Smoothing Calabi-Yau toric hypersurfaces using the Gross-Siebert algorithm

Abstract

We explain how to form a novel dataset of simply connected Calabi-Yau threefolds via the Gross-Siebert algorithm. We expect these to degenerate to Calabi-Yau toric hypersurfaces with certain Gorenstein (not necessarily isolated) singularities. In particular, we explain how to `smooth the boundary' of a class of $4$-dimensional reflexive polytopes to obtain a polarised tropical manifolds. We compute topological invariants of a compactified torus fibration over each such tropical manifold, expected to be homotopy equivalent to the general fibre of the Gross-Siebert smoothing. We consider a family of examples related to the joins of elliptic curves. Among these we find $14$ topological types with $b_2=1$ which do not appear in existing lists of known rank one Calabi-Yau threefolds.

arXiv commentWe have added 5 tables of examples and additional Magma source code

References in corpus (8)

• Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
• Fano 3-folds in codimension 4, Tom and Jerry, Part I
• Differential equations, mirror maps and zeta values
• Smoothing toroidal crossing spaces
• Calabi-Yau Threefolds With Small Hodge Numbers
• GLSMs, joins, and nonperturbatively-realized geometries
• Algebraic Topology of Calabi-Yau Threefolds in Toric Varieties
• Lagrangian torus fibration models of Fano threefolds

Cited by in corpus (3)

• A homology theory for tropical cycles on integral affine manifolds and a perfect pairing
• GLSMs, joins, and nonperturbatively-realized geometries
• On genus one fibered Calabi-Yau threefolds with 5-sections