The first link between log geometry and tropical geometry comes from an elementary combinatorial construction. Given a log scheme $X^{\dagger}$, we can construct the tropicalization of $X^{\dagger}$, as follows. For each geometric point $\bar{\eta}$ of $X$, we have a monoid $\overline{\mathcal{M}}_{X,\bar{\eta}}$, and hence a cone $C_{\bar{\eta}}:=\operatorname{Hom}(\overline{\mathcal{M}}_{X,\bar{\eta}},\mathbb{R}_{\geq 0})$ (where here $\operatorname{Hom}$ denotes monoid homomorphisms and $\mathbb{R}_{\geq 0}$ is given the additive monoid structure). Further, if $\bar{\eta}$ is in the closure of $\bar{\eta}^{\prime}$, there is a generization map $\overline{\mathcal{M}}_{X,\bar{\eta}}\rightarrow\overline{\mathcal{M}}_{X,\bar{\eta}^{\prime}}$.¹¹ 1 Since we need to work in the étale topology, there can actually be a number of generization maps. For example, if $X$ is a nodal cubic, then there are two generization maps from $\bar{\eta}$ the node to $\bar{\eta}^{\prime}$ the generic point. Dualizing, this gives maps $C_{\bar{\eta}^{\prime}}\rightarrow C_{\bar{\eta}}$. If the log structure on $X$ is fine, then these maps are inclusions of faces of strictly convex rational polyhedral cones. We can then form a cell complex by making identifications given by these inclusions of faces, obtaining a polyhedral cone complex $\mathrm{Trop}(X^{\dagger})$. Actually, in general this may not really make sense as a cell complex because the generization maps may induce many strange self-identifications on faces, but in the situations we want to describe here, this will not cause a problem.

This construction is functorial, so if $f:X^{\dagger}\rightarrow Y^{\dagger}$ is a morphism of log schemes, then we obtain $\mathrm{Trop}(f):\mathrm{Trop}(X^{\dagger})\rightarrow\mathrm{Trop}(Y^{\dagger})$.

For example, consider the case of a toric degeneration $\mathcal{X}\rightarrow D$. As we saw in the previous section, this gives a morphism of log schemes $\mathcal{X}_{0}^{\dagger}\rightarrow 0^{\dagger}$. The bad set $Z\subseteq\mathcal{X}_{0}$ is precisely the locus where the log structure on $\mathcal{X}_{0}$ is not fine. Thus we can apply the above tropicalization construction to $\mathcal{X}_{0}^{\dagger}\setminus Z\rightarrow 0^{\dagger}$. Now $\mathrm{Trop}(0^{\dagger})=\mathbb{R}_{\geq 0}$ is a ray. On the other hand, if $x\in\mathcal{X}_{0}$ is a zero-dimensional stratum, locally a neighbourhood of $x$ looks like the fibre over $0$ of $f_{x}:Y_{x}\rightarrow\mathbb{C}$ where $Y_{x}$ is a toric variety defined by $C(\sigma_{x})\subseteq M_{\mathbb{R}}\oplus\mathbb{R}$ for a lattice polytope $\sigma_{x}\subseteq M_{\mathbb{R}}$, and the morphism $Y_{x}\rightarrow\mathbb{C}$ is given by the projection $M_{\mathbb{R}}\oplus\mathbb{R}\rightarrow\mathbb{R}$. Then $\overline{\mathcal{M}}_{\mathcal{X}_{0},x}=C(\sigma_{x})^{\vee}\cap(N\oplus\mathbb{Z})$, as follows from Exercise 8.3, and $C_{x}=C(\sigma_{x})$. In particular, the induced map $\mathrm{Trop}(f):C_{x}\rightarrow\mathrm{Trop}(0^{\dagger})$ has fibre $\mathrm{Trop}(f)^{-1}(1)=\sigma_{x}$. From this, one checks easily that

$$\mathrm{Trop}(f):\mathrm{Trop}(\mathcal{X}_{0}^{\dagger}\setminus Z)\rightarrow\mathrm{Trop}(0^{\dagger})=\mathbb{R}_{\geq 0}$$

has fibre

$$\mathrm{Trop}(f)^{-1}(1)=B,$$

with $B$ coming with the polyhedral decomposition $\mathscr{P}$. So the dual intersection complex comes from a very general construction. In particular, note that $B$ only depends on $\mathcal{X}_{0}^{\dagger}$, not on $\mathcal{X}$ (although this is obvious without knowing this general construction).

