Mirror symmetry got its start in 1989 with work of Greene and Plesser [17] and Candelas, Lynker and Schimmrigk [9]. These two works first observed the existence of pairs of Calabi-Yau manifolds exhibiting an exchange of Hodge numbers. Recall that by Yau’s proof of the Calabi conjecture [76], a Calabi-Yau manifold is an $n$-dimensional complex manifold $X$ with a nowhere vanishing holomorphic $n$-form $\Omega$ and a Ricci-flat Kähler metric with Kähler form $\omega$. Ricci-flatness is equivalent to $\omega^{n}=C\Omega\wedge\bar{\Omega}$ for a constant $C$.

The most famous example of a Calabi-Yau manifold is a smooth quintic three-fold $X\subseteq\mathbb{P}^{4}$. The Hodge numbers of $X$ are $h^{1,1}(X)=1$ and $h^{1,2}(X)=101$, with topological Euler characteristic $-200$. The original construction of Greene and Plesser gave a mirror to $X$, as follows. Let $Y\subseteq\mathbb{P}^{4}$ be given by the equation

and let

An element $(a_{0},\ldots,a_{4})\in G$ acts on $Y$ by

for $\xi$ a primitive fifth root of unity. The quotient $Y/G$ is highly singular, but there is a resolution of singularities $\check{X}\rightarrow Y/G$ such that $\check{X}$ is also Calabi-Yau, and one finds $h^{1,1}(\check{X})=101$ and $h^{1,2}(\check{X})=1$, so that $\check{X}$ has topological Euler characteristic $200$.

The relationship between these two Calabi-Yau manifolds proved to be much deeper than just this exchange of Hodge numbers. Pioneering work of Candelas, de la Ossa, Greene and Parkes [10] performed an amazing calculation, following string-theoretic predictions which suggested that certain enumerative calculations on $X$ should give the same answer as certain period calculations on $\check{X}$. The calculations on $\check{X}$, though subtle, could be carried out: these involved integrals of the holomorphic form on $\check{X}$ over three-cycles as the complex structure on $\check{X}$ is varied. On the other hand, the corresponding calculations on $X$ involved numbers of rational curves on $X$ of each degree. For example, the number of lines on a generic quintic threefold is $2875$ and the number of conics is $609250$. String theory thus gave predictions for these numbers for every degree, an astonishing feat given that most of these numbers seemed far beyond the reach of algebraic geometry at the time.

More generally, string theory introduced the concepts of the $A$-model and $B$-model. The $A$-model involves the symplectic geometry of Calabi-Yau manifolds. Properly defined, the counts of rational curves are in fact symplectic invariants, now known as Gromov-Witten invariants. The $B$-model, on the other hand, involves the complex geometry of Calabi-Yau manifolds. Holomorphic forms of course depend on the complex structure, so the period calculations mentioned above can be thought of as $B$-model calculations. Ultimately, string theory predicts an isomorphism between the $A$-model of a Calabi-Yau manifold $X$ and the $B$-model of its mirror, $\check{X}$. The equality of numerical invariants is then a consequence of this isomorphism.

Proofs of these string-theoretic predictions of curve-counting invariants were given by Givental [15] and Lian, Liu and Yau [58], with successively simpler proofs by many other researchers. However, all the proofs relied on the geometry of the ambient space $\mathbb{P}^{4}$ in which the quintic is contained. Roughly speaking, one considers all rational curves in $\mathbb{P}^{4}$, and tries to understand how to compute how many of these are contained in a given quintic hypersurface.

This raised the question: is there some underlying intrinsic geometry to mirror symmetry?

Historically the first approach to an intrinsic formulation of mirror symmetry is Kontsevich’s Homological Mirror Symmetry conjecture, stated in 1994 in [51]. This made mathematically precise the notion of an isomorphism between the $A$- and $B$-models. The homological mirror symmetry conjecture posits an isomorphism between two categories, the Fukaya category of Lagrangian submanifolds of $X$ (the $A$-model) and the derived category of coherent sheaves on the mirror $\check{X}$ (the $B$-model). Morally, this states that the symplectic geometry of $X$ is the same as the complex geometry of $\check{X}$. At the time this conjecture was made, however, there was no clear idea as to how such an isomorphism might be realised, nor did this conjecture state how to construct mirror pairs.

The second approach is due to Strominger, Yau and Zaslow in their 1996 paper [74]. They made a remarkable proposal, based on new ideas in string theory, which gave a very concrete geometric interpretation for mirror symmetry.

