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From real affine geometry to complex geometry

Mark Gross·Bernd Siebert
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Abstract

We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical order-by-order description of the degeneration via families of tropical trees. This gives complete control of the B-model side of mirror symmetry in terms of tropical geometry. For example, we expect our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves. We anticipate this will lead to a proof of mirror symmetry via tropical methods. This paper is the key step of the program we initiated in math.AG/0309070.

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Mark Gross, Bernd Siebert. 2012-07-29. From real affine geometry to complex geometry. https://arxiv.org/abs/math/0703822

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