Deformations of Calabi-Yau hypersurfaces arising from deformations of toric varieties

One can deform the complex structure of Calabi-Yau hypersurfaces in toric varieties by changing the coefficients of the defining polynomial. However, there must also exist non-polynomial deformations of the Calabi-Yau hypersurfaces which cannot be realized inside the ambient toric variety. In this paper, we have constructed the missing non-polynomial deformations, which are induced by an automorphism on the open part of the ambient toric variety.