Let me summarize, very roughly, the physical argument they used here. Developments in string theory in the mid-1990s introduced the notion of Dirichlet branes, or $D$-branes. These are submanifolds of space-time, with some additional data, which serve as boundary conditions for open strings, i.e., we allow open strings to propagate with their endpoints constrained to lie on a $D$-brane. Remembering that space-time, according to string theory, looks like $\mathbb{R}^{1,3}\times X$, where $\mathbb{R}^{1,3}$ is ordinary space-time and $X$ is a Calabi-Yau three-fold, we can split a $D$-brane into a product of a submanifold of $\mathbb{R}^{1,3}$ and one on $X$. It turned out, simplifying a great deal, that there are two particular types of submanifolds on $X$ of interest: holomorphic $D$-branes, i.e., holomorphic submanifolds with a holomorphic line bundle, and special Lagrangian $D$-branes, which are special Lagrangian submanifolds with flat $U(1)$-bundle:

Holomorphic $D$-branes can be viewed as $B$-model objects, and special Lagrangian $D$-branes as $A$-model objects. The isomorphism between the $B$-model on $X$ and the $A$-model on $\check{X}$ then suggests that the moduli space of holomorphic $D$-branes on $X$ should be isomorphic to the moduli space of special Lagrangian $D$-branes on $\check{X}$. (This is now seen as a physical manifestation of the homological mirror symmetry conjecture). Now $X$ itself is the moduli space of points on $X$. So each point on $X$ should correspond to a pair $(M,\nabla)$, where $M\subseteq\check{X}$ is a special Lagrangian submanifold and $\nabla$ is a flat $U(1)$-connection on $M$.

A theorem of McLean [60] tells us that the tangent space to the moduli space of special Lagrangian deformations of a special Lagrangian submanifold $M\subseteq\check{X}$ is $H^{1}(M,\mathbb{R})$. Of course, the moduli space of flat $U(1)$-connections modulo gauge equivalence on $M$ is the torus $H^{1}(M,\mathbb{R})/H^{1}(M,\mathbb{Z})$. In order for this moduli space to be of the correct dimension, we need $\dim H^{1}(M,\mathbb{R})=n$, the complex dimension of $X$. This suggests that $X$ consists of a family of tori which are dual to a family of special Lagrangian tori on $\check{X}$. An elaboration of this argument yields the following conjecture:

This conjecture motivated a great deal of work in the five years following its introduction in 1996, some of which will be summarized in the following sections. There was a certain amount of success, as we shall see, with the conjecture proved for some cases, including the quintic three-fold, at the topological level. Further, the conjecture gave a solid framework for thinking about mirror symmetry at an intuitive level. However, work of Dominic Joyce demonstrated that the conjecture was unlikely to be literally true. Nevertheless, it is possible that weaker limiting forms of the conjecture still hold.

In the first several sections of this survey, I will clarify the conjecture, review what is known about it, and state a weaker form which seems accessible. Most importantly, I will explain how the SYZ conjecture leads to the study of affine manifolds (manifolds with transition functions being affine linear) and hence to an algebro-geometric interpretation of the conjecture, developed by me and Bernd Siebert. This removes the hard analysis, and gives a powerful framework for understanding mirror symmetry at a conceptual level.

The bulk of the paper is devoted to outlining this framework as developed over the last ten years. I explain how affine manifolds are related to degenerations of Calabi-Yau manifolds. Once one begins to consider degenerations, log geometry of K. Kato and Fontaine–Illusie comes into the picture. Conjecturally, the base of the SYZ fibration incorporates key combinatorial information about log structures on degenerations of Calabi-Yau manifolds. Log geometry then gives a connection with tropical geometry and log Gromov-Witten theory, which theoretically allows a description of $A$-model curve counting using tropical geometry. On the mirror side, we explain how again tropical geometry is used to describe complex structures. This identifies tropical geometry as the geometry underlying both sides of mirror symmetry, and guides us towards a conceptual understanding of mirror symmetry. We end with a description of recent work with Pandharipande and Siebert [27] which provides a snapshot of the relationship between the two sides of mirror symmetry.

I would like to thank the organizers of Current Developments in Mathematics 2012 for inviting me to take part in the conference, and Bernd Siebert, my collaborator on much of the work described here. Some of the material appearing in this article was first published in my article “The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations,” in Algebraic Geometry: Seattle 2005, edited by D. Abramovich, et al., Proceedings of Symposia in Pure Mathematics Vol. 80, part 1, 149-192, published by the American Mathematical Society. (c) 2009 by the American Mathematical Society. Finally, I would like to thank Lori Lejeune and the Clay Institute for Figure 3.

