\delta>0\) such that \(P_{\pi,\gamma,t}(w)\in N_{ij}\) for all \(w\in W\) and \(t\in[-\delta,0)\). The conclusion follows since \(z\in\mathcal{X}_{\gamma(t)}\) satisfies \(\lim_{s\to 0}P_{\pi,\gamma|_{[s,0)},s}(z)\in\hat{U}\) if and only if \(P_{\pi,\gamma,t}^{-1}(z)\in W\). ◻

Proof. We wish to obtain the estimate 3 with \(y(t)\) the horizontal lift with respect to \(\tilde{\pi}\). The first two estimates assumed in the statement of the present lemma allow to deduce the estimate 2 for \(\tilde{\pi}\) from the same estimate for \(\pi\). To obtain the middle estimate of 3 we first derive a formula for the horizontal lift for a non-holomorphic function. We claim that equation 1 for the horizontal lift generalizes to \[y'(t)=\frac{\gamma'_1(t)J\nabla \pi_2(y(t))+\gamma'_2(t)J\nabla \pi_1(y(t))}{\langle \nabla\tilde{\pi}_1, J\nabla\tilde{\pi}_2\rangle.}\] Indeed, one verifies this is symplectically orthogonal to \(\ker d\tilde{\pi}\). Moreover, using \[d\pi=\langle \nabla\tilde{\pi}_1, \cdot\rangle+i\langle \nabla\tilde{\pi}_2, \cdot\rangle\] one verifies it projects to \(\gamma'(t)\) under \(d\tilde{\pi}\). We now readily deduce the estimate 3 for \(\tilde{\pi}\). The rest of the proof is the same. ◻

Proof. For any triple intersection point we fix an admissible local symplectic chart \(\psi: U\subset\mathbb{C}^3\to \mathcal{X}\). Consider the function \(g=\pi\circ \psi-\pi_0.\) For each \(I\subset \{1,2,3\}\) introduce the notation \(\pi_I=\Pi_{i\in I}z_i\Pi_{i\not\in I}\overline{z}_i\). We claim \(g\) can be written WLOG in the form \[g=\sum_{I\subset\{1,2,3\}} \pi_I h_I\] where \(h_I:U\to\mathbb{C}\) are smooth function which vanish at the origin at least to order \(1\). To see this let \(\psi_{hol}:U\to \mathcal{X}\) be holomorphic coordinates pulling back \(\pi\) to the function \(\pi_0\). These exist by the normal crossing assumption. WLOG \(\pi\) is normalized so \(d\psi_{hol}=d\psi\). Let \(f=\pi\circ\psi\) then \(f=\pi_0\circ (\psi_{hol}^{-1}\circ \psi).\) Let \(\phi_i\) be the \(i\)th component of \(\psi_{hol}^{-1}\circ \psi\). Since \(\psi_{hol}^{-1}\circ \psi=Id+O(|z|^2)\), we have \(\phi_i(z_1,z_2,z_3)=z_i+O(|z|^2)\). Moreover, since \(\phi_i(z_1,z_2,z_3)-z_i\) vanishes identically on the plane \(z_i=0\) we actually have \(\phi_i=z_i+z_ia_i+\overline{z}_ib_i\) for \(a_i,b_i:U\to\mathbb{C}\) are smooth functions which are \(O(|z|)\). Rearranging \(f=\phi_1\phi_2\phi_3\), we get the desired form.

Let \(\rho:\mathbb{R}_+\to[0,1]\) be a bump function identically \(0\) near the origin and identically \(1\) near \(1\). For \(r\in(0,1)\) let \(\rho_r(z_1,z_2,z_3)=\rho(|z|/r).\) We claim that for \(r\) small enough, the function \(\pi_r=\pi_0+\rho_rg\) has symplectic fibres. For this recall that according to [20], the criterion for having symplectic fibres is \(|\overline{\partial}\pi_r|<|\partial \pi_r|\).

Let \(C\) be an estimate for \(\max_{I\subset \{1,2,3\}}\{\partial h_I,\overline{\partial}h_I\}\) on some precompact neighborhood of the origin in \(U\) and for \(\rho'\) on the unit interval. Then for sufficiently small \(r\) we get \[|\partial\pi_r|\geq |\partial \pi_0|-\sum_I\left(|\rho h_I\partial \pi_I|+C|\rho\pi_I|\right)- \left|\sum h_I\pi_I\right|C/r\] and, \[|\overline{\partial}\pi_r|\leq \sum_I\left(|\rho h_I\overline{\partial} \pi_I|+C|\rho\pi_I|\right)+\left|\sum h_I\pi_I\right|C/r .\] All the terms can be made arbitrarily small relative to \(|\partial \pi_0|\) by considering small enough neighborhood of the origin. Indeed, we have \(\frac{|\partial\pi_I|}{|\partial \pi_0|}\) \(\frac{|\overline{\partial}\pi_I|}{|\partial \pi_0|}\) are \(O(1)\), \(h_I\) and \(\frac{|\pi_I|}{|\partial \pi_0|}\) are \(O(|z|)\) and \(\frac{|h_I\pi_I|}{|\partial \pi_0|}\) are \(O(|z|^2)\). The estimates of Lemma 5 are also immediate. ◻

Proof. (1) is Proposition 16. We then modify \(\pi\) in accordance with Lemma 7. The modified map satisfies all the properties of Lemma 5 by construction; by the corollary to Lemma 5, \(\hat{\mu}_1 = \hat{\mu}_0\circ P_\gamma\) is therefore continuous, proving (2).

For (3), that \(\hat{\mu}_1\) is an almost toric fibration over the interior of the top stratum follows from Proposition 16. Near each vertex \(B_{ijk}\), Lemma 7 provides an equivariant symplectomorphism \(\psi_{ijk}\) of a neighborhood of \(X_{ijk}\) with a neighborhood of the origin in \(\mathbb{C}^3\) intertwining \(\tilde{\pi}\) with \(\pi_0(z_1,z_2,z_3)=z_1z_2z_3\). The moment map \(\mu_0\) of the standard torus action on \(\mathbb{C}^3\) preserving the holomorphic volume form is invariant under the flow of \(\pi_0\), hence by Lemma 6 is preserved by parallel transport. Thus \(\hat{\mu}_1=\mu_0\circ\psi_{ijk}^{-1}\) near the preimage of \(B_{ijk}\), giving an almost toric fibration over a neighborhood of \(B_{ijk}\) whose Arnold-Liouville coordinates coincide with the canonical affine structure \(\phi_{ijk}\) from Definition 15.

