I will now introduce the basic objects of the program developed by myself and Siebert to understand mirror symmetry in an algebro-geometric context. This program was announced in [28], and has been developed further in a series of papers [30], [31], [32], [22], [33], [27].

The motivation for this program came from two different directions. The first, which was largely my motivation, was the discussion of the limiting form of the SYZ conjecture of the previous sections. The second arose in work of Schröer and Siebert [71], [72], which led Siebert to the idea that log structures on degenerations of Calabi-Yau manifolds would allow one to view mirror symmetry as an operation performed on degenerate Calabi-Yau varieties. Siebert observed that at a combinatorial level, mirror symmetry exchanged data pertaining to the log structure and a polarization. This will be explained more clearly in the following section, when I introduce log structures. Together, Siebert and I realised that the combinatorial data he was considering could be encoded naturally in the dual intersection complex of the degeneration, which we saw in the previous section appears to be the base of the SYZ fibration. The combinatorial interchange of data necessary for mirror symmetry then corresponded to a discrete Legendre transform on the dual intersection complex. It became apparent that this approach provided an algebro-geometrization of the SYZ conjecture.

To set this up properly, one has to consider what kind of degenerations to allow. They should be maximally unipotent, of course, but there can be many different birational models of degenerations. Below we define the notion of toric degeneration. The class of toric degenerations may seem rather restrictive, but it appears to be the largest class of degenerations closed under mirror symmetry: one can construct the mirror of a toric degeneration as a toric degeneration. It does not appear that there is any other natural family of degenerations with this property. Much of the material in this section comes from [30], §4.

Roughly put, a toric degeneration of Calabi-Yau varieties is a degeneration whose central fibre is a union of toric varieties glued along toric strata, and the total space of the degeneration is, off of some well-behaved set $Z$ contained in the central fibre, locally toric with the family locally given by a monomial. The precise technical definition is as follows.

Given a toric degeneration $f:\mathcal{X}\rightarrow D$, we can build the dual intersection complex $(B,\mathscr{P})$ of $f$, as follows. Here $B$ is an integral affine manifold with singularities, and $\mathscr{P}$ is a polyhedral decomposition of $B$, i.e., a decomposition of $B$ into lattice polytopes. In fact, we will construct $B$ as a union of lattice polytopes. Specifically, let the normalisation of $\mathcal{X}_{0}$, $\widetilde{\mathcal{X}}_{0}$, be written as a disjoint union $\coprod X_{i}$ of toric varieties $X_{i}$, $\nu:\widetilde{\mathcal{X}}_{0}\rightarrow\mathcal{X}_{0}$ the normalisation. The strata of $\mathcal{X}_{0}$ are the elements of the set

$$Strata(\mathcal{X}_{0})=\{\nu(S)\,|\,\hbox{$S$ is a toric stratum of $X_{i}$ for some $i$}\}.$$

Here by toric stratum we mean the closure of a $(\mathbb{C}^{*})^{n}$ orbit.

Let $\{x\}\in Strata(\mathcal{X}_{0})$ be a zero-dimensional stratum. Applying Definition 7.1,(4), to a neighbourhood of $x$, there is a toric variety $Y_{x}$ such that in a neighbourhood of $x$, $f:\mathcal{X}\rightarrow D$ is locally isomorphic to $f_{x}:Y_{x}\rightarrow\mathbb{C}$, where $f_{x}$ is given by a monomial. Now the condition that $f_{x}$ vanishes precisely once along each toric divisor of $Y_{x}$ is the statement that $Y_{x}$ is Gorenstein, and as such, it arises as in Exercise 6.1. Indeed, let $M,N$ be given in Exercise 6.1, with $\operatorname{rank}M=\dim\mathcal{X}_{0}$. Then there is a lattice polytope $\sigma_{x}\subseteq M_{\mathbb{R}}$ such that $C(\sigma_{x})=\{(rm,r)|m\in\sigma,r\geq 0\}$ is the cone defining the toric variety $Y_{x}$. As we saw in Exercise 6.1, a small neighbourhood of $x$ in $\mathcal{X}$ should contribute a copy of $\sigma_{x}$ to $B$, which provides the motivation for our construction. We can now describe how to construct $B$ by gluing together the polytopes

$$\{\sigma_{x}\,|\,\{x\}\in Strata(\mathcal{X}_{0})\}.$$

We will do this in the case that every irreducible component of $\mathcal{X}_{0}$ is in fact itself normal so that $\nu:X_{i}\rightarrow\nu(X_{i})$ is an isomorphism. The reader may be able to imagine the more general construction. With this normality assumption, there is a one-to-one inclusion reversing correspondence between faces of $\sigma_{x}$ and elements of $Strata(\mathcal{X}_{0})$ containing $x$. We can then identify faces of $\sigma_{x}$ and $\sigma_{x^{\prime}}$ if they correspond to the same strata of $\mathcal{X}_{0}$. Some argument is necessary to show that this identification can be done via an integral affine transformation, but again this is not difficult.