Now let us turn to the $A$-model, which for the purposes of this discussion means counting curves on Calabi-Yau manifolds. Suppose we have a toric degeneration $\mathcal{X}\rightarrow D$. We would like to count curves on the general fibre. Can we do so by counting curves on $\mathcal{X}_{0}$ instead, where the problem might have a more combinatorial nature?

This question has a long history. The first work on this kind of question was due to Li and Ruan [57] and Ionel and Parker [45],[46]. Essentially they considered a situation where one has a degeneration $\mathcal{X}\rightarrow D$ where the special fibre $\mathcal{X}_{0}=X_{1}\cup X_{2}$ is a normal crossings union of two smooth irreducible components. They showed that there was a theory of Gromov-Witten invariants of $\mathcal{X}_{0}$, and that it gave the same answer as Gromov-Witten theory on a general fibre. Further, they gave gluing formulas, which stated that the Gromov-Witten invariants of $\mathcal{X}_{0}$ could be computed using the Gromov-Witten invariants of the two pairs $(X_{i},X_{1}\cap X_{2})$, $i=1,2$. Here the Gromov-Witten theory associated to a pair $(X,D)$ where $D\subseteq X$ is a smooth divisor is the theory of relative Gromov-Witten invariants, where one considers curves in $X$ with some imposed orders of tangency at points on the curve with $D$. This gluing formula has proven to be a very powerful tool in Gromov-Witten theory.

In 2001, Bernd Siebert [73] proposed using log geometry to generalize these results. Meanwhile, Jun Li was working on an algebro-geometric approach to the Li-Ruan and Ionel-Parker theories (which were carried out using symplectic techniques). He gave a satisfactory algebro-geometric definition of relative Gromov-Witten invariants and reproved the gluing formula, using a few techniques from log geometry. However, the theory possesses a technical difficulty. In Gromov-Witten theory, it is standard that one allows the domain curves to develop bubbles. But in relative Gromov-Witten theory, it is also necessary to allow the target space $X$ to develop bubbles. This occurs when an irreducible component of the domain curve falls into the divisor $D$, so that the order of tangency with $D$ becomes meaningless. So the actual target space for a relative stable map might be $X$ with a chain of $\mathbb{P}^{1}$-bundles over $D$ glued to $D\subseteq X$. This often makes the analysis more difficult, and was a major stumbling block for extending these techniques to more complicated degenerations.

Several solutions to this problem were completed in 2011. Brett Parker in [65], [66] provided a completely new category, the category of exploded manifolds, in which to study Gromov-Witten theory. These manifolds carry information similar to log spaces, but is a somewhat more flexible and “softer” category in which to work. In [66] he provides a definition of Gromov-witten invariants in this setting and gives a gluing formula. Also, Siebert and I [33] completed a theory of logarithmic Gromov-Witten invariants, as did Abramovich and Chen [11],[1], working with Siebert’s original suggestion. I will summarize the basic ideas here.

Here log smoothness implies that the geometric fibres of $C\rightarrow W$ are nodal curves, which is pleasant as this is precisely the sort of curve which is allowed as the domain of a stable map. The log structure can also be viewed as incorporating marked points. For example, given a smooth curve $C$ over $W=\operatorname{Spec}\mathbb{C}$, one can take a finite number of points $x_{1},\ldots,x_{k}\in C$ and give $C$ the divisorial log structure associated to the subset $\{x_{1},\ldots,x_{k}\}\subseteq C$. Then $C^{\dagger}$ is log smooth over $W$ with the trivial log structure $\mathcal{M}_{W}=\mathcal{O}_{W}^{\times}$.

The main work of [33] is to construct a well-behaved moduli space of stable log maps. There is a technical issue which arises whenever one tries to construct a moduli space of log objects; this was explored by Martin Olsson in his thesis [64]. The problem is as follows. Suppose we are given a stable log map with domain $\pi:(C,\mathcal{M}_{C})\rightarrow(W,\mathcal{M}_{W})$. Then $\pi^{\prime}:(C,\mathcal{M}_{C}\oplus\mathbb{N}^{r})\rightarrow(W,\mathcal{M}_{W}\oplus\mathbb{N}^{r})$ also gives the domain of a stable log map. Here the structure map $\alpha_{C}$ (or $\alpha_{W}$) takes the value $0$ on the non-zero elements of the constant sheaf $\mathbb{N}^{r}$, and the map $\pi^{\prime}$ acting on monoids just takes $\mathbb{N}^{r}$ isomorphically to $\mathbb{N}^{r}$. The new map $f^{\#}$ is the composition of the old $f^{\#}:f^{-1}\mathcal{M}_{X}\rightarrow\mathcal{M}_{C}$ and the inclusion $\mathcal{M}_{C}\rightarrow\mathcal{M}_{C}\oplus\mathbb{N}^{r}$. As a result, a single stable log map gives rise to a countable number of other maps, so the stack of stable log maps has no chance of being finite type, and hence cannot be proper.