As the modifications in Proposition 16 preserve the integral affine structure near the vertices, the Arnold-Liouville structure on \(\Delta_{\mathcal{X}}\setminus A\) coincides with the restriction of the one from Theorem 20, proving (4). ◻

Proof. By Remark 21, the monodromy of the integral affine structure around the loop \(\gamma\) is the identity. ◻

Proof. Fix a tubular neighborhood \(N\) of the cylinder \(C=\hat{\mu}_0^{-1}(\hat{c})\). Let \(U\subset \overline{U}\subset N\cap\mathcal{X}_0\) and assume \(R\) is such that \(\pi^{-1}(R)\subset U\). Thicken to \(\hat{R}'\) so that \(R':=q^{-1}_B(\hat{R}')\) is convex with smooth boundary and so that \(\hat{\mu}_0^{-1}(\hat{R}')\subset U\). Then Corollary 24 immediately implies that \(\hat{\mu}_1^{-1}(\hat{R}')\) is strong symplectic filling of \((\mathbb{T}^3,\xi_{\mathop{\mathrm{std}}})\). For minimality it suffices to prove that \(\hat{\mu}_1^{-1}(\hat{R}')\) is topologically aspherical. For this note that by Lemma 4 the image of \(\hat{\mu}_1^{-1}(\hat{R})\) under parallel transport to \(\mathcal{X}_{\gamma(t)}\) for \(t\) close enough to \(0\) is contained in the normal neighborhood \(N\) of \(C\) which a cylinder (the sphere with two little discs removed) in the total space of the fibration. But this parallel transport is a diffeomorphism as long \(t\) does not reach \(0\). ◻

Proof of Lemma 8. Letting \(W=\hat{\mu}_1^{-1}(\hat{R}')\) as in Lemma 9 and attaching a cone to obtain \(W_{\infty}\), Corollary 26 gives a diffeomorphism \(T^*\mathbb{T}^2\to W_{\infty}\) which intertwines \(\hat{\mu}_1\) with the projection to the cotangent fiber outside of a compact set. The claim about \(\hat{\mu}_1^{-1}(\hat{R})\) follows by inflating a neighborhood of \(\hat{\mu}_1^{-1}(\partial \hat{R})=\mu_1^{-1}(\partial R)\) in a fiber preserving manner. ◻

Proof of Theorem 5 assuming Proposition 27. By the last item of Proposition 23 there is no monodromy around the components of \(A\). See Remark 21. Thus each point \(\hat{c}\) of \(\hat{A}\) admits a neighborhood \(\hat{R}_{\hat{c}}\) in \(\hat{B}\) whose preimage \(R_{\hat{c}}\subset B\) is rectangular in the above sense. By Lemma 8 there is a diffeomorphism \(\phi:\hat{\mu}_1^{-1}(\hat{R}_{\hat{c}})\to R_{\hat{c}}\times \mathbb{T}^2\). We can pick \(\phi\) which agrees near \(\partial \hat{\mu}_1^{-1}(\hat{R}_{\hat{c}})\) with Arnold-Liouville coordinates induced by \(\hat{\mu}_1\). In particular, \(\phi\) is a symplectomorphism near the boundary. Let \(\Psi:R_{\hat{c}}\times \mathbb{T}^2\to R_{\hat{c}}\times \mathbb{T}^2\) be the result of applying Proposition 27 to the symplectic form \((\phi^{-1})^*\omega\) (viewing \(R_{\hat{c}}\) as a rectangle in \(\mathbb{R}^2\) via an integral affine chart). Then \(\Psi\circ\phi:\hat{\mu}_1^{-1}(\hat{R}_{\hat{c}})\to R_{\hat{c}}\times\mathbb{T}^2\) is a symplectomorphism and the composition \(\mu':=\mu_{\mathop{\mathrm{std}}}\circ\Psi\circ\phi\) agrees with \(\hat{\mu}_1\) on a neighborhood of \(\partial \hat{\mu}_1^{-1}(\hat{R}_{\hat{c}})\) (where \(\mu_{\mathop{\mathrm{std}}}\) is defined to be the standard projection map onto \(R_{\hat{c}}\) in \(R_{\hat{c}} \times \mathbb{T}^2\)). We thus define \[\mu(b)=\begin{cases} \hat{\mu}_1(b),&\quad b\in \hat{B}\setminus\cup_{\hat{c}}\hat{R}_{\hat{c}}=B\setminus \cup_{\hat{c}}R_{\hat{c}}\\ \mu'(b),&\quad b\in \cup_{\hat{c}}R_{\hat{c}}. \end{cases}\] By construction in Proposition 23, the IAS induced by \(\hat{\mu}_1\) agrees with the one from Theorem 20. Similarly, by construction, the IAS induced by \(\mu'\) is the standard one on \(R_{\hat{c}}\times\mathbb{T}^2\) with \(R_{\hat{c}}\) an affine rectangle in \(\Delta_{\mathcal{X}}\) with the same IAS. Thus \(\mu\) is as required. ◻

Proof of Proposition 27 assuming Proposition 29. Compactify \(R \times \mathbb{T}^2\) to \(S^2 \times S^2\) by symplectic reduction on the boundary rectangle. Let \(D:= B_1 \cup F_1 \cup B_2 \cup F_2\) denote the configuration of boundary divisors. Let \(\omega,\omega_0\) still denote the induced symplectic forms on the compactification. Here \(B_1\) and \(B_2\) are in the class of the base \(B= [S^2 \times \{*\}]\) and \(F_1\) and \(F_2\) are in the class of the fibre \(F:= [\{*\} \times S^2]\).

Since \(\omega\) coincides with \(\omega_0\) on \(U\) we have that the induced form on \(S^2\times S^2\) still coincides with a standard product form in a neighborhood of the boundary divisors. In particular, these are all symplectically orthogonal. Furthermore, as the Chern class of the tangent bundle is determined by its restriction to a neighbourhood of the boundary divisors of \(S^2 \times S^2\), we also have that the first Chern classes are equal for both \(\omega\) and \(\omega_0\), and hence the boundary divisor \(D\) is log Calabi-Yau for both the forms. By the same reasoning, we have \([\omega]=[\omega_0]\). By Theorem 28, there exists a diffeomorphism \(\psi:S^2\times S^2\to S^2\times S^2\) and so that \(\psi^*\omega_0=\omega\) and \(\psi(D)=D\). Restricting the diffeomorphism to the complement of \(D\) gives us a diffeomorphism, still denoted by \(\psi\) and mapping \(R \times \mathbb{T}^2\to R\times \mathbb{T}^2,\) such that \(\Phi^*\omega_0=\omega\).