Making these identifications, one obtains $B$. One can then prove

See [30], Proposition 4.10 for a proof.

Now so far $B$ is just a topological manifold, constructed by gluing together lattice polytopes. Let

$$\mathscr{P}=\{\sigma\subseteq B|\hbox{$\sigma$ is a face of $\sigma_{x}$ for some zero-dimensional stratum $x$}\}.$$

There is a one-to-one inclusion reversing correspondence between strata of $\mathcal{X}_{0}$ and elements of $\mathscr{P}$.

It only remains to give $B$ an affine structure with singularities. In fact, I shall describe somewhat more structure on $B$ derived from $\mathcal{X}_{0}$ which in particular gives an affine structure with singularities on $B$.

First, for $\tau\in\mathscr{P}$, let

$$U_{\tau}:=\bigcup_{\{\sigma\in\mathscr{P}\,|\,\tau\subseteq\sigma\}}\operatorname{Int}(\sigma).$$

A fan structure along $\tau\in\mathscr{P}$ is a continuous map $S_{\tau}:U_{\tau}\rightarrow\mathbb{R}^{k}$ such that

1. (1)

   $S_{\tau}^{-1}(0)=\operatorname{Int}(\tau)$.

2. (2)

   If $e:\tau\rightarrow\sigma$ is an inclusion then $S_{\tau}|_{\operatorname{Int}\sigma}$ is an integral affine submersion onto its image.

3. (3)

   The collection of cones

   $$\{K_{e}:=\mathbb{R}_{\geq 0}S_{\tau}(\sigma\cap U_{\tau})\,|\,e:\tau\rightarrow\sigma\}$$

   defines a finite fan $\Sigma_{\tau}$ in $\mathbb{R}^{k}$.

Two fan structures $S_{\tau},S^{\prime}_{\tau}:U_{\tau}\rightarrow\mathbb{R}^{k}$ are considered equivalent if they differ only by an integral linear transformation of $\mathbb{R}^{k}$.

If $S_{\tau}:U_{\tau}\rightarrow\mathbb{R}^{k}$ is a fan structure along $\tau\in\mathscr{P}$ and $\sigma\supseteq\tau$ then $U_{\sigma}\subseteq U_{\tau}$. The fan structure along $\sigma$ induced by $S_{\tau}$ is the composition

$$U_{\sigma}\smash{\mathop{\longrightarrow}\limits}U_{\tau}\smash{\mathop{\longrightarrow}\limits^{S_{\tau}}}\mathbb{R}^{k}\smash{\mathop{\longrightarrow}\limits}\mathbb{R}^{k}/L_{\sigma}\cong\mathbb{R}^{\ell}$$

where $L_{\sigma}\subseteq\mathbb{R}^{k}$ is the linear span of $S_{\tau}(\sigma)$.

Such data gives $B$ the structure of an integral affine manifold with singularities. Let $\Gamma\subseteq B$ be the union of those cells of $\operatorname{Bar}(\mathscr{P})$ (the first barycentric subdivision of $\mathscr{P}$) which are not contained in maximal cells of $\mathscr{P}$ nor contain vertices of $\mathscr{P}$. Then $B_{0}:=B\setminus\Gamma$ can be covered by

$$\{\operatorname{Int}(\sigma)\,|\,\sigma\in\mathscr{P}_{\max}\}\cup\{W_{v}\,|\,\hbox{$v\in\mathscr{P}$ a vertex}\}$$

for certain open neighbourhoods $W_{v}$ of $v\in B$ contained in $U_{v}$. We define an affine structure on $B_{0}$ by giving $\operatorname{Int}(\sigma)$ the natural affine structure given by $\sigma$ being a lattice polytope, while $S_{v}:U_{v}\rightarrow\mathbb{R}^{n}$ restricts to an affine chart on $W_{v}$.

