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

$$X(L):=\mathcal{T}_{L}/(\mathcal{T}_{L}\cap\Lambda)\subseteq X(B).$$

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

$$\sum_{j}c_{ij}y_{j}=d_{i},\quad 1\leq i\leq p,$$

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

$$\prod_{j}q_{j}^{c_{ij}}=e^{-2\pi d_{j}},\quad 1\leq i\leq p.$$

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.

This basic picture for curves in algebraic tori is now very well studied. In particular, this study spawned the subject of tropical geometry. The word tropical is motivated by the role that the tropical semiring plays. This is the semiring $(\mathbb{R},\oplus,\odot)$ where addition and multiplication are given by

	$\displaystyle a\oplus b:={}$	$\displaystyle\min(a,b)$
	$\displaystyle a\odot b:={}$	$\displaystyle a+b.$

The word “tropical” is used in honor of the Brazilian mathematician Imre Simon, who pioneered use of this semi-ring.

We now consider polynomials over the tropical semiring, as follows. Let $S\subseteq\mathbb{Z}^{n}$ be a finite subset, and consider tropical polynomials on $\mathbb{R}^{n}$ of the form

$$g:=\sum_{(p_{1},\ldots,p_{n})\in S}c_{p_{1}\ldots p_{n}}x_{1}^{p_{1}}\cdots x_{n}^{p_{n}}$$

where the coefficients lie in $\mathbb{R}$ and the operations are in the tropical semiring. Then $g$ is a convex piecewise linear function on $\mathbb{R}^{n}$, and the locus where $g$ is not linear is called a tropical hypersurface. In particular, in the case $n=2$, we obtain a tropical curve. In the example of Figure 2, the relevant tropical polynomial could be taken to be $0\oplus x_{1}\oplus x_{2}$.

While the tropical semiring has been used extensively in tropical geometry, it is not so convenient for us to view our tropical curves on $B$ as being defined by equations, since typically these curves will be of high codimension. Instead, it is better to follow Mikhalkin [61] and use parameterized tropical curves.

The domain of a parameterized tropical curve will be a weighted graph. In what follows, $\overline{\Gamma}$ will denote a connected graph. Such a graph can be viewed in two different ways. First, it can be viewed as a purely combinatorial object, i.e., a set $\overline{\Gamma}^{[0]}$ of vertices and a set $\overline{\Gamma}^{[1]}$ of edges consisting of unordered pairs of elements of $\overline{\Gamma}^{[0]}$, indicating the endpoints of an edge. We can also view $\overline{\Gamma}$ as the topological realization of the graph, i.e., a topological space which is the union of line segments corresponding to the edges. We shall confuse these two viewpoints at will. We will then denote by $\Gamma$ the topological space obtained from $\overline{\Gamma}$ by deleting the univalent vertices of $\overline{\Gamma}$, so that $\Gamma$ may have some non-compact edges.

We also take $\overline{\Gamma}$ to come with a weight function, a map

$$w:\overline{\Gamma}^{[1]}\rightarrow\mathbb{N}=\{0,1,2,\ldots\}.$$

Replacing $\mathbb{R}^{n}$ with a general tropical affine manifold $B$, we now arrive at the following definition:

Here the balancing condition is just expressing the topological requirement that the boundaries of the various cylinders $X(h(E_{i}))\subseteq X(B)$ can be connected up with a surface contained in the fibre of $X(B)\rightarrow B$ over $h(V)$. The weights can be interpreted as taking the cylinders $X(h(E_{i}))$ with multiplicity.

An important question then arises:

This question has attracted a great deal of attention when $B=\mathbb{R}^{n}$, with completely satisfactory results in the case $n=2$ (Answer: always), and less complete results when $n\geq 3$. The $n=2$ case was first treated by Mikhalkin [61], and resuts in all dimensions were first obtained by Nishinou and Siebert [63]. In particular, Mikhalkin proved that in this two-dimensional case, one can calculate numbers of curves of a given degree and genus passing through a fixed set of points, showing that difficult holomorphic enumerative problems can be solved by a purely combinatorial approach. This work gives hope that one can really count curves combinatorially in much more general settings. In the two-dimensional case, again, my own work [24] showed that the mirror side (for mirror symmetry for $\mathbb{P}^{2}$) could also be interpreted tropically, giving a completely tropical interpretation of mirror symmetry for $\mathbb{P}^{2}$.

So far we have not considered the case that $B$ has singularities. In case $B$ has singularities, we expect that one should be able to relax the balancing condition when a vertex falls inside of a point of the singular locus, and in particular one can allow univalent vertices which map to the singular locus. The reason for this is that once we compactify $X(B_{0})$ to $X(B)$, one expects to find holomorphic disks fibering over line segments emanating from singular points: see Figure 3 for a depiction of this when $B$ is two-dimensional, having isolated singularities.

We will avoid giving a precise definition of what a tropical curve should mean in the case that $B$ has singularities, largely because it is not clear yet what the precise definition should be. Hopefully, though, this discussion makes it clear that at an intuitive level, counting curves should be something which can be done on $B$.