It remains to modify \(\Phi\) further so it preserves \(\mu_{\mathop{\mathrm{std}}}\) near the boundary. This is achieved by postcomposing \(\Phi\) with an appropriate \(\Psi\) as in Proposition 29. ◻

Proof. Consider the graph \(\Gamma(\Phi)\) of \(\Phi\) in \((U \times M, \omega\oplus (-\omega))\). \(\Gamma(\Phi)\) is a properly embedded Lagrangian submanifold of \(U\times M\). By the assumption regarding \(\Phi|_S\) and \(d\Phi\) restricted to \(TM|_S\) is the identity, by taking arbitrarily small neighborhoods \(U'\) of \(S\), the intersection of \(\Gamma(\Phi)\) with neighborhoods \(U'\times M\) of \(S\times S\) is arbitrarily \(C^1\) close to the diagonal Lagrangian \(\Delta\simeq U'\). Identifying a Weinstein neighborhood of \(\Delta\) with an open neighborhood of the \(0\) section in \(T^*U'\), the Lagrangian \(\Gamma(\Phi)\) becomes the graph of a closed \(1\)-form \(\alpha\) over \(U'\). The latter vanishes on \(S\) by assumption. We may assume \(U'\) deformation retracts to \(S\). Therefore, the class of \(\alpha\) vanishes, and in particular, \(\Phi\) is Hamiltonian isotopic to the identity. Moreover, the primitive \(f\) is constant along \(S\) and may be taken to be \(0\) there. Suppose \(\Phi\) is identity along an open neighborhood of \(A\). We may pass to a smaller neighborhood which deformation retracts to a subset of \(S\). Since the Hamiltonian vanishes on \(S\), it vanishes on that neighborhood as well. ◻

Proof. The proof is exactly as in Lemma 31.16 in [22] rephrased for 1-parameter families. ◻

Proof of Lemma 10. Since \(d\phi\) preserves the tangent space to the symplectic submanifold \(S\), it also preserves the normal space, and so defines a symplectic automorphism of \(N(S)\). The structure group \(Sp(2, \mathbb{R})\) of \(N(S)\) is homotopy equivalent to \(U(1)\). Since \(U(1)\) is Abelian, one deduces from this that the space of symplectic automorphisms of \(N(S)\) is homotopy equivalent to the space of maps \(S^2\to S^1\) which connected, since \(\pi_2(S^1) = 0\). Therefore, we can isotope the automorphism \(A:= d\phi|_{N(S)}\) to the identity via a path \(A_t\) of fiberwise symplectic automorphisms. Consider the 1-parameter family of diffeomorphisms \(\phi_t:= \exp \circ A_t \circ \exp^{-1}\) where \(\exp\) is the exponential map from the normal bundle to an open neighbourhood of \(S\). Note that \(d\phi_t|_{N(S)} = A_t\). Consider the 1-parameter family of symplectic forms \(\omega_t:= \phi_t^*\omega\). As \(A_t\) are symplectic automorphisms, we see that \(\omega_t|_{T_S M} = \omega\) for all \(t\) (where \(T_S M\) denotes the restriction of \(TM\) to \(S\)). By Lemma 11, we can find a 1-parameter family of primitives \(\sigma_t\) such that \(d\sigma_t = \frac{d}{dt}\omega_t\) and \(\sigma_t\) vanishes along \(S\). Mimicking the proof of Moser’s theorem we then obtain a time dependent vector fields \(X_t\) whose flow gives us a path of diffeomorphisms \(\psi_t\) such that \(\psi_t^*\omega_t =\omega\) and moreover \(d\psi_t|_{T_S M} = \mathop{\mathrm{id}}\). Then \(d(\phi_t \circ \psi_t)|_{T_S M} = A_t\) and \(\phi_t\circ\psi_t\) is a path of symplectomorphisms connected to the identity. Let \(Y_t\) be the symplectic vector field which is the time derivative of \(\phi_t\circ\psi_t\). Then in a small simply connected neighbourhood \(U\) of the sphere \(S\), \(Y_t\) is Hamiltonian and is generated by a function \(H_t\). Considering the time 1-flow of the bumped-off Hamiltonian \(\rho H_t\) gives us the required global symplectomorphism \(\beta: M \rightarrow M\) whose derivative by construction on the normal of \(S\) is \(A_1 = d\phi|_{N(S)}\). For the last sentence in the statement, note that the path \(\phi_t\) is identity on a neighborhood of \(B\) by construction. Therefore, \(\sigma_t\) can be chosen to vanish on a neighborhood of \(B\). ◻

Proof. Let \(\omega_0\) denote the standard symplectic form on the sphere \(S^2\) and let \(\Omega_{x,y}\) denote the space of symplectic forms on \(S^2\) that are in the same cohomology class as \(\omega_0\) and are equal to the standard symplectic form in some fixed small neighbourhood \(V_x\) and \(V_y\) of the points \(x,y \in S^2\) respectively. Note that all elements of \(\Omega_{x,y}\) tame the standard complex structure. Thus \(\Omega_{x,y}\) is convex. By Moser’s theorem we have the following fibration \[\mathop{\mathrm{Symp}}_{\mathop{\mathrm{id}},x,y}(S^2) \rightarrow \mathop{\mathrm{Diff}}_{\mathop{\mathrm{id}},x,y}(S^2) \twoheadrightarrow \Omega_{x,y}(S^2)\] where \(\mathop{\mathrm{Diff}}_{\mathop{\mathrm{id}},x,y}\) denotes the space of diffeomorphisms that are the identity on \(V_x\) and \(V_y\) and \(\mathop{\mathrm{Symp}}_{\mathop{\mathrm{id}},x,y}\) are the symplectomorphisms that are the identity on the same open neighbourhoods. As \(\Omega_{x,y}\) is contractible, we get a homotopy equivalence \(\mathop{\mathrm{Symp}}_{\mathop{\mathrm{id}},x,y} \simeq \mathop{\mathrm{Diff}}_{\mathop{\mathrm{id}},x,y}\). By Proposition 2.4 in [23], we know that \(\pi_0(\mathop{\mathrm{Diff}}_{\mathop{\mathrm{id}},x,y}) \simeq \mathbb{Z}\) and furthermore each component contains a representative of a Dehn twist. As the Dehn twists are in fact symplectomorphisms identity on \(V_x\) and \(V_y\), the above discussion implies that each isotopy class of \(\mathop{\mathrm{Symp}}_{\mathop{\mathrm{id}},x,y}\) is represented by a Dehn twist.

