Degenerations and mirror contractions of Calabi–Yau complete intersections

We construct deformations of Fano toric varieties which induce deformations of Calabi–Yau hypersurfaces realized by complete intersections in a higher dimensional Fano toric variety. Equivalently, we get a degeneration of a (minimal) Calabi–Yau complete intersection to a singular Calabi–Yau hypersurface, which can be blown up to a nonsingular Calabi–Yau. A geometric transition from a minimal Calabi–Yau variety to another minimal Calabi–Yau is a contraction followed by deformation (correspondingly, a degeneration followed by blow up). Applying Batyrev–Borisov mirror symmetry construction, we found a geometric transition between the mirror partners of the minimal Calabi–Yau complete intersection and the minimal Calabi–Yau hypersurface consistently with the Morrison's conjecture on the existence of the mirror geometric transitions. Physicists expect that all nonsingular Calabi–Yau 3-folds can be connected into a single web by geometric transitions and there is possibly a finite number of distinct nonsingular Calabi–Yau 3-folds up to deformation. Our construction gives a strong evidence that all Calabi–Yau complete intersections in toric varieties can be connected by explicit geometric transitions.