It is expected that the symplectic structure that we get using semi-flat SYZ mirror symmetry can naturally be compactified to give a global symplectic structure on the mirror. Indeed the work of Castaño-Bernard and Matessi [14] have shown that the topological Calabi-Yau compactifications constructed by Gross in [69] can be made into symplectic compactifications, hence producing pairs of compact symplectic 6-folds which are homeomorphic to known mirror pairs of Calabi-Yau 3-folds (such as the quintic 3-fold and its mirror) and equipped with Lagrangian torus fibrations whose bases are Legendre dual integral affine manifolds with singularities.

On the other hand, as we have mentioned before, it was already anticipated in the original SYZ proposal [153] that the Ricci-flat metric on the mirror should differ from the semi-flat Calabi-Yau metric [64] by instanton corrections from holomorphic disks whose boundaries wrap non-trivial 1-cycles in the fibers of an SYZ fibration. Since the metric on the mirror is determined uniquely by its symplectic and complex structures, it is natural to expect that the instanton corrections are all contributing to perturbations of the complex structure on the mirror. This is indeed the key idea underlying the SYZ conjecture. As holomorphic disks can be glued to give holomorphic curves, this explains why mirror symmetry can lead to enumerative predictions.

Now, given an affine manifold with singularities $B$. Let $\Gamma\subset B$ be the singular locus and denote by $B_{0}=B\setminus\Gamma$ the smooth part. Then one would like to construct a complex manifold which is a compactification of a small deformation of $X_{0}:=TB_{0}/\Lambda$. This is called the reconstruction problem, which lies at the heart of the algebraic-geometric SYZ program of Gross and Siebert [75, 76, 77, 78].

The reconstruction problem was first attacked by Fukaya [49]. He attempted to find suitable perturbations by directly solving the Maurer-Cartan equation which governs the deformations of complex structures on $X_{0}$. In the dimension two case, his heuristic arguments suggested that the perturbations should come from gradient flow trees in $B$ whose ends emanate from the singular set $\Gamma$. The latter should be limits of holomorphic disks bounding Lagrangian fibers and singularities of an SYZ fibration when one approaches a large complex structure limit. Fukaya made a series of beautiful conjectures explaining how quantum corrections are contributing to the complex structure on the mirror and provided an intuitively clear picture, but unfortunately the analysis required to make his arguments rigorous seemed out of reach.

Kontsevich and Soibelman [106] got around the analytic difficulties in Fukaya’s arguments by working with rigid analytic spaces. They started with an integral affine structure on $S^{2}$ with 24 singular points such that the monodromy of the affine structure around each singular point is the simplest one: $\left(\begin{array}[]{cc}1&1\\ 0&1\end{array}\right)$, and managed to construct a non-Archimedean analytic $K3$ surface. The basic idea is to attach an automorphism to each gradient flow line in Fukaya’s construction, and modify the gluing between complex charts in the mirror by these automorphisms, thereby resolving the incompatibilities between charts which arise from the nontrivial monodromy of the affine structure around the discriminant locus. A crucial step in their argument is a key lemma showing that when two gradient flow lines intersect, one can always add new lines together with new automorphisms attached so that the composition around each intersection point is the identity. This is called the scattering phenomenon, which assures that the composition of automorphisms attached to lines crossed by a path is independent of the path chosen, hence guaranteeing that the modified gluings are consistent.

At around the same time, Gross and Siebert launched their spectacular program [75, 76, 77, 78] aiming at an algebraic-geometric approach to the SYZ conjecture. Motivated by the limiting version of the SYZ conjecture we discussed in the previous section and the observation by Kontsevich that the Gromov-Hausdorff limit will be roughly the dual intersection complex of the degeneration, they formulated an algebraic-geometric SYZ procedure to construct the mirror. In more details, starting with a toric degeneration of Calabi-Yau manifolds, the first step is to construct the dual intersection complex. Then one takes the (discrete) Legendre transform and try to reconstruct the mirror toric degeneration of Calabi-Yau manifolds from the Legendre dual. In this way, they can completely forget about SYZ fibrations. The claim is that all information are encoded in the tropical geometry of the dual intersection complex, which is an integral affine manifold with singularities and plays the role of the base of an SYZ fibration.

Using the above key lemma of Kontsevich and Soibelman, together with many new ideas such as employing log structures and techniques from tropical geometry, Gross and Siebert eventually succeeded in giving a solution to the reconstruction problem in any dimension [78]. More precisely, given any integral affine manifold with singularities satisfying certain assumptions and equipped with some additional structures like a polyhedral decomposition, they constructed a toric degeneration of Calabi-Yau manifolds which can be described explicitly and canonically via tropical trees in $B$. Furthermore, the Calabi-Yau manifolds they constructed are defined over $\mathbb{C}$, instead of being rigid analytic spaces.

Now the goal is to acquire a conceptual understanding of mirror symmetry by going through the tropical world. On the B-side, Gross and Siebert conjectured that the deformation parameter in their construction is a canonical coordinate and period integrals of the family of Calabi-Yau manifolds can be expressed in terms of tropical disks in $B$. They have already given some evidences in the local cases (such as the local $\mathbb{P}^{2}$ example in [78, Remark 5.1]) and are working out the general case.

On the A-side, one would like to understand the Gromov-Witten theory of a smooth fiber by working entirely on the central fiber of a toric degeneration, whose dual intersection complex is precisely the affine manifold that Gross and Siebert started with. The recent independent works of Abramovich and Chen [30, 4] and Gross and Siebert [80], which developed the theory of log Gromov-Witten invariants, generalizing previous works of A.-M. Li and Ruan [113], Ionel and Parker [92, 93], and Jun Li [114] on relative Gromov-Witten theory, constituted a significant step towards this goal. If furthermore one can prove a general correspondence theorem between tropical and holomorphic curves/disks, in the same vein as the works of Mikhalkin [128, 129], Nishinou-Siebert [137] and Nishinou [136, 135], then we would be able to connect the A-side (i.e. Gromov-Witten theory) of a Calabi-Yau manifold to the tropical world.

Albeit much work needs to be done, this lays out a satisfying picture explaining the geometry of mirror symmetry via tropical geometry. We refer the reader to the beautiful survey articles of the inventors [79, 65] for overviews of the Gross-Siebert program.