The solution is to identify log structures on $W$ which are universal in a suitable sense. In the above example, all the log curves in question arise as a cartesian diagram of log schemes:

Thus all these extraneous log curves can be viewed as obtained by pull-back from the initial choice of log curve via a logarithmic base-change.

To solve this problem, we introduce a property of stable log maps called basic. I do not wish to give the definition here, as it is very involved, but the important properties of basic stable log maps are universality and boundedness, as expressed in the following two theorems, a summation of the main results of [33]:

Similar results were also obtained by Abramovich and Chen in [1],[11].

This is a promising start to the problem of understanding the $A$-model by working entirely on the central fibre of a toric degeneration. There are, however, still two major gaps in the theory which need to be filled.

First, one needs an analogue of the gluing formula. This should allow us to break down a calculation of curves on the central fibre of a degeneration into simpler pieces. This is expected to be quite subtle, however, and is still work in progress. I will say a bit more shortly about what one expects such a formula to look like.

Second, as observed earlier, the central fibre of a toric degeneration $\mathcal{X}_{0}^{\dagger}\rightarrow 0^{\dagger}$ is only fine saturated off of the set $Z$. As a result none of the above theorems about stable log maps apply. It is quite likely that even the definition of stable log map is not the correct one in this case. So the theory still needs to be extended. This is also work in progress of Michael Kasa.

Let us return to the tropicalization functor. Suppose we have a degeneration $q:\mathcal{X}\rightarrow D$, which we assume to be log smooth, so that we don’t have to worry about the singular set $Z$. As usual, this gives $\mathcal{X}_{0}^{\dagger}\rightarrow 0^{\dagger}$. Suppose we have a basic stable log map over a point, i.e., a diagram

Here $Q$ is a monoid given by $Q=\sigma_{Q}\cap\mathbb{Z}^{n}$ for a strictly convex rational polyhedral cone $\sigma_{Q}$, and the log structure on $W$ is given by $\alpha:Q\oplus\mathbb{C}^{\times}\rightarrow\mathbb{C}$ defined by

$$\alpha(p,s)=\begin{cases}s&p=0\\ 0&p\not=0\end{cases}$$

Here, the monoid $Q$ is determined by the fact the curve is basic. We then tropicalize this, so get a diagram

The fibres of $\mathrm{Trop}(\pi)$ are in general one-dimensional graphs, while $\mathrm{Trop}(q)^{-1}(1)$ is the dual intersection complex $B$ of $\mathcal{X}_{0}^{\dagger}$. (In general, this is only a polyhedral complex and does not carry an affine structure in codimension one, unlike the case of a toric degeneration.) Thus $\mathrm{Trop}(g)^{-1}(1)\subseteq\sigma_{Q}^{\vee}$ can be viewed as a space parameterizing maps from graphs (fibres of $\mathrm{Trop}(\pi)$) into $B$. These will be tropical curves. In fact, where $B$ does carry an affine structure, these curves satisfy the tropical balancing condition.

The fundamental property that the monoid $Q$ associated with the basic log structure must satisfy is that $\mathrm{Trop}(g)^{-1}(1)$ must parameterize all tropical curves in $B$ of the same “combinatorial type”. This makes precise the correspondence between tropical curves and log curves.

We can also describe the expected shape of a gluing formula, in keeping with the formula developed by Brett Parker in his setting [66]. One considers tropical curves in $B$ as above. These in general move in families, but there will be, for any given set of data $\beta$, a finite number of tropical curves representing $\beta$ which cannot be deformed without changing the domain graph. We call such tropical curves rigid. The actual moduli space $\mathscr{M}(\mathcal{X}_{0}^{\dagger}/0^{\dagger},\beta)$ can then be viewed to have a “decomposition into virtual irreducible components” indexed by these rigid curves. Furthermore, the “virtual irreducible component” associated to any rigid curve can be further related to moduli spaces of curves associated to each vertex of the tropical curve. This should ultimately allow an expression for the Gromov-Witten invariants of $\mathcal{X}_{0}^{\dagger}/0^{\dagger}$, and hence the Gromov-Witten invariants of a smoothing of $\mathcal{X}_{0}^{\dagger}$, in terms of much simpler invariants. This is an ongoing joint project with Abramovich, Chen and Siebert.