In the generic region $f:M\to B$ is a smooth proper submersion with $T^{3}$ fibres. A torus fibration is called semiflat if the metric restricts to flat metrics on the tori. The Calabi-Yau structure $(g,\omega,\Omega)$ on a semiflat SYZ $T^{3}$-fibration can be locally described in action-angle coordinates as

Here $g_{ij}$ is the Hessian of a real valued function $\varphi$ on $B$ solving the real Monge-Ampère equation:

and $g_{ij}$ defines a metric on the base $g_{B}=g_{ij}d\mu_{i}d\mu_{j}$ such that $f:M\to B$ is a Riemannian submersion.

The Calabi-Yau structure induce two sets of affine structures on the base $B$: the symplectic moment coordinates $\mu_{i}$ satisfying $d\mu_{i}=-\omega(\frac{\partial}{\partial\theta_{i}},\cdot)$, and the complex affine coordinates $y_{i}$ satisfying $dy_{i}=\text{Im}\Omega(\frac{\partial}{\partial\theta_{j}},\frac{\partial}{\partial\theta_{k}},\cdot)$ for cyclic indices $i,j,k$. These coordinates are related by the Legendre transform $y_{i}=-\frac{\partial\varphi}{\partial\mu_{i}}.$

Semiflat mirror symmetry is the observation that if over the same base $B$ we fibrewise replace $T^{3}$ by the dual tori, then there is a canonical Calabi-Yau structure exhibiting this dual torus fibration as a semiflat SYZ fibration. Furthermore, the base metric $g_{B}$ is unchanged while the roles of the two affine structures are interchanged.

A major part of the SYZ Conjecture 1.1 is that Calabi-Yau metrics on (polarised) manifolds near the large complex structure limit are asymptotically described by such semiflat SYZ fibrations in the generic region up to exponentially small errors.

Along an edge $\mathfrak{\Delta}\subset\mathfrak{D}$, the singular fibres have the topology of $T^{3}$ with $T^{2}$ collapsed to $S^{1}$, alternatively written as $I_{1}\times S^{1}$, where $I_{1}$ refers to the nodal elliptic curve or equivalently $S^{2}$ with two points identified. Notice the singularity on the fibre is not isolated. These singular fibres have Betti numbers $(b_{1},b_{2})=(2,2)$ and Euler characteristic 0. The $T^{3}$-fibration is locally described as the Kodaira type $I_{1}$ degenerating family of elliptic curves over a disc $D^{2}\subset\mathbb{C}$, Cartesian product with the trivial $S^{1}$-bundle $S^{1}\times\mathbb{R}\to\mathbb{R}$. The monodromy around the edge $\mathfrak{\Delta}\subset\mathfrak{D}$ acting on $H_{1}(T^{3})\simeq\mathbb{Z}^{3}$ can be written in a suitable basis as

For an alternative viewpoint which ties in better with the Gibbons-Hawking construction (see later Sections 1.2), the base is $B\subset\mathbb{R}^{3}$, and the total space $M$ is a singular $S^{1}$-bundle over $S^{1}\times B\times S^{1}$, where the $S^{1}$-fibres collapse to points along the codimension 3 locus $\{0\}\times\mathfrak{\Delta}\times S^{1}\subset S^{1}\times B\times S^{1}$. In the 3 transverse directions, the singular $S^{1}$-bundle structure is topologically modelled on the Hopf map

The Chern class $c_{1}\in H^{2}(S^{1}\times B\times S^{1}\setminus(\{0\}\times\mathfrak{\Delta}\times S^{1}),\mathbb{Z})$ evaluates to 1 on a suitably oriented $S^{2}$-cycle linking $\{0\}\times\mathfrak{\Delta}\times S^{1}$ inside $S^{1}\times B\times S^{1}$.

Let $\mathfrak{D}\subset B$ be a graph with one vertex emitting 3 edges. Topologically, we can present $\mathfrak{D}$ as