The first step in understanding the SYZ conjecture is to examine the structures which arise on the base of a special Lagrangian fibration. These structures arise from McLean’s theorem on the moduli space of special Lagrangian submanifolds [60], and these structures and their relationships were explained by Hitchin in [41]. We outline some of these ideas here. McLean’s theorem says that the moduli space of deformations of a compact special Lagrangian submanifold of a compact Calabi-Yau manifold $X$ is unobstructed. Further, the tangent space at the point of moduli space corresponding to a special Lagrangian $M\subseteq X$ is canonically isomorphic to the space of harmonic $1$-forms on $M$. This isomorphism is seen explicitly as follows. Let $\nu\in\Gamma(M,N_{M/X})$ be a normal vector field to $M$ in $X$. Then the restriction of the contractions $(\iota(\nu)\omega)|_{M}$ and $(\iota(\nu)\operatorname{Im}\Omega)|_{M}$ are both seen to be well-defined forms on $M$: one needs to lift $\nu$ to a vector field but the choice is irrelevant because $\omega$ and $\operatorname{Im}\Omega$ restrict to zero on $M$. McLean shows that if $M$ is special Lagrangian then

where $*$ denotes the Hodge star operator on $M$. Furthermore, $\nu$ corresponds to an infinitesimal deformation preserving the special Lagrangian condition if and only if $d(\iota(\nu)\omega)=d(\iota(\nu)\operatorname{Im}\Omega)=0$. This gives the correspondence between harmonic $1$-forms and infinitesimal special Lagrangian deformations.

Let $f:X\rightarrow B$ be a special Lagrangian fibration with torus fibres, and assume for now that all fibres of $f$ are non-singular. Then we obtain three structures on $B$, namely two affine structures and a metric, as we shall now see.

Hessian and Monge-Ampére metrics were first discussed by Cheng and Yau in [12].

We obtain the three structures as follows:

Affine structure 1. For a normal vector field $\nu$ to a fibre $X_{b}$ of $f$, $(\iota(\nu)\omega)|_{X_{b}}$ is a well-defined $1$-form on $X_{b}$, and we can compute its periods as follows. Let $U\subseteq B$ be a small open set, and suppose we have submanifolds $\gamma_{1},\ldots,\gamma_{n}\subseteq f^{-1}(U)$ which are families of 1-cycles over $U$ and such that $\gamma_{1}\cap X_{b},\ldots,\gamma_{n}\cap X_{b}$ form a basis for $H_{1}(X_{b},\mathbb{Z})$ for each $b\in U$. Consider the $1$-forms $\omega_{1},\ldots,\omega_{n}$ on $U$ defined by fibrewise integration:

for $\nu$ a tangent vector on $B$ at $b$, which we can lift to a normal vector field of $X_{b}$. We have $\omega_{i}=f_{*}(\omega|_{\gamma_{i}})$, and since $\omega$ is closed, so is $\omega_{i}$. Thus there are locally defined functions $y_{1},\ldots,y_{n}$ on $U$ with $dy_{i}=\omega_{i}$. Furthermore, these functions are well-defined up to the choice of basis of $H_{1}(X_{b},\mathbb{Z})$ and constants. Finally, they give well-defined coordinates, as follows from the fact that $\nu\mapsto\iota(\nu)\omega$ yields an isomorphism of $\mathcal{T}_{B,b}$ with $H^{1}(X_{b},\mathbb{R})$ by McLean’s theorem. Thus $y_{1},\ldots,y_{n}$ define local coordinates of a tropical affine structure on $B$.

Affine structure 2. We can play the same trick with $\operatorname{Im}\Omega$: choose submanifolds

which are families of $n-1$-cycles over $U$ and such that $\Gamma_{1}\cap X_{b},\ldots,\Gamma_{n}\cap X_{b}$ form a basis for $H_{n-1}(X_{b},\mathbb{Z})$. We define $\lambda_{i}$ by $\lambda_{i}=-f_{*}(\operatorname{Im}\Omega|_{\Gamma_{i}})$, or equivalently,

Again $\lambda_{1},\ldots,\lambda_{n}$ are closed $1$-forms, with $\lambda_{i}=d\check{y}_{i}$ locally, and again $\check{y}_{1},\ldots,\check{y}_{n}$ are affine coordinates for a tropical affine structure on $B$.

