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 1/3(1,1,1) and 1/5(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.
Related
- Non-smoothable Calabi-Yau threefolds from reflexive polytopes
- Deformations of toric pairs and the smoothing of Batyrev Calabi-Yau threefolds
- Mirror symmetry for gCICY threefolds (I): a [3,−1] block and a non-Gorenstein toric phase
- Mirror symmetry for gCICY threefolds (II): [3,−1] blocks, VGIT, and toric phases
- A unified family of counterexamples to Batyrev's non-negativity conjecture on stringy Hodge numbers
BibTeX
@misc{wuebben2026doubly,
title = {{Doubly isolated Batyrev mirror pairs and non-smoothable Calabi-Yau threefolds}},
author = {Bernd Johannes Wuebben},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03753},
eprint = {2610.03753},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03753},
}
RIS
TY - UNPB
TI - Doubly isolated Batyrev mirror pairs and non-smoothable Calabi-Yau threefolds
AU - Bernd Johannes Wuebben
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.03753
UR - https://arxiv.org/abs/2610.03753
AB - 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 1/3(1,1,1) and 1/5(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.
ER -