By assumption \(\phi|_S \in \mathop{\mathrm{Symp}}_{\mathop{\mathrm{id}},x,y}\). Let \(\Lambda_n\) denote the Dehn twist isotopic to \(\phi|_S\). Since \(\Lambda_n\) is identity on \(U_x\cap S\) and \(U_y\cap S\) we can pick a trivialization of the normal bundle over the complement of \(U_x,U_y\), assuming WLOG they are invariant, and extend \(\Lambda_n\) to an equivariant symplectomorphism of the normal bundle as \(\tilde{\Lambda}_n (x,v):= (\Lambda(x),v)\) for all \((x,v) \in (S\setminus (U_x\cup U_y)) \times \mathbb{R}^2\). We take \(\psi=\tilde{\Lambda}^{-1}\). Clearly, \(\psi\) maps a neighborhood \(U\) to itself and preserves \(\mu\). Moreover, \(\phi \circ \psi\) is isotopic to the identity via symplectomorphism that are identity on \(U_x\) and \(U_y\). ◻

Proof of Proposition 29. By picking Lagrangian sections \(\sigma,\sigma'\) of \(\mu,\mu'\) respectively over \(V\), Arnold-Liouville provides us with a symplectic map \(\Phi_0:U\to X\) such \(\mu'=\mu\circ\Phi_0\) over \(V\). The claim will follow if we show that the map \(\Phi_0\) is Hamiltonian, at least after composing with a fiber preserving local symplectomorphism. Indeed, bumping off the Hamiltonian away from the boundary divisors, we obtain a map \(\Phi:X\to X\) which agrees with \(\Phi_0\) near the boundary divisors. This \(\Phi\) satisfies the requirements of the Proposition.

To proceed, let \(D_i\) denote the irreducible components of the boundary divisor \(D\). Let \(D_{ij}:= D_i \cap D_j\) (\(i \neq j\)). It is automatic that \(\Phi_0(D_{ij})= D_{ij}\). Using the argument in the proof of 2 we obtain a Hamiltonian diffeomorphism \(\Psi_{vert}:X\to X\) so that \(\Theta_{vert}:= \Psi_{vert}\circ\Phi_0\) is the identity on a neighbourhood of the points \(D_{ij}\) and still preserves \(D\). Namely, we symplectically identify a small ball \(B_{ij}\) centered at \(D_{ij}\) with the standard ball in \(\mathbb{C}^2\) so that \(D_i,D_j\) map to the coordinate axes. We use the Alexander trick to connect \(\Phi^{-1}|_{B_{ij}}\) to the identity via symplectomorphisms which preserve the coordinate axes, and then modify to get a global symplectomorphism which is supported in \(\cup_{i,j}B_{ij}\), preserves the divisor \(D\) and agrees with \(\Phi_0^{-1}\) on \(B_{ij}\) (after slightly shrinking). The details are exactly the same as in the proof of Lemma 2.

For each \(i\), let \(\psi_i:U\to U\) be a map satisfying the properties guaranteed by Proposition 31 with respect to \(\Theta_{vert}\circ\psi_i\). We may extend \(\psi_i\) as identity in a neighborhood of the divisors \(D_j\) for \(j\neq i\). We can take this neighborhood to still be \(U\). Let \(\psi\) be the composition of the \(\psi_i\) taken in any order. Then \(\Theta_{vert}\circ\psi|_{D_i}\) is isotopic to the identity via symplectomorphisms that are identity near a fixed neighbourhood of \(D_{ij}\). Since \(\psi\) preserves \(\mu'\), we adjust our initial choice of map \(\Phi_0\) to \(\Phi_0\circ\psi\). We thus assume WLOG that \(\Theta_{vert}|_{D_i}\) is isotopic to the identity via symplectomorphisms that are identity near a fixed neighbourhood of \(D_{ij}\).

We extend this Hamiltonian isotopy of \(\Theta_{vert}|_{D_i}\)to a neighborhood of \(D\) using the normal projection. Call the time \(1\) map of this isotopy \(\alpha\). Then \(\Theta_{nbhd}:=\alpha^{-1}\circ\Theta_{vert} =\alpha^{-1}\circ \Psi_{vert}\circ\Phi_0\) is a local symplectomorphism which is identity on \(D\) and on a neighborhood of the \(D_{ij}\) and is Hamiltonian isotopic to \(\Phi_0\).

Lemma 10 gives us an additional Hamiltonian symplectomorphism \(\beta:X\to X\) such that \(d\beta|_{D_i} = d\Theta_{nbhd}|_{D_i}\). Let \(\Theta_{der}:=\beta^{-1}\circ \Theta_{nbhd}\). The local symplectomorphism \(\Theta_{der}\) is Hamiltonian isotopic to \(\Phi_0\) and satisfies \(\Theta_{der}|_{D_i} = \mathop{\mathrm{id}},\) \(d\Theta_{der}|_{D_i} = \mathop{\mathrm{id}},\) and \(\Theta_{der}\) is the identity near \(D_{ij}.\)

Thus the restriction of \(\Theta_{der}\) to a neighborhood of any of the components \(D_i\) satisfies the conditions of Proposition 30 with \(A\) a neighbourhood of the vertices \(D_{ij}\). Therefore, \(\Theta_{der}\) is Hamiltonian in that neighborhood and the Hamiltonian generating it can be taken to be \(0\) in a neighbourhood of \(\bigcup_{i,j}D_{ij}\). Thus \(\Theta_{der}\) is Hamiltonian isotopic to the identity on a neighborhood of \(D\). Since \(\Phi_0\) is Hamiltonian isotopic to \(\Theta_{der}\), the claim follows. ◻

