Algebraic topology of Calabi–Yau threefolds in toric varieties

One of the most fruitful sources of Calabi–Yau threefolds is hypersurfaces, or more generally complete intersections, in toric varieties. This is especially true since there is a proposal for the mirror of any such Calabi–Yau threefold. Usually the toric varieties associated to convex lattice polytopes are singular, causing the Calabi–Yau threefolds in them also to be singular, so that to get smooth Calabi–Yau threefolds we must resolve the ambient singularities and take the preimage in the resolution of the singular Calabi–Yau threefold. This can be done torically by the combinatorial device of taking a triangulation of the boundary of the convex lattice polytope defining the toric variety where the vertices of the triangulation are exactly the lattice points contained in the boundary of the polytope. In general, there will be many such triangulations of a given lattice polytope, leading to different ambient resolutions producing different families of Calabi–Yau threefolds associated with the original toric variety.