To deal with singular fibres, we define

Here we will give a relatively simple construction of such affine manifolds with singularities; a broader class of examples is given in [22]; see also [39] and [40].

Let $\Delta$ be a reflexive polytope in $M_{\mathbb{R}}=M\otimes_{\mathbb{Z}}\mathbb{R}$, where $M=\mathbb{Z}^{n}$. This means that $\Delta$ is a lattice polytope with a unique interior integral point $0\in\Delta$, and the polar dual polytope

$$\nabla:=\{n\in N_{\mathbb{R}}|\hbox{$\langle m,n\rangle\geq-1$ for all $m\in\Delta$}\}$$

is also a lattice polytope.

Let $B=\partial\Delta$, and let $\mathscr{P}$ be a decomposition of $B$ into lattice polytopes, i.e., $\mathscr{P}$ is a set of lattice polytopes contained in $B$ such that (1) $B=\bigcup_{\sigma\in\mathscr{P}}\sigma$; (2) $\sigma_{1},\sigma_{2}\in\mathscr{P}$ implies $\sigma_{1}\cap\sigma_{2}$ lies in $\mathscr{P}$ and is a face of both $\sigma_{1}$ and $\sigma_{2}$; (3) if $\sigma\in\mathscr{P}$, any face of $\sigma$ lies in $\mathscr{P}$.

We now define a structure of integral affine manifold with singularities on $B$, with discriminant locus $\Gamma\subseteq B$ defined as follows. Let $\operatorname{Bar}(\mathscr{P})$ denote the first barycentric subdivision of $\mathscr{P}$ and let $\Gamma\subseteq B$ be the union of all simplices of $\operatorname{Bar}(\mathscr{P})$ not containing a vertex of $\mathscr{P}$ (a zero-dimensional cell) or intersecting the interior of a maximal cell of $\mathscr{P}$. Setting $B_{0}:=B\setminus\Gamma$, we define an affine structure on $B_{0}$ as follows. $B_{0}$ has an open cover

$$\{W_{\sigma}|\hbox{$\sigma\in\mathscr{P}$ maximal}\}\cup\{W_{v}|\hbox{$v\in\mathscr{P}$ a vertex}\}$$

where $W_{\sigma}=\operatorname{Int}(\sigma)$, the interior of $\sigma$, and

$$W_{v}=\bigcup_{\tau\in\operatorname{Bar}(\mathscr{P})\atop v\in\tau}\operatorname{Int}(\tau)$$

is the (open) star of $v$ in $\operatorname{Bar}(\mathscr{P})$. We define an affine chart

$$\psi_{\sigma}:W_{\sigma}\rightarrow\mathbb{A}^{n-1}\subseteq N_{\mathbb{R}}$$

given by the inclusion of $W_{\sigma}$ in $\mathbb{A}^{n-1}$, the affine hyperplane containing $\sigma$. Also, take $\psi_{v}:W_{v}\rightarrow M_{\mathbb{R}}/\mathbb{R}v$ to be the projection. One checks easily that for $v\in\sigma$, $\psi_{\sigma}\circ\psi_{v}^{-1}$ is integral affine linear (integrality follows from reflexivity of $\Delta$!) so $B$ is an integral affine manifold with singularities.

One main result of [20] is

We won’t give any details here of how this is carried out, but it is not particularly difficult, as long as one restricts to the category of topological (not $C^{\infty}$) manifolds. However, it is interesting to look at the singular fibres we need to add.

If $b\in\Gamma$ is a point which is not a vertex of $\Gamma$, then $f^{-1}(b)$ is homeomorphic to $I_{1}\times S^{1}$, where $I_{1}$ denotes a Kodaira type $I_{1}$ elliptic curve, i.e., a pinched torus.

If $v$ is a vertex of $\Gamma$, with monodromy of the first type, then $f^{-1}(v)=S^{1}\times S^{1}\times S^{1}/\sim$, with $(a,b,c)\sim(a^{\prime},b^{\prime},c^{\prime})$ if $(a,b,c)=(a^{\prime},b^{\prime},c^{\prime})$ or $a=a^{\prime}=1$, where $S^{1}$ is identified with the unit circle in $\mathbb{C}$. This is the three-dimensional analogue of a pinched torus, and $\chi(f^{-1}(v))=+1$. We call this a positive fibre.

If $v$ is a vertex of $\Gamma$, with monodromy of the second type, then $f^{-1}(v)$ can be described as $S^{1}\times S^{1}\times S^{1}/\sim$, with $(a,b,c)\sim(a^{\prime},b^{\prime},c^{\prime})$ if $(a,b,c)=(a^{\prime},b^{\prime},c^{\prime})$ or $a=a^{\prime}=1$, $b=b^{\prime}$, or $a=a^{\prime},b=b^{\prime}=1$. The singular locus of this fibre is a figure eight, and $\chi(f^{-1}(v))=-1$. We call this a negative fibre.

So we see a very concrete local consequence of SYZ duality: in the compactifications $X(B)$ and $\check{X}(B)$, the positive and negative fibres are interchanged. Of course, this results in the observation that the Euler characteristic changes sign under mirror symmetry for Calabi-Yau threefolds.

W.-D. Ruan in [69] gave a description of Lagrangian torus fibrations for hypersurfaces in toric varieties using a symplectic flow argument, and his construction should coincide with a symplectic compactification of the symplectic manifolds $\check{X}(B_{0})$. In the three-dimensional case, such a symplectic compactification has been constructed by Ricardo Castaño-Bernard and Diego Matessi [8]. If this compactification is applied to the affine manifolds with singularities described here, the resulting symplectic manifolds should be symplectomorphic to the corresponding toric hypersurface, but this has not yet been shown.