Finally, the point is that the structure of $\mathcal{X}_{0}$ gives rise to an integral tropical manifold structure on $(B,\mathscr{P})$. Indeed, each vertex $v\in\mathscr{P}$ corresponds to an irreducible component $X_{v}$ of $\mathcal{X}_{0}$ and this irreducible component is a toric variety with fan $\Sigma_{v}$ in $\mathbb{R}^{n}$. Furthermore, there is a one-to-one correspondence between $p$-dimensional cones of $\Sigma_{v}$ and $p$-dimensional cells of $\mathscr{P}$ containing $v$ as a vertex, as they both correspond to strata of $\mathcal{X}_{0}$ contained in $X_{v}$. There is then a continuous map

$$\psi_{v}:U_{v}\rightarrow\mathbb{R}^{n}$$

which takes $U_{v}\cap\sigma$, for any $\sigma\in\mathscr{P}$ containing $v$ as a vertex, into the corresponding cone of $\Sigma_{v}$ integral affine linearly. Such a map is uniquely determined by the combinatorial correspondence and the requirement that it be integral affine linear on each cell. These maps define a fan structure at each vertex. Furthermore, these fan structures are compatible in the sense that if $v,w\in\tau$, the two induced fan structures on $U_{\tau}$ are equivalent. This follows because there is a well-defined fan $\Sigma_{\tau}$ defining the stratum corresponding to $\tau$.

Is the dual intersection complex the right affine manifold with singularities? The following theorem provides evidence for this, and gives the connection between this construction and the SYZ conjecture.

We will not be precise here about what we mean by small deformation; by twist, we mean a twist of the complex structure of $X_{\epsilon}(U)$ by a $B$-field. See [28] for a much more precise statement; the above statement is meant to give a feel for what is true. The proof, along with much more precise statements, will eventually appear in [29].

If $\mathcal{X}\rightarrow D$ is a polarized toric degeneration, i.e., if there is a relatively ample line bundle $\mathcal{L}$ on $\mathcal{X}$, then we can construct another integral tropical manifold $(\check{B},\check{\mathscr{P}})$, which we call the intersection complex, as follows.

For each irreducible component $X_{i}$ of $\mathcal{X}_{0}$, $\mathcal{L}|_{X_{i}}$ is an ample line bundle on a toric variety. Let $\check{\sigma}_{i}\subseteq N_{\mathbb{R}}$ denote the Newton polytope of this line bundle. There is then a one-to-one inclusion preserving correspondence between strata of $\mathcal{X}_{0}$ contained in $X_{i}$ and faces of $\check{\sigma}_{i}$. We can then glue together the $\check{\sigma}_{i}$’s in the obvious way: if $Y$ is a codimension one stratum of $\mathcal{X}_{0}$, it is contained in two irreducible components $X_{i}$ and $X_{j}$, and defines faces of $\check{\sigma}_{i}$ and $\check{\sigma}_{j}$. These faces are affine isomorphic because they are both the Newton polytope of $\mathcal{L}|_{Y}$, and we can then identify them in the canonical way. Thus we obtain a topological space $\check{B}$ with a polyhedral decomposition $\check{\mathscr{P}}$.

To define the fan structure at a vertex $v\in\mathscr{P}$, note that such a vertex corresponds to a zero-dimensional stratum of $\mathcal{X}_{0}$, giving rise to a maximal cell $\sigma_{v}$ of the dual intersection complex. Take the fan structure at $v$ to be defined using the normal fan $\check{\Sigma}_{v}$ to $\sigma_{v}$. Then there is a one-to-one inclusion preserving correspondence between cones in $\check{\Sigma}_{v}$ and strata of $\mathcal{X}_{0}$ containing the stratum corresponding to $v$. This correspondence allows us to define a fan structure

$$S_{v}:U_{v}\rightarrow\mathbb{R}^{n}$$

which takes $U_{v}\cap\check{\sigma}$, for any $\check{\sigma}\in\check{\mathscr{P}}$ containing $v$ as a vertex, into the corresponding cone of $\check{\Sigma}_{v}$. One checks easily that this set of fan structures satisfies the definition of integral tropical manifold, and hence defines the intersection complex $(\check{B},\check{\mathscr{P}})$.

Analogously to Theorem 7.8, we expect

I don’t expect this to be particularly difficult: it should be amenable to the techniques of W.-D. Ruan [70], but such an approach has not been carried out in general.