The total space $M^{+}$ is built as a singular $T^{2}$-bundle over $B\times S^{1}=\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{R}\times S^{1}$ with discriminant locus $\mathfrak{D}\times\{0\}\subset B\times S^{1}$. Let $e_{1},e_{2}$ denote a basis of $H_{1}(T^{2},\mathbb{Z})$, and denote $T(ae_{1}+be_{2})$ as the subtorus with homology class $ae_{1}+be_{2}$. Over $(B\times S^{1})\setminus(\mathfrak{D}\times\{0\})$ the space $M^{+}$ is a principal $T^{2}$-bundle, whose Chern class $c_{1}\in H^{2}(B\times S^{1}\setminus(\mathfrak{D}\times\{0\}),\mathbb{Z}e_{1}\oplus\mathbb{Z}e_{2})$ evaluates to $e_{1},-e_{2},-e_{1}+e_{2}$ respectively on the $S^{2}$-cycles linking $\mathfrak{D}_{1}\times\{0\},\mathfrak{D}_{2}\times\{0\},\mathfrak{D}_{3}\times\{0\}$ inside $B\times S^{1}$. Over the codimension 3 loci $\mathfrak{D}_{1}\times\{0\}$, $\mathfrak{D}_{2}\times\{0\}$, $\mathfrak{D}_{3}\times\{0\}$ inside $B\times S^{1}$, the $T^{2}$-fibres collapse to circle fibres $T^{2}/T(e_{1})$, $T^{2}/T(-e_{2})$, $T^{2}/T(-e_{1}+e_{2})$ respectively. Finally, over the origin $\{0\}\subset B\times S^{1}$, the $T^{2}$-fibre collapses to a point. The singular $T^{2}$-bundle over a small neighbourhood of the origin $D^{4}\subset B\times S^{1}$ is topologically modelled on

whose discriminant locus is compatible with $\mathfrak{D}$.

By construction $M^{+}$ fibres over $B$ with generic fibre $T^{3}$. The singular fibre over $0\in B$ has the topology of $T^{3}$ with $T^{2}$ collapsed to a point, so has Betti numbers $(b_{1},b_{2})=(1,2)$ and Euler characteristic $+1$ (hence the name ‘positive vertex’). A basis of $H_{1}(T^{3})$ is given by $e_{1},e_{2}\in H_{1}(T^{2})\subset H_{1}(T^{3})$ and an $S^{1}$-cycle $e_{0}$ on the total space lifting the cycle $S^{1}\subset B\times S^{1}$. The monodromies around the 3 edges $\mathfrak{D}_{1}$, $\mathfrak{D}_{2}$, $\mathfrak{D}_{3}$ acting on $H_{1}(T^{3})$ are given in the basis $e_{0},e_{1},e_{2}$ as

We recount here the historical perspective of Gross and Ruan on negative vertices, to be modified in Section 1.1.5. Let $\mathfrak{D}\subset B$ be a graph with one vertex emitting 3 edges. Topologically, we present $\mathfrak{D}$ as

Let $e_{1},e_{2}$ be a basis of $H_{1}(T^{2},\mathbb{Z})$. Let $S\subset B\times T^{2}$ be a ‘pair of pants’ (namely a surface homeomorphic to the complement of 3 points in $S^{2}$) sitting over $\mathfrak{D}\subset B$, such that $S\cap(\mathfrak{D}_{i}\times T^{2})$ is a cylinder $\mathfrak{D}_{i}\times S^{1}$, where the $S^{1}$ factor inside $T^{2}$ has homology class $e_{2},e_{1},-e_{1}-e_{2}$ for $i=1,2,3$ respectively. The fact that these 3 classes add up to zero means the 3 cylinders $\mathfrak{D}_{i}\times S^{1}$ can be joined together over $\{0\}\times T^{2}$. The fibre of $S\to\mathfrak{D}$ over $0\in\mathfrak{D}$ is a ‘figure 8 diagram’.

Then the total space $M^{-}$ is built as a singular $S^{1}$-bundle over $B\times T^{2}$, which restricts to a principal $S^{1}$-bundle over the complement of the codimension 3 locus $S\subset B\times T^{2}$, and along $S$ the $S^{1}$-fibres collapse to points. The first Chern class of the $S^{1}$-bundle evaluates trivially on $T^{2}\subset B\times T^{2}$ but nontrivially on the $S^{2}$-cycle wrapping $S$. In the 3 transverse directions, the $S^{1}$ fibration is modelled topologically on (1.1).

By construction $M^{-}$ fibres over $B$ with generic fibre $T^{3}$, where $T^{3}$ itself is an $S^{1}$-bundle over $T^{2}$. The class of this $S^{1}\subset T^{3}$ is denoted $e_{3}$. The singular fibre of $M^{-}\to B$ over $0\in B$ is obtained by taking the bundle $T^{3}\to T^{2}$, and collapse down its $S^{1}$-fibres over a ‘figure 8 diagram’ inside $T^{2}$. This singular fibre has Betti numbers $(b_{1},b_{2})=(2,1)$ and Euler characteristic $-1$ (hence the name ‘negative vertex’). The homology classes $e_{1},e_{2}$ lift to $H_{1}(T^{3})$. The monodromies around the edges $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$ acting on $H_{1}(T^{3},\mathbb{Z})$ are given in the basis $e_{1},e_{2},e_{3}$ of $H_{1}(T^{3},\mathbb{Z})$ as

Joyce [14] gave reasons that the above topological picture of Gross-Ruan cannot literally describe a special Lagrangian fibration in a generic Calabi-Yau 3-fold, based on his study of local $U(1)$-invariant special Lagrangian submanifolds inside $\mathbb{C}^{3}$. Joyce’s critique hinges on two geometric observations:

• •

  Special Lagrangian fibrations need not be defined by a smooth map, and the discriminant locus needs not have codimension 2.

