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.