The relationship between the intersection complex and the dual intersection complex can be made more precise by introducing multi-valued piecewise linear functions, in analogy with the multi-valued convex functions of Definition 1.3.

The rather technical condition on the local behaviour of each $\varphi_{i}$ on $U_{\tau}$ comes from the idea that such a multi-valued piecewise linear function is really just a collection of piecewise linear functions on the fans $\Sigma_{\tau}$ given by the fan structure of $(B,\mathscr{P})$. These functions need to satisfy some compatibility conditions, and this compatibility is motivated by the following discussion.

Suppose we are given a polarized toric degeneration $\mathcal{X}\rightarrow D$. We in fact obtain a multi-valued piecewise linear function $\varphi$ on the dual intersection complex $(B,\mathscr{P})$ as follows. Restricting to any toric stratum $X_{\tau}$, $\mathcal{L}|_{X_{\tau}}$ is determined completely by an integral piecewise linear function $\varphi_{\tau}$ on $\Sigma_{\tau}$, well-defined up to a choice of linear function. Pulling back this piecewise linear function via $S_{\tau}$ to $U_{\tau}$, we obtain a collection of piecewise linear functions $\{(U_{\tau},\varphi_{\tau}\circ S_{\tau})\,|\,\tau\in\mathscr{P}\}$. The fact that $(\mathcal{L}|_{X_{\tau}})|_{X_{\sigma}}=\mathcal{L}|_{X_{\sigma}}$ for $\tau\subseteq\sigma$ implies that on overlaps $\varphi_{\sigma}\circ S_{\sigma}$ and $\varphi_{\tau}\circ S_{\tau}$ differ by at most an affine linear function. So $\{(U_{\tau},\varphi_{\tau}\circ S_{\tau})\}$ defines a multi-valued piecewise linear function. The last condition in the definition of multi-valued piecewise linear function then reflects the need for the function to be locally a pull-back of a function via $S_{\sigma}$ in a neighbourhood of $\sigma$.

If $\mathcal{L}$ is ample, then the piecewise linear function determined by $\mathcal{L}|_{X_{\sigma}}$ is strictly convex. So we say a multi-valued piecewise linear function is strictly convex if $\varphi_{\tau}$ is strictly convex for each $\tau\in\mathscr{P}$.

As a consequence, if $\mathcal{X}\rightarrow D$ is a polarized toric degeneration, we will write $(B,\mathscr{P},\varphi)$ for the data of the dual intersection complex and the induced multi-valued function $\varphi$. We call this triple the dual intersection complex of the polarized degeneration.

Now suppose we are given abstractly a triple $(B,\mathscr{P},\varphi)$ with $(B,\mathscr{P})$ an integral tropical manifold and $\varphi$ a strictly convex multi-valued piecewise linear function on $B$. Then we construct the discrete Legendre transform $(\check{B},\check{\mathscr{P}},\check{\varphi})$ of $(B,\mathscr{P},\varphi)$ as follows.

$\check{B}$ will be constructed by gluing together Newton polytopes. If we view, for $v$ a vertex of $\mathscr{P}$, the fan $\Sigma_{v}$ as living in $M_{\mathbb{R}}$, then the Newton polytope of $\varphi_{v}$ is

$$\check{v}:=\{x\in N_{\mathbb{R}}\,|\,\langle x,y\rangle\geq-\varphi_{v}(y)\quad\forall y\in M_{\mathbb{R}}\}.$$

There is a one-to-one inclusion reversing correspondence between faces of $\check{v}$ and cells of $\mathscr{P}$ containing $v$. Furthermore, if $\sigma$ is the smallest cell of $\mathscr{P}$ containing two vertices $v$ and $v^{\prime}$, then the corresponding faces of $\check{v}$ and $\check{v}^{\prime}$ are integral affine isomorphic, as they are both isomorphic to the Newton polytope of $\varphi_{\sigma}$. Thus we can glue $\check{v}$ and $\check{v}^{\prime}$ along this common face. After making all these identifications, we obtain a cell complex $(\check{B},\check{\mathscr{P}})$, which is really just the dual cell complex of $(B,\mathscr{P})$. This is given an integral tropical structure by taking the fan structure at a vertex $\check{\sigma}$, for $\sigma\in\mathscr{P}_{\max}$, to be given by the normal fan to $\sigma$.