• •

  The $I_{1}\times S^{1}$ singular fibres have non-isolated special Lagrangian singularities, which is an infinite codimensional phenomenon in the parameter space, namely the singularity structure cannot persist under almost any perturbation of the Kähler structure or the boundary data of the special Lagrangian.

Furthermore, in the $U(1)$-invariant setting, Joyce constructed examples illustrating the possibility that fibres with $I_{1}\times S^{1}$ singularities can break up into fibres with a pair of special Lagrangian $T^{2}$ cones. Such fibres lie over a thickened version of the original edges in $\mathfrak{D}$, and in particular the discriminant locus of the SYZ fibration now has codimension 1.

As suggested by Morrison [22] this thickening picture is linked to the description of the negative vertex (Section 1.1.4) as follows. We can view $B\times T^{2}$ as $(\mathbb{C}^{*})_{z_{1},z_{2}}^{2}\times\mathbb{R}\simeq\mathbb{R}^{2}\times\mathbb{R}\times T^{2}$. The ‘pair of pants’ $S$ is realised topologically by

The algebraic 2-torus $(\mathbb{C}^{*})^{2}$ maps to $\mathbb{R}^{2}_{y_{1},y_{2}}$ via

The image of $S$ in $\mathbb{R}^{2}_{y_{1},y_{2}}$ under this map is an amoeba which can be thought as a thickend version of $\mathfrak{D}\subset\mathbb{R}^{2}_{y_{1},y_{2}}$. Along the 3 directions defined by $\mathfrak{D}$, the asymptotic geometry of $S$ near infinity approaches 3 cylinders. In the modified construction $M^{-}$ is a singular $S^{1}$-bundle over $(\mathbb{C}^{*})^{2}\times\mathbb{R}$ whose fibres collapse to points along the codimension 3 locus $S\subset(\mathbb{C}^{*})^{2}\times\mathbb{R}$. The natural smooth map $M^{-}\to(\mathbb{C}^{*})^{2}\times\mathbb{R}\to B=\mathbb{R}^{2}_{y_{1},y_{2}}\times\mathbb{R}$ cannot be exactly a special Lagrangian fibration since its discriminant locus is of codimension 1, but it is still possible to be an approximate special Lagrangian fibration.

We also wish to resolve a paradox here in advance. Part of our plan is to construct a family of $T^{2}$-symmetric Ooguri-Vafa type Calabi-Yau metrics on the positive vertex, which admit a special Lagrangian fibration with all the topological features predicted by Gross and Ruan, and in particular the singular fibres will have non-isolated singularities. We emphasize there is no contradiction with Joyce’s critique: it is possible for the Ooguri-Vafa type metrics to be a good metric model for a generic Calabi-Yau 3-fold near the large complex structure limit, while the singularity structure of the SYZ fibration changes drastically. Joyce’s critique does not rule out the Gross-Ruan picture as a limiting description of SYZ fibrations.

A familiar picture from Riemann surface theory is that higher genus algebraic curves can be obtained topologically by patching together ‘pairs of pants’ along cylindrical necks. There is a similar picture for Calabi-Yau toric hypersurfaces approaching a large complex structure limit, well studied in tropical geometry. The discussions below are loosely based on Zharkov [32][31], and are included to predict the holomorphic structure of the positive and the negative vertices.

Let $\mathbb{P}_{\triangle}$ be a toric manifold whose moment polytope is the reflexive integral polytope $\triangle$ in $\mathbb{R}^{4}$, so the integral points $v\in\triangle$ correspond to a basis $\{s_{v}\}$ for anticanonical sections. Let $\lambda$ be a (suitably generic) function on $\triangle\cap\mathbb{Z}^{4}$ whose piecewise linear extension is a convex function on $\mathbb{R}^{4}$ minimized at $0\in\triangle$ with minimum value 0. We consider a polarised family of hypersurfaces $X_{t}$ defined by

where $a_{v}$ are fixed nonzero complex numbers and $t$ is a small positive parameter. The holomorphic volume form is determined from the adjunction formula.

The key point is that when $t$ is very small, the hypersurface $X_{t}$ decompose into a finite number of regions, on each of which only a small number of monomial functions $\frac{s_{v}}{s_{0}}$ dominate the rest. Thus up to scaling coordinates by powers of $t$, there are only a small number of complex geometric local models, typically with some torus symmetry. Furthermore there is some combinatorial structure which controls how these local models patch together to give $X_{t}$ as a complex manifold.

In this paper we are primarily interested in the positive and negative vertices. These are relevant for certain regions near the intersection of $X_{t}$ with some higher depth strata of the toric boundary of $\mathbb{P}_{\triangle}$.