Non-smoothable Calabi-Yau threefolds from reflexive polytopes
Abstract
We construct projective Calabi-Yau threefolds with isolated Gorenstein canonical singularities that admit no smoothing, as anticanonical hypersurfaces in Gorenstein Fano toric fourfolds. One example has a unique singularity, the anticanonical cone over the first Hirzebruch surface. Another has a unique singularity whose reduced miniversal base is a smooth curve but whose deformations are all singular. The construction combines Altmann's deformation theory with the smoothing criterion of Corti-Filip-Petracci: a two-dimensional face of the ambient fan polytope with primitive edges obstructs smoothing if it has no Minkowski decomposition into unit segments and unimodular triangles. We also classify the 217 isolated Gorenstein toric threefold singularities whose polygon edge vectors have coordinates in [−2,2] in some lattice basis, recording their deformation types and smoothing components. Subject to completeness and absence of repetitions in the database copy used, a scan of the Kreuzer-Skarke classification finds this obstruction in 39,175,536 of its 473,800,776 polytopes (approximately 8.27%), whose generic anticanonical hypersurfaces therefore admit no smoothing.
Related
- Doubly isolated Batyrev mirror pairs and non-smoothable Calabi-Yau threefolds
- The Tate Conjecture for Ordinary Calabi-Yau Threefolds
- Unimodular defect and antipodal lattice points of smooth reflexive polytopes
- Integral Bott--Chern cohomology of Generalized Calabi--Eckmann threefolds
- Degeneration of Calabi-Yau threefolds with Galois action induced by modular form of weight 3
BibTeX
@misc{wuebben2026nonsmoothable,
title = {{Non-smoothable Calabi-Yau threefolds from reflexive polytopes}},
author = {Bernd Johannes Wuebben},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28499},
eprint = {2609.28499},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28499},
}
RIS
TY - UNPB
TI - Non-smoothable Calabi-Yau threefolds from reflexive polytopes
AU - Bernd Johannes Wuebben
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.28499
UR - https://arxiv.org/abs/2609.28499
AB - We construct projective Calabi-Yau threefolds with isolated Gorenstein canonical singularities that admit no smoothing, as anticanonical hypersurfaces in Gorenstein Fano toric fourfolds. One example has a unique singularity, the anticanonical cone over the first Hirzebruch surface. Another has a unique singularity whose reduced miniversal base is a smooth curve but whose deformations are all singular. The construction combines Altmann's deformation theory with the smoothing criterion of Corti-Filip-Petracci: a two-dimensional face of the ambient fan polytope with primitive edges obstructs smoothing if it has no Minkowski decomposition into unit segments and unimodular triangles. We also classify the 217 isolated Gorenstein toric threefold singularities whose polygon edge vectors have coordinates in [−2,2] in some lattice basis, recording their deformation types and smoothing components. Subject to completeness and absence of repetitions in the database copy used, a scan of the Kreuzer-Skarke classification finds this obstruction in 39,175,536 of its 473,800,776 polytopes (approximately 8.27%), whose generic anticanonical hypersurfaces therefore admit no smoothing.
ER -