Doubly isolated Batyrev mirror pairs and non-smoothable Calabi–Yau threefolds

Abstract

We classify reflexive four-polytopes for which the generic anticanonical hypersurfaces associated with the face fans of the polytope and its polar both have at most isolated singularities. There are exactly 590 such polytopes up to lattice equivalence in the Kreuzer–Skarke classification. Their two-dimensional faces belong to twelve lattice-isomorphism classes, all triangles, zonotopes, or reflexive polygons. The only nonsmoothable local models are the cyclic quotients $\frac13(1,1,1)$ and $\frac15(1,1,3)$ and the anticanonical cone over the first Hirzebruch surface; no singularity has a positive-dimensional reduced deformation space without a smoothing component. Within this classification, the Hirzebruch cone occurs in a unique mirror pair, associated with polytopes having 22 and 26 vertices. We exhibit this pair explicitly: one hypersurface is nonsmoothable, while all singularities of its mirror are locally smoothable. The crepant resolutions have Hodge numbers $(20,26)$ and $(26,20)$. At the opposite extreme, the self-dual 24-cell is the unique member of the classification for which both hypersurfaces are smooth before crepant resolution.