Analyticity of Gross--Siebert Calabi--Yau families

Gross and Siebert gave an algorithm to produce from toric degeneration data a canonical formal
Calabi--Yau family. Siebert and I prove that this family is in fact the completion of an analytic
family. In particular, its nearby fibres are decent Calabi-Yau manifolds over the complex
numbers. Furthermore, the family is semi-universal, i.e. is in a sense locally the moduli
space of Calabi--Yaus. The key result on the route to analyticity is the computation of canonical
coordinates on the base by explicit integration of a holomorphic volume form over topological
cycles that we construct from tropical $1$-cycles in the base of the SYZ-fibration.