Proof. Let \(C_{ij}\) be the curve \(D_i \cap D_j\). The anti-canonical bundle is ample. Therefore, so is its pullback to \(C_{ij}\). So \(-K_X|_{C_{ij}}\) has strictly positive degree. It follows that the restriction of \(s_1|_{C_{ij}}\) has a zero. ◻

Proof. In a neighborhood of the point at infinity \(\lambda_0=0\), the defining equation can be written as \(\lambda s_0+s_1=0\) for \(\lambda\) the coordinate \(\lambda_0/\lambda_1\). In particular on the fiber over infinity, \(s_1=0\), and the differential coincides with \(ds_1+s_0d\lambda\) which has full rank by smoothness of \(H\) and independence of \(d\lambda\) from \(ds_1\). So, we focus on the complement of the fiber over infinity. Set \(\lambda_0=1\) and write now \(\lambda=\lambda_1\). The differential is now \(ds_0+\lambda ds_1+s_1d\lambda\). Since \(d\lambda\) is independent of \(ds_0,ds_1\), singular points can only occur when \(s_1=0\) which implies also \(s_0=0\). Since \(H\) meets each component transversely, the relation \(ds_0+\lambda ds_1=0\) can only hold if \(\lambda=0\) and therefore \(ds_0=0\). So, singular points occur only at intersections of \(H\) with higher codimension strata of the toric boundary. By Lemma 12 these are precisely the intersections with the \(1\)-dimensional strata. ◻

Proof. The proof is given as part of the proof of Theorem 2.9 in [6]. We recall the details for self containment. Let \(\tilde{\alpha}_D\) denote a 2-form in the class Poincaré dual to \(D\). For each exceptional curve \(C\) fix a holomorphic isomorphism of the normal bundle to \(C\) with the holomorphic vector bundle to \(\mathcal{O}(-1) \oplus \mathcal{O}(-1)\). In a tubular neighborhood \(U_C\) we can pull back the standard Kähler form on the total space of the bundle \(\mathcal{O}(-1) \oplus \mathcal{O}(-1)\) for which the area of \(\mathbb{CP}^1\) is \(1\). Shrinking the neighborhood \(U_C\) if necessary, the pulled back form \(\omega_C\) tames \(J_{sb}\) on \(U_C\). Let \(\lambda_C:=[D]\cdot[C]>0\). Then the form \(\lambda_C\omega_C\) still tames \(J_{sb}\) on \(U_C\). Moreover, since the homology of \(U_C\) is generated by \([C]\), the closed form \(\tilde{\alpha}_D-\lambda_C\omega_C\) is exact. We can thus interpolate \(\tilde{\alpha}_D\) with \(\lambda_C\omega_C\) by cutting off a primitive. We obtain a closed two form \(\alpha_D\) which agrees with \(\lambda_C\omega_C\) in a sufficiently small neighborhood of each exceptional curve \(C\). More importantly, it tames \(J_{sb}\) in an open neighborhood \(V\) of the exceptional locus. By compactness of \(M_{sb}\setminus V\) the closed two-form \(\pi_{sb}^*\omega+t\alpha_D\) is non-degenerate for \(t\) small enough. On the other hand, since both forms tame \(J_{sb}\) on \(V\), we have that \(\pi_{sb}^*\omega+t\alpha_D\) is non-degenerate on \(V\) for any \(t\). ◻

Proof. The first claim follows from the second. To prove the latter let us focus on the local model near one singular point \(p\). In a neighborhood of the double point \(p\) the nodal projective variety \(E\) is described up to units by the equation \(z_iz_j=\lambda s_1\) where \(z_i,z_j\) define the divisors \(D_i,D_j\), and \(\lambda,s_1\) are as before. Then \(\tilde{\Sigma}_i\) is the proper transform of \(\{z_i=s_1=0\}\) which intersects \(C\) transversally and thus intersects \(C\) with intersection number \(+1\) by holomorphicity. Similarly, \(\tilde{\Sigma}_j\) is the proper transform of \(\{z_j=s_1=0\}\) which contains \(C\) and thus has intersection number \(-1\) with \(C\). See Figure 2. ◻

Proof. The claim follows if we prove it for each component \(\tilde{\Sigma}_i\). Note that the restriction of the Poincaré dual of \(\tilde{\Sigma}_i\) to the generic fiber \(\tilde{\pi}^{-1}(t)\) represents the class \(\mathop{\mathrm{PD}}[ \tilde{\Sigma}_i \cap \tilde{\pi}^{-1}(t)]\) where \(\mathop{\mathrm{PD}}\) stands for Poincaré dual in the cohomology of the fibre \(\tilde{\pi}^{-1}(t)\) and \(\cap\) is the homological intersection. As the small blow up operation is local and is only performed on the degenerate fibre, we have that the homological intersection in the small blow up is the same as the intersection in the original manifold i.e \(\mathop{\mathrm{PD}}[ \tilde{\Sigma}_i \cap \tilde{\pi}^{-1}(t)] = \mathop{\mathrm{PD}}[\Sigma_i \cap \pi^{-1}({t})]\). But \(\mathop{\mathrm{PD}}[\Sigma_i \cap \pi^{-1}({t})] = \mathop{\mathrm{PD}}[D_i \cap \pi^{-1}(t)]\). Indeed, \(\Sigma_i= B_i \times \mathbb{CP}^1\) where \(B_i\) was the component of the base locus meeting \(D_i\). ◻

Proof. Write \(\omega_0=\omega_{\vec{\lambda}}=\sum \lambda_i\beta_i\) where \(\beta_i\) is Poincaré dual to \(D_i\). We want to show that we can pick \(\vec{t}=(t_1\dots t_N)\) and a Poincaré dual \(\alpha\) to \(\Sigma_{\vec{t}}\) such that \(\omega_1:=\tilde{\pi}^*\omega_{\vec{\lambda}-\vec{t}}+\alpha\) tames \(J_{sb}\).

For this we revisit the construction in Lemma 14. Fix open neighborhoods \(U_j\) of the exceptional spheres \(C_j\) which are biholomorphic to open neighborhoods of the zero section in \(\mathcal{O}(-1)\oplus\mathcal{O}(-1)\), and let \(\omega_{U_i}\) be a Kahler form on \(\mathcal{O}(-1)\oplus\mathcal{O}(-1)\) for which the zero section has unit area. For each face \(D_i\) let \(\alpha_i\) be a Poincaré dual to \(\tilde{\Sigma}_i\) which coincides with \(\pm\omega_{U_j}\) on some smaller neighborhood \(V_j\) of \(C_j\) whose closure is in \(U_i\). The sign is determined by the intersection number. This can be assumed as shown in the proof of Lemma 14.

