  We generalized the construction of deformations of affine toric varieties of
K. Altmann and our previous construction of deformations of weak Fano toric
varieties to the case of arbitrary toric varieties by introducing the notion of
Minkowski sum decompositions of polyhedral complexes. Our construction embeds
the original toric variety into a higher dimensional toric variety where the
image is given by a prime binomial complete intersection ideal in Cox
homogeneous coordinates. The deformations are realized by families of complete
intersections. For compact simplicial toric varieties with at worst Gorenstein
terminal singularities, we show that our deformations span the infinitesimal
space of deformations by Kodaira-Spencer map. For Fano toric varieties, we show
that their deformations can be constructed in higher-dimensional Fano toric
varieties related to the Batyrev-Borisov mirror symmetry construction.
