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.

67 pages, 8 figures. v2: The introduction is updated. The results on Poincaré duality are removed

References in corpus (7)

• Integral affine structures on spheres and torus fibrations of Calabi-Yau toric hypersurfaces I
• A homology theory for tropical cycles on integral affine manifolds and a perfect pairing
• Euler Characteristic of real nondegenerate tropical complete intersections
• Tropical surfaces
• Homology of tropical fans
• Toric geometry and integral affine structures in non-archimedean mirror symmetry
• Tropical $\psi$ classes

Cited by in corpus (1)

• A Note on $h^{2,1}$ of Divisors of CY Fourfolds I