Tropical contractions to integral affine manifolds with singularities

Abstract: We consider a toric degeneration of Calabi--Yau complete intersections of Batyrev--Borisov in the Gross--Siebert program. One can associate two types of tropical spaces with it. One is a tropical variety obtained by tropicalization. The other one is an integral affine manifold with singularities, which arises as the dual intersection complex of the toric degeneration. In this article, we show that the latter is contained in the former as a subset, and construct an integral affine contraction map from the former to the latter. We also show that the contraction preserves tropical cohomology groups, and sends the eigenwave to the radiance obstruction.