The McLean metric. The Hodge metric on $H^{1}(X_{b},\mathbb{R})$ is given by

for $\alpha$, $\beta$ harmonic $1$-forms, and hence induces a metric on $B$, which can be written as

A crucial observation of Hitchin [41] is that these structures are related by the Legendre transform:

Thus we introduce the notion of the Legendre transform of an affine manifold with a multi-valued convex function.

Curiously, this Legendre transform between affine manifolds with Hessian metric seems to have first appeared in a work in statistics predating mirror symmetry, see [2].

Let’s forget about special Lagrangian fibrations for the moment. Instead, we will look at how the structures found on $B$ in the previous section give a toy version of mirror symmetry.

We write $f:X(B)\rightarrow B$ and $\check{f}:\check{X}(B)\rightarrow B$ for these torus fibrations; these are clearly dual.

Now suppose in addition we have a Hessian metric $g$ on $B$, with local potential function $K$. Then the following propositions show that in fact both $X(B)$ and $\check{X}(B)$ become Kähler manifolds.

We write this Kähler manifold as $X(B,K)$.

Dually we have

As before, we call this Kähler manifold $\check{X}(B,K)$.

This motivates the definition

Monge-Ampère manifolds were first studied by Cheng and Yau in [12].

There is a key extra parameter which appears in mirror symmetry known as the $B$-field. This appears as a field in the non-linear sigma model with Calabi-Yau target space, and is required mathematically to make sense of mirror symmetry. Mirror symmetry roughly posits an isomorphism between the complex moduli space of a Calabi-Yau manifold $X$ and the Kähler moduli space of $\check{X}$. If one interprets the Kähler moduli space to mean the space of all Ricci-flat Kähler forms on $\check{X}$, then one obtains only a real manifold as moduli space, and one needs a complex manifold to match up with the complex moduli space of $X$. The $B$-field is interpreted as an element ${\bf B}\in H^{2}(\check{X},\mathbb{R}/\mathbb{Z})$, and one views ${\bf B}+i\omega$ as a complexified Kähler class on $\check{X}$ for $\omega$ a Kähler class on $\check{X}$.

In the context of our toy version of mirror symmetry, we view the $B$-field as an element ${\bf B}\in H^{1}(B,\Lambda_{\mathbb{R}}/\Lambda)$, where $\Lambda_{\mathbb{R}}=\Lambda\otimes_{\mathbb{Z}}\mathbb{R}$. This does not quite agree with the above definition of the $B$-field, as this group does not necessarily coincide with $H^{2}(\check{X},\mathbb{R}/\mathbb{Z})$. However, in many important cases, such as for simply connected Calabi-Yau threefolds with torsion-free integral cohomology, these two groups do coincide. More generally, including the case of K3 surfaces and abelian varieties, one would need to pass to generalized complex structures [42], [38], [7], [3], [43], which we do not wish to do here.

Noting that a section of $\Lambda_{\mathbb{R}}/\Lambda$ over an open set $U$ can be viewed as a section of $\mathcal{T}_{U}/\Lambda|_{U}$, such a section acts on $\mathcal{T}_{U}/\Lambda|_{U}$ via translation, and this action is in fact holomorphic with respect to the semi-flat complex structure. Thus a Čech 1-cocycle $(U_{ij},\beta_{ij})$ representing ${\bf B}$ allows us to reglue $X(B)$ via translations over the intersections $U_{ij}$. This is done by identifying the open subsets $f^{-1}(U_{ij})\subseteq f^{-1}(U_{i})$ and $f^{-1}(U_{ij})\subseteq f^{-1}(U_{j})$ via the automorphism of $f^{-1}(U_{ij})$ given by translation by the section $\beta_{ij}$. This gives a new complex manifold $X(B,{\bf B})$. If in addition there is a multi-valued potential function $K$ defining a metric, these translations preserve the metric and yield a Kähler manifold $X(B,{\bf B},K)$.

Thus the full toy version of mirror symmetry is as follows:

This provides a reasonably fulfilling picture of mirror symmetry in a simple context. Many more aspects of mirror symmetry can be worked out in this semi-flat context, see [54] and [3], Chapter 6. This semi-flat case is an ideal testing ground for concepts in mirror symmetry. However, ultimately this only sheds limited insight into the general case. The only compact Calabi-Yau manifolds with semi-flat Ricci-flat metric which arise in this way are complex tori (shown by Cheng and Yau in [12]). To deal with more interesting cases, we need to allow singular fibres, and hence, singularities in the affine structure of $B$. The existence of singular fibres are fundamental for the most interesting aspects of mirror symmetry.

