From real affine geometry to complex geometry

TL;DR

This paper constructs degenerations of Calabi-Yau manifolds from tropical manifolds, providing a canonical tropical geometric framework to understand mirror symmetry and its B-model side.

Contribution

It introduces an explicit, order-by-order tropical geometric approach to degenerations of Calabi-Yau manifolds, advancing the understanding of mirror symmetry.

Findings

Constructs degenerations of Calabi-Yau manifolds from tropical manifolds.

Provides a canonical, explicit description of degenerations via tropical trees.

Enables potential proof of mirror symmetry using tropical geometry.

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.