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We generalized the construction of deformations of affine toric varieties ofK. Altmann and our previous construction of deformations of weak Fano toricvarieties to the case of arbitrary toric varieties by introducing the notion ofMinkowski sum decompositions of polyhedral complexes. Our construction embedsthe original toric variety into a higher dimensional toric variety where theimage is given by a prime binomial complete intersection ideal in Coxhomogeneous coordinates. The deformations are realized by families of completeintersections. For compact simplicial toric varieties with at worst Gorensteinterminal singularities, we show that our deformations span the infinitesimalspace of deformations by Kodaira-Spencer map. For Fano toric varieties, we showthat their deformations can be constructed in higher-dimensional Fano toricvarieties related to the Batyrev-Borisov mirror symmetry construction.

Deformations of Toric Varieties via Minkowski Sum Decompositions of Polyhedral Complexes