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

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

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

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

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.

Recalling that mirror symmetry is supposed to allow us to count curves, let us discuss at an intuitive level how the picture so far gives us insight into this question. Let $B$ be a tropical affine manifold. Then as we saw, $X(B)$ carries the semi-flat complex structure, and it is easy to describe some complex submanifolds of $X(B)$ as follows. Let $L\subseteq B$ be a linear subspace with rational slope, i.e., the tangent space $\mathcal{T}_{L,b}$ to $L$ at any $b\in L$ can be written as $M\otimes_{\mathbb{Z}}\mathbb{R}$ for some sublattice $M\subseteq\Lambda_{b}$. Then we obtain a submanifold

One checks easily that this is a complex submanifold. For example, if $B=\mathbb{R}^{n}$, so that $X(B)$ is just an algebraic torus $(\mathbb{C}^{*})^{n}$ with coordinates $q_{1},\ldots,q_{n}$, and $L\subseteq B$ is a codimension $p$ affine linear subspace defined by equations

with $c_{ij}\in\mathbb{Z}$, $d_{i}\in\mathbb{R}$, then the corresponding submanifold of $X(B)$ is the subtorus given by the equations

Of course, subtori of tori are not particularly interesting. How might we build more complicated submanifolds? Let us focus on curves, where we take the linear submanifolds of $B$ to be of dimension one. Then if we take $L$ to be a line segment, ray, or line, $X(L)$ is a cylinder, with or without boundary in the various cases. We can then try to glue such cylinders together to obtain more complicated curves. For example, imagine we are given rays meeting at a point $b\in B=\mathbb{R}^{2}$ as pictured in Figure 2. Take primitive integral tangent vectors $v_{1},v_{2},v_{3}\in\mathbb{R}^{2}$ to $L_{1},L_{2}$ and $L_{3}$ pointing outwards from the point $b$ where the three segments intersect. Now we have the three cylinders $X(L_{i})$ which do not match up over $b$: the fibre $f^{-1}(b)=\mathbb{R}^{2}/\mathbb{Z}^{2}$ intersects $X(L_{i})$ in a circle $\mathbb{R}v_{i}/\mathbb{Z}v_{i}$. These circles are represented in $H_{1}(f^{-1}(b),\mathbb{Z})=\Lambda_{b}$ precisely by the vectors $v_{1},v_{2},v_{3}$, and so the condition that the circles bound a surface in $f^{-1}(b)$ is that $v_{1}+v_{2}+v_{3}=0$. Thus, if this condition holds, we can glue in a surface $S$ contained in $f^{-1}(b)$ so that $X(L_{1})\cup X(L_{2})\cup X(L_{3})\cup S$ now has no boundary at $b$. Of course, it is very far from being a holomorphic submanifold. The expectation, however, is that this sort of object can be deformed to a nearby holomorphic curve.

Precisely, continuing with the above example, suppose $b=0$ and $v_{1}=(1,0)$, $v_{2}=(0,1)$ and $v_{3}=(-1,-1)$. With holomorphic coordinates $q_{1},q_{2}$ on $X(B)=(\mathbb{C}^{*})^{2}$, consider the curve $C\subseteq(\mathbb{C}^{*})^{2}$ defined by $1+q_{1}+q_{2}=0$. Look at the image of this curve under the map $f:X(B)\rightarrow B$, which here can be written explicitly as $(q_{1},q_{2})\mapsto{-1\over 2\pi}(\log|q_{1}|,\log|q_{2}|)$. One finds that one obtains a thickening of the trivalent graph above, typically known as an amoeba. Further, if one considers not $X(B)$ but $X_{\epsilon}(B)$, where now holomorphic coordinates are given by $q_{j}=e^{2\pi i(x_{j}+iy_{j})/\epsilon}$ and $f_{\epsilon}:X_{\epsilon}(B)\rightarrow B$ is given by $(q_{1},q_{2})\mapsto-{\epsilon\over 2\pi}(\log|q_{1}|,\log|q_{2}|)$, one finds that as $\epsilon\rightarrow 0$, $f_{\epsilon}(C)$ converges to the trivalent graph in the above figure. In this sense the trivalent graph on $B$ is a limiting version of curves on a family of varieties tending towards a large complex structure limit.