There is an open neighborhood \(W\) of \(\vec{\lambda}\) so that \(\omega_{\vec{\lambda'}}\) is symplectic for all \(\lambda'\in W\). Shrinking \(W\) slightly and fixing a background metric on \(E_{sb}\) we can find an \(\epsilon>0\) such that on \(E_{sb}\setminus \cup V_j\), we have that if \(v\) is any unit tangent vector then \(\omega_{\lambda'}(v,Jv)>\epsilon\) for all \(\lambda'\in W.\) Then for \(\vec{t}\) small enough with respect to \(\epsilon\) and such that \(\vec{\lambda}-\vec{t}\in W\), we have that \(\tilde{\pi}^*\omega_{\vec{\lambda}-\vec{t}}+\sum_{i}t_i\alpha_i\) tames \(J\). By Lemma 16, we have the cohomology class of \(\tilde{\pi}^*\omega_{\vec{\lambda}-\vec{t}}+\sum_{i}t_i\alpha_i\) is the same as \(\omega_0\) thus proving the claim. ◻

Proof. By Theorem 36, the cohomology class of \(\omega\) is given by \(\frac{1}{2 \pi} [\omega] = -\sum_{i=1}^d \lambda_i C_i\) for support constants \(\lambda_i\). Perturb the cohomology class by perturbing the support constants \(\lambda_i\) defining the facets \(F_i\) slightly by \(h_i[C_i]\) (\(h_i \in \mathbb{R}\)). For small enough perturbations of \(\lambda_i\) the resulting polytope still satisfies the Delzant conditions. Call this new perturbed polytope \(\Delta'\) with boundary divisors \(C_i'\) and the symplectic manifold associated to it by \((M_{\Delta'},\omega_{\Delta'})\). The fan structures associated to \(\Delta\) and \(\Delta'\) respectively are isomorphic and hence the complex projective varieties \(X_{\Delta}\) and \(X_{\Delta'}\) associated to \(\Delta\) and \(\Delta'\) respectively are equal i.e \(X_{\Delta} =X_{\Delta'}\) and have \(\bigcup_i C_i\) as their torus invariant codimension 1 divisors. We also have the following equivariant orientation preserving diffeomorphisms \(\phi_{\Delta}: M_{\Delta}\rightarrow X_{\Delta},\) and \(\phi_{\Delta'}: M_{\Delta'} \rightarrow (X_{\Delta'}= X_{\Delta})\) that send the toric strata in \(M_{\Delta}\) and \(M_{\Delta'}\) to the toric strata in \(X_{\Delta}\). Thus \(\omega':= (\phi_{\Delta}^{-1})^* \circ \phi_{\Delta'}^*\omega_{\Delta'}\) is a toric symplectic form (for the action pulled back from \(M_{\Delta'}\) via \(\phi_{\Delta'}^{-1}\circ \phi_{\Delta}\)) in the cohomology class \([\omega]+ h_i[C_i]\) and the boundary toric divisor for the pulled back action is \(\phi_{\Delta}^{-1}\circ \phi_{\Delta'}(C'_i)\) which is equal to \(C_i\) as required. By suitable perturbations of the support constants defining \(C_i\) we can obtain such a toric symplectic form in any class in a small enough neighbourhood of \([\omega]\). ◻

Proof. Let \(\{E^i_1,\cdots, E^i_k\}\) denote the exceptional spheres on \(\tilde{D}_i\) coming from performing the small blow-up (in the complex category) and let \(s_j^t=\tilde{\omega}_t(E^i_j)\). By construction, for a given \(j\), the size of the exceptional curves that come from blowing up points on \(D_{ij}\) are all the same. We may choose for each \(i\) a symplectic form on \(D_i\) with the property that \(\omega_t(D_{ij}) = \omega(D_{ij}) + l_js_j^t\) for each boundary divisor \(D_{ij}\) of \(D_i\). Here \(l_j\) is the number of blown up points on \(D_{ij}\). Furthermore, each \(D_{ij}\) is still symplectic for \(\omega_t\). To see this, note the symplectic blow down is diffeomorphic to the complex blow down which reproduces \(D\). This is realized by identifying a neighborhood of each exceptional curve with the tautological disc bundle. \(\omega_t\) can be taken as the the pullback of the symplectic form on the blow-down via this diffeomorphism.

We proceed to construct a toric structure on \(D_i\). By making \(t\) small \(\omega_t\) is arbitrarily close to \(\omega\). By Lemma 19, there exists a form \(\Omega_t\) cohomologous to \(\omega_t\) such that \(\Omega_t\) admits a toric action with toric boundary \(\cup_j D_{ij}\). Any symplectic toric manifold in dimension 4 is diffeomorphic to either \(\mathbb{CP}^2\# k\overline{\mathbb{CP}^2}\) for some \(k \in \mathbb{N}_{>0}\) or \(S^2 \times S^2\). Thus by Theorem 37, there exists a diffeomorphism \(\varphi: (D_i,\omega_t) \to (D_i,\Omega_t)\) that acts trivially on homology. Pulling back the toric action on \((D_i,\Omega_t)\) via \(\varphi\), gives us a toric action on \((D_i,\omega_t)\) with toric boundary \(\varphi^{-1}(\cup_j D_{ij})\). As \(\cup_j D_{ij}\) is symplectic for \(\omega_t\) by construction and \(\varphi\) acts trivially on homology, we have that \(\cup_j D_{ij}\) is in fact log Calabi-Yau for the symplectic form \(\omega_t\). Thus we can compose \(\varphi\) with a symplectomorphism coming from the Torelli theorem (Theorem 28), to obtain a toric structure for \(\omega_t\) with \(\cup_j D_{ij}\) as the toric boundary.