Finally, the function $\varphi$ has a discrete Legendre transform $\check{\varphi}$ on $(\check{B},\check{\mathscr{P}})$. We have no choice but to define $\check{\varphi}$ in a neighbourhood of a vertex $\check{\sigma}\in\check{\mathscr{P}}$ dual to a maximal cell $\sigma\in\mathscr{P}$ to be a piecewise linear function whose Newton polytope is $\sigma$, i.e.,

$$\check{\varphi}_{\check{\sigma}}(y)=-\inf\{\langle y,x\rangle\,|\,x\in\sigma\subseteq M_{\mathbb{R}}\}.$$

This gives $(\check{B},\check{\mathscr{P}},\check{\varphi})$, the discrete Legendre transform of $(B,\mathscr{P},\varphi)$. If $B$ is $\mathbb{R}^{n}$, then this coincides with the classical notion of discrete Legendre transform. The discrete Legendre transform has several relevant properties:

• •

  The discrete Legendre transform of $(\check{B},\check{\mathscr{P}},\check{\varphi})$ is $(B,\mathscr{P},\varphi)$.

• •

  If we view the underlying topological spaces $B$ and $\check{B}$ as identified by being the underlying space of dual cell complexes, then $\Lambda_{B_{0}}\cong\check{\Lambda}_{\check{B}_{0}}$ and $\check{\Lambda}_{B_{0}}\cong\Lambda_{\check{B}_{0}}$, where the subscript denotes which affine structure is being used to define $\Lambda$ or $\check{\Lambda}$.

This hopefully makes it clear that the discrete Legendre transform is a suitable replacement for the duality provided by the Legendre transform of §2.

Note in particular that if $\mathcal{X}\rightarrow D$ is a polarized toric degeneration, with dual intersection complex $(B,\mathscr{P},\varphi)$, then the discrete Legendre transform $(\check{B},\check{\mathscr{P}},\check{\varphi})$ satisfies the condition that $(\check{B},\check{\mathscr{P}})$ is the intersection complex of the polarized degeneration. The function $\check{\varphi}$ is some extra information on $\check{B}$, which from the definition of discrete Legendre transform encodes the cells of $\mathscr{P}$. These cells of the dual intersection complex were defined using the log structure on $\mathcal{X}_{0}^{\dagger}$. So $\check{\varphi}$ can be seen as carrying information about the log structure. We will say $(\check{B},\check{\mathscr{P}},\check{\varphi})$ is the intersection complex of the polarized toric degeneration $\mathcal{X}\rightarrow D$.

So we see that for $(B,\mathscr{P},\varphi)$, $\mathscr{P}$ carries information about the log structure and $\varphi$ carries information about the polarization, but for $(\check{B},\check{\mathscr{P}},\check{\varphi})$, $\check{\mathscr{P}}$ carries information about the polarization and $\check{\varphi}$ carries information about the log structure. Mirror symmetry interchanges these two pieces of information!

We can now state an algebro-geometric SYZ procedure. In analogy with the procedure suggested in §5, we could follow these steps:

1. (1)

   We begin with a toric degeneration of Calabi-Yau manifolds $\mathcal{X}\rightarrow D$ with an ample polarization.

2. (2)

   Construct the dual intersection complex $(B,\mathscr{P},\varphi)$ from this data, as explained above.

3. (3)

   Perform the discrete Legendre transform to obtain $(\check{B},\check{\mathscr{P}},\check{\varphi})$.

4. (4)

   Try to construct a polarized degeneration of Calabi-Yau manifolds $\check{\mathcal{X}}\rightarrow D$ whose dual intersection complex is $(\check{B},\check{\mathscr{P}},\check{\varphi})$, or whose intersection complex is $(B,\mathscr{P},\varphi)$.

The only step missing in this mirror symmetry algorithm is the last:

One could hope to solve this problem via naive deformation theory, by constructing the central fibre $\mathcal{X}_{0}$ from the data $(B,\mathscr{P},\varphi)$, and then deforming this to find a smoothing. However, as initially observed in the normal crossings case by Kawamata and Namikawa in [50], one needs to put some additional structure on $\mathcal{X}_{0}$ before it has good deformation theory. This structure is a log structure, and introducing log structures allows us to study many aspects of mirror symmetry directly on the degenerate fibre itself. So let us turn to a review of the theory of logarithmic structures.