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From real affine geometry to complex geometry. (English) Zbl 1266.53074
This is one of the most important papers in mirror symmetry. It solves a fundamental problem in SYZ mirror symmetry which is known as reconstruction of the mirror manifold. It is known as the Gross-Siebert program, and it involves important algebro-geometric techniques such as toric degeneration and wall-crossing which is a realization of the theory developed by Kontsevich-Soibelman.
Strominger-Yau-Zaslow (SYZ) proposed that mirror symmetry can be understood in terms of duality of special Lagrangian torus fibrations. In particular, the mirror manifold can be constructed by taking a fiberwise torus dual of the original manifold. There are two main problems, namely, construction of Lagrangian fibrations and handling quantum corrections contributed from singular fibers.
The Gross-Siebert program uses toric degenerations to avoid the difficult problem of constructing Lagrangian fibrations. Namely, given a toric degeneration of a variety $$X$$ whose central fiber is a union of toric varieties glued at toric prime divisors, the authors use the dual intersection complex of the central fiber to construct an affine manifold $$B$$ with singularities and a polyhedral decomposition, which is conjecturally the base of a Lagrangian fibration on $$X$$. This produces real affine geometry from complex geometry.
This paper mainly deals with the reverse process known as reconstruction, namely, producing complex geometry from real affine geometry. The key is to capture the quantum corrections by a scattering diagram on the affine manifold $$B$$, which is basically a union of tropical trees in $$B$$ with leaves on the singularities of $$B$$, attached with certain ring automorphisms. Conjecturally such data is equivalent to counting holomorphic discs bounded by a torus fiber of the (conjectural) Lagrangian fibration. Gross and Siebert compute explicitly the scattering diagram order-by-order by employing wall-crossing techniques. The scattering diagram divides $$B$$ into chambers, and they produce the toric degeneration by gluing complexifications of the chambers according to the ring automorphisms.
The main result is Theorem 0.1. For any polarized affine manifold $$B$$ with singularities with positive and simple monodromy, the authors construct a toric degeneration whose dual intersection complex gives $$B$$. The outcome is conjectured to be the mirror of $$X$$ written in terms of flat coordinates in the moduli (Conjecture 0.2), which gives a deep relation between tropical discs in $$X$$ and periods of the mirror. They also speculate that homological mirror symmetry could be understood by considering a ‘tropical Morse category’ on $$B$$.
This paper is a magnificent breakthrough in the study of mirror symmetry via tropical geometry. Everyone who is interested in a geometric approach to mirror symmetry should read this paper.

##### MSC:
 53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category 14J33 Mirror symmetry (algebro-geometric aspects) 14T05 Tropical geometry (MSC2010)
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