We now proceed to construct an almost toric fibration on \((\tilde{D}_i,\tilde{\omega}_t|_{\tilde{D}_i})\). Let \((D'_i,\omega'_t)\) be the almost toric blow up of \((D_i,\omega_t)\) obtained by removing triangles of capacity \(s^t_j\) from the moment polytope of \((D_i,\omega_t)\). Note that we use the fact that \(s_j^t\) is small to guarantee the existence of disjoint triangles to perform the almost toric blow up. We can thus endow \(D'_i\) with an almost toric fibration \(\mu'_i:D'_i\to \tilde{B}_i\). Let \(\bigcup_{j}D'_{ij}\) denote the boundary of this almost toric fibration. Since \(\tilde{D}_i\) obtained by blowing up \(i_j\) points on \(D_{ij}\) in the complex category we obtain a diffeomorphism \(\tilde{D}_i\to D'_i\) which maps \(\tilde{D}_{ij}\) to \(D'_{ij}\). Moreover, it maps the class of \(\tilde{\omega}_t|_{\tilde{D}_i}\) to the class of \(\omega'_t\) by equality of the areas of the exceptional spheres and the boundary spheres.

Furthermore, by Remark 34 we know that \(\tilde{D}_{ij}\) is symplectically orthogonal with respect to \(\tilde{\omega}_t|_{D_i}\) and \(\phi^{-1}(D'_{ij})\) is symplectically orthogonal with respect to \(\phi^*\omega'_t\) by construction of the almost toric blow up. Applying the Torelli Theorem 28, we obtain a symplectomorphism \(\psi:(\tilde{D}_i,\tilde{\omega}_t,\cup_j\tilde{D}_{ij})\to (D_i',\omega'_t ,\cup_jD'_{ij})\).

Pulling back the almost toric fibration on \(D'_i\) via \(\psi\) gives us the required almost toric fibration on \((\tilde{D}_i, \tilde{\omega}_t|_{\tilde{D}_{i}})\) with boundary divisors \(\cup_j \tilde{D}_{ij}\) as required. ◻

Proof. This follows as in Theorem 20 but done for each face of the hybrid degeneration. ◻

Proof of Proposition 43. Order the components of \(\partial P\) as \(F_1,F_2\dots F_m\). For \(k=1,\dots m\) let \(\Delta^{(k)}\) be the boundary \(\partial P_{\omega}\) of the moment polytope endowed with the piecewise nodal integral affine structure obtained by Proposition 21 from the hybrid degeneration associated with carrying out the first \(k\) steps of the greedy blow-up. Note \(\Delta^{(k)}\) is obtained from \(\Delta^{(k-1)}\) by putting on \(F_k\) the nodal integral affine structure whose edges have the same affine length as that of the toric integral affine structure with nodes placed an affine distance of \(t_k\) from the boundary. Thus, it attaches continuously to \(\Delta^{(k-1)}\setminus F_k.\) Theorem 21 then tells us the piecewise integral affine structure extends smoothly across \(\partial F_k\).

Let \(Q^{(k)}\) be the boundary of the polytope obtained from \(P_{\omega}\) by replacing the first \(k\) defining inequalities \(\langle \nu_i,x\rangle\leq b_i\) with \(\langle \nu_i,x\rangle\leq b_i+t_i\). We inductively construct piecewise integral affine isomorphisms \[f_k:Q^{(k)}\to \Delta^{(k)}\] such that the following hold:

1. The restriction of \(f_k\) to any face is a strict integral affine isomorphism.

2. \(f_k\) agrees with the restriction of \(f_{k-1}\) to \(Q^{(k-1)}\setminus \partial_k Q^{(k-1)}\). Here \(\partial_k Q^{(k-1)}\) is the face normal to \(\nu_k\). For \(k=1\) we take \(f_0:Q^{(0)}\to \Delta^{(0)}\) to be the identity.

3. The image under \(f_k\) of any edge occurring in the first \(k\) faces of \(Q^{(k)}\) contains at least one node.

The claim then follows by putting \(k=m\). Indeed, \(f_m\) is a piecewise integral affine isomorphism from \(\partial P_{\omega_{\vec{t}}}\) with the affine structure induced from \(N_{\mathbb{R}}\) to \(\Delta_{\mathcal{X}_{\vec{t}}}\) with the nodal integral affine structure \(\mathcal{A}_{\vec{t}}\). Moreover, \(f_m\) is strictly integral affine on the faces of \(Q^{(m)}=\partial P_{\omega_{\vec{t}}}\). Since the integral affine structure in the neighborhood of each vertex of \(\Delta\) is the canonical one, we have \(\mathcal{A}_{\vec{t}}=\mathcal{A}_{GS}\).

For the base case start by blowing up the double points mapping to edges of \(F_1\) by an amount \(t_1\). To construct the map \(f_1\) we start with the obvious inclusion \(i:Q^{(0)}\setminus\partial_1Q^{(0)}\hookrightarrow Q^{(1)}\setminus \partial_1Q^{(1)}\), illustrated in Figure 4. We have the identification \(f_0: Q^{(0)} \xrightarrow{=} \Delta^{(0)}\). So we take \(f_1\circ i = f_0\).

For a \(j\neq 1\) such that the face \(\partial_jQ^{(1)}\) meets \(\partial_1Q^{(1)}\) we extend \(f_1\) to all of \(\partial_jQ^{(1)}\) by the following procedure. Consider inside \(F_1\) the polygon \(p_1\subset F_1\) whose edges are monodromy invariant segments through the nodes (e.g., the polygon defined by the union of the green line segments in Figure 5). Let \(V_j\subset F_1\) be the polygon cut out by the segment \(\ell_{1,j}=\partial_jQ^{(0)}\cap F_1\), the edge \(\ell_2\) of \(p_1\) parallel to \(\ell_{1,j}\) and the segments \(e_1,e_2\) connecting the vertices of \(\ell_{1,j}\) with those of \(\ell_2\) (See Figure 5).

