We now have two notions of limit: the familiar algebro-geometric notion of a degenerating family $\mathcal{X}\rightarrow D$ over a disk on the one hand, and the Gromov-Hausdorff limit on the other. In 2000 Kontsevich and Soibelman had an important insight (see [52]) into the connection between these two. In this section I will give a rough idea of how and why this works.

Very roughly speaking, the Gromov-Hausdorff limit $(\mathcal{X}_{t_{i}},g_{t_{i}})$ as $t_{i}\rightarrow 0$, or equivalently, the base of the putative SYZ fibration, should coincide, topologically, with the dual intersection complex of the singular fibre $\mathcal{X}_{0}$. More precisely, in a relatively simple situation, suppose $f:\mathcal{X}\rightarrow D$ is relatively minimal (in the sense of Mori) and normal crossings, with $\mathcal{X}_{0}$ having irreducible components $X_{1},\ldots,X_{m}$. The dual intersection complex of $\mathcal{X}_{0}$ is the simplicial complex with vertices $v_{1},\ldots,v_{m}$, and which contains a simplex $\langle v_{i_{0}},\ldots,v_{i_{p}}\rangle$ if $X_{i_{0}}\cap\cdots\cap X_{i_{p}}\not=\emptyset$. The idea that the dual intersection complex should play a role in describing the base of the SYZ fibration was perhaps first suggested by Leung and Vafa in [55].

Let us explain roughly why this should be, first by looking at a standard family of degenerating elliptic curves with periods $1$ and ${n\over 2\pi i}\log t$ for $n$ a positive integer. Such a family over the punctured disk is extended to a family over the disk by adding a Kodaira type $I_{n}$ (a cycle of $n$ rational curves) fibre over the origin.

Taking a sequence $t_{i}\rightarrow 0$ with $t_{i}$ real and positive gives a sequence of elliptic curves of the form $X_{\epsilon_{i}}(B)$ where $B=\mathbb{R}/n\mathbb{Z}$ and $\epsilon_{i}=-{2\pi\over\ln t_{i}}$. In addition, the metric on $X_{\epsilon_{i}}(B)$, properly scaled, comes from the constant Hessian metric on $B$. So we wish to explain how $B$ is related to the geometry near the singular fibre. To this end, let $X_{1},\ldots,X_{n}$ be the irreducible components of $\mathcal{X}_{0}$; these are all $\mathbb{P}^{1}$’s. Let $P_{1},\ldots,P_{n}$ be the singular points of $\mathcal{X}_{0}$.

We’ll consider two sorts of open sets in $\mathcal{X}$. For the first type, choose a coordinate $z$ on $X_{i}$, with $P_{i}$ given by $z=0$ and $P_{i+1}$ given by $z=\infty$. Let $U_{i}\subseteq X_{i}$ be the open set $\{z\,|\,\delta\leq|z|\leq 1/\delta\}$ for some small fixed $\delta$. Then one can find a neighbourhood $\widetilde{U}_{i}$ of $U_{i}$ in $\mathcal{X}$ such that $\widetilde{U}_{i}$ is biholomorphic to $U_{i}\times D_{\rho}$ for $\rho>0$ sufficiently small, $D_{\rho}$ a disk of radius $\rho$ in $\mathbb{C}$, and $f|_{\widetilde{U}_{i}}$ is the projection onto $D_{\rho}$.

On the other hand, each $P_{i}$ has a neighbourhood $\widetilde{V}_{i}$ in $\mathcal{X}$ biholomorphic to a polydisk $\{(z_{1},z_{2})\in\mathbb{C}^{2}\,|\,|z_{1}|\leq\delta^{\prime},|z_{2}|\leq\delta^{\prime}\}$ on which $f$ takes the form $z_{1}z_{2}$.

If $\delta$ and $\delta^{\prime}$ are chosen correctly, then for $t$ sufficiently close to zero,

form an open cover of $\mathcal{X}_{t}$. Now each of the sets in this open cover can be written as $X_{\epsilon}(U)$ for some $U$ a one-dimensional (non-compact) affine manifold and $\epsilon=-2\pi/\ln|t|$. If $U$ is an open interval $(a,b)\subseteq\mathbb{R}$, then $X_{\epsilon}(U)$ is biholomorphic to the annulus

as $q=e^{2\pi i(x+iy)/\epsilon}$ is a holomorphic coordinate on $X_{\epsilon}((a,b))$. Thus

with $\epsilon=-2\pi/\ln|t|$. As $t\rightarrow 0$, the interval $(\epsilon\ln\delta/2\pi,-\epsilon\ln\delta/2\pi)$ shrinks to a point. So $\widetilde{U}_{i}\cap\mathcal{X}_{t}$ is a smaller and smaller open subset of $\mathcal{X}_{t}$ as $t\rightarrow 0$ when we view things in this way. This argument suggests that every irreducible component should be associated to a point on $B$.

Now look at $\widetilde{V}_{i}\cap\mathcal{X}_{t}$. This is

with $\epsilon=-2\pi/\ln|t|$. This interval approaches the unit interval $(0,1)$ as $t\rightarrow 0$. So the open set $\widetilde{V}_{i}\cap\mathcal{X}_{t}$ ends up being a large portion of $\mathcal{X}_{t}$. We end up with $\mathcal{X}_{t}$, for small $t$, being a union of open sets of the form $X_{\epsilon}((i+\epsilon^{\prime},i+1-\epsilon^{\prime}))$ (i.e., $\widetilde{V}_{i}\cap\mathcal{X}_{\epsilon}$) and $X_{\epsilon}((i-\epsilon^{\prime\prime},i+\epsilon^{\prime\prime}))$ (i.e., $\widetilde{U}_{i}\cap\mathcal{X}_{t}$) for $\epsilon^{\prime}$, $\epsilon^{\prime\prime}$ sufficiently small. These should glue, at least approximately, to give $X_{\epsilon}(B)$. So we see that irreducible components of $\mathcal{X}_{0}$ seem to coincide with points on $B$, but intersections of components coincide with lines. In this way we see the dual intersection complex emerge.

Let us make one more observation before beginning with rigorous results in the next section. Suppose more generally we had a Gorenstein toroidal crossings degeneration of Calabi-Yau manifolds $f:\mathcal{X}\rightarrow D$ (see [72]). This means that every point $x\in\mathcal{X}$ has a neighbourhood isomorphic to an open set in an affine Gorenstein (i.e., the canonical class is a Cartier divisor) toric variety, with $f$ given locally by a monomial which vanishes exactly to order $1$ on each codimension one toric stratum. This is a generalization of the notion of normal crossings. Very roughly, the above argument suggests that each irreducible component of the central fibre will correspond to a point of the Gromov-Hausdorff limit. The following exercise shows what kind of contribution to $B$ to expect from a point $x\in\mathcal{X}_{0}$ which is a zero-dimensional stratum in $\mathcal{X}_{0}$.

This observation hopefully motivates the basic construction of the next section.