Consider the union \(U_j= f_0(\partial_jQ^{(0)})\cup V_j\subset \Delta^{(k)}\). We show now that \(f_1\) extends uniquely to an affine isomorphism \(\partial_jQ^{(1)}\to U_j\). By Lemma 21 the nodal piecewise integral affine structure on \(U_j\) is integral affine with no nodes. Since \(U_j\) is simply connected there is a developing map \(D_j:U_j \rightarrow \mathbb{R}^2\). Consider the composition map \(D_j \circ f_1\) defined thus far on \(i(\partial_jQ^{(0)})\) and extend to \(\partial_jQ^{(1)}\) linearly. Call this extended map \(\widetilde{\left(D_j \circ f_0\right)}: \partial_j Q^{(1)} \rightarrow \mathbb{R}^2\). We claim the image of this map is exactly \(D_j(U_j)\). Namely, we argue that the two images are bounded by the same lines in the affine space. For this we first observe that, by Lemma 12 that the base locus intersects each codimension 1 strata and hence after performing the small blow-up on each of the edges of \(F_1\) in the hybrid degeneration, each edge of \(p_1\) is parallel to an edge in the boundary of \(F_1\) and is an affine distance of \(t_1\) away from it. Furthermore, by the description of the integral affine structure near the vertices as in Definition 15, it follows that the segments \(e_1,e_2\) connect smoothly with edges of \(F_j\) incident on the vertices \(v_1,v_2\) of \(\ell_{1,j}= F_j\cap F_1\). In other words, the union of \(e_i\) with the edges of \(F_j\) incident on \(v_j\) is a straight line in this affine chart. Let us denote these two lines by \(\tilde{e}_1\) and \(\tilde{e}_2\) respectively. Now, \(f_0\) maps the boundary edges of \(\partial_jQ^{(0)}\) that transversely intersect \(\partial_1Q^{(0)}\) into the lines \(\tilde{e}_1\) and \(\tilde{e}_2\). This is depicted by the images of the edges labelled purple and orange in Figure 6. The proof that \(D_j(U_j) = \widetilde{\left(D_j \circ f_0\right)}(\partial_j Q^{(1)})\) is then complete if we show that \(\widetilde{\left(D_j \circ f_0\right)}(\partial_1 Q^{(1)} \cap \partial_j Q^{(1)})\) maps to the affine line \(D_j (\ell_2)\) (denoted by the blue edges in Figure 6). We prove this by showing that the above two affine lines are the level sets of an affine function at the same level. Consider the following two affine functions: \[\begin{align} H_1: \partial_j Q^{(1)} &\longrightarrow \mathbb{R}\\ x &\mapsto \langle x,\nu_1\rangle, \end{align}\] and, \[\begin{align} G_1:=\begin{cases} \langle x,(f_0)_*\nu_1\rangle \text{~~if x\in f_0(\partial_jQ^{(0)})} \\ \langle x,\vec{\ell}_{1,j}\rangle \text{~~if x \in V_j}, \end{cases} \end{align}\] where \(\vec{\ell}_{1,j}\) denotes the primitive integral normal to the line \(\ell_{1,j}\). We take the direction of \(\vec{\ell}_{1,j}\) to be in the same direction as primitive normal to the level sets of \(\langle \cdot,(f_0)_*\nu_1\rangle\). Note that the line \(\partial_1 Q^{(1)} \cap \partial_j Q^{(1)}\) is defined by \(H_1^{-1}(b_1+t_1)\). With these choices, it follows that \(G_1\) is an integral affine function on \(U_j\) having the monodromy invariant line \(\ell_2\) as a level set with value \(b_1 + t_1\). Pushing forward the maps \(H_1\) and \(G_1\) to \(\mathbb{R}^2\) by \(\widetilde{\left(D_j \circ f_0\right)}\) and \(D_j\) respectively, we see that the pushforward maps agree on the overlap \(D_j(f_0(\partial_jQ^{(0)})) = \widetilde{\left(D_j \circ f_0\right)}(\partial_jQ^{(0)})\). As the two maps are integral affine maps on \(\mathbb{R}^2\) that agree on an open set, they agree everywhere and hence their level sets at \(b_1+t_1\) is the same set. This completes the proof that \(\widetilde{\left(D_j \circ f_0\right)}(\partial_1 Q^{(1)} \cap \partial_j Q^{(1)})\) maps to the affine line \(D_j (\ell_2)\).

It follows that after blowing up \(F_1\), the integral affine structure in a neighborhood of \(F_1\), is the required one and the map \(f_1\) is defined as follows: \(f_1(x) := (D_j)^{-1}\circ \widetilde{\left(D_j \circ f_0\right)}(x)\).

Repeating this for all \(j\neq 1\) such that the face \(\partial_jQ^{(1)}\) meets \(\partial_1Q^{(1)}\), this extends \(f_1\) to all of \(Q^(1)\setminus\partial_1Q^{(1)}\). But then \(\partial_1Q^{(1)}\) maps onto \(p_1\) in the obvious way. The map \(f_1\) thus defined satisfies the first two inductive assumptions by construction. It satisfies the third since the edges in \(\partial_1Q^{(1)}\) map to the edges of \(p_1\) each of which contains a node as already noted.

Suppose inductively we have proven the claim for \(k-1\). To deduce it for \(k\) observe the face \(F_k\) is bordered by segments which either have singularities that have not yet been resolved, or which contain nodes. The effect of resolving the singularities is to push all the nodes into the interior of \(F_k\). Indeed, for the new nodes this is by definition and Lemma 15. For the nodes from the previous step, note that by Lemma 15 we are decreasing the size of the exceptional curve by an amount \(t_k\). Let \(p_k\) be the polygon formed by monodromy invariant segments through the nodes inside \(F_k\). Then \(p_k\) has affine distance \(t_k\) from \(\partial F_k\) and the rest of the proof proceeds as in the base case. ◻

Proof of Theorem 7. Start with a symplectic form \(\omega_0\) on \(X\) inducing a given form on the fiber \(V\). Pick any smooth anti-canonical divisor \(H\) which intersects the toric divisors transversely. Construct \(E_{sb}\) as in the discussion preceding Lemma 15. By Lemma 17 we can find a symplectic form \(\omega\) on \(E_{sb}\) taming \(J\) and whose restriction to \(H\) is cohomologous to that of \(\omega_0\). Two cohomologous forms which tame the same \(J\) are symplectomorphic. By Corollary 39 \((E_{sb},\omega)\) is a symplectic Kulikov model.

The given hypersurface \(V\) can be connected to \(H\) by a path of smooth anti-canonical hypersurfaces. Thus \((V,\omega_0|_V)\) is symplectomorphic to \((H,\omega_0|_H)\) by symplectic parallel transport. The latter is symplectomorphic to the fiber \((H,\omega|_H)\) by the argument above. Thus \(V\) admits a Kulikov model by definition.

The claim concerning the integral affine structure is the content of Proposition 43. ◻