A vanishing-cycle obstruction to smoothing Calabi-Yau threefolds

@misc{wuebben2026vanishingcycle,
  title = {{A vanishing-cycle obstruction to smoothing Calabi-Yau threefolds}},
  author = {Bernd Johannes Wuebben},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35911},
  eprint = {2610.35911},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35911},
}

TY  - UNPB
TI  - A vanishing-cycle obstruction to smoothing Calabi-Yau threefolds
AU  - Bernd Johannes Wuebben
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.35911
UR  - https://arxiv.org/abs/2610.35911
AB  - We prove a homological obstruction to smoothing projective Calabi-Yau threefolds with nodes and anticanonical cones over smooth del Pezzo surfaces of degrees six and seven. On a crepant resolution that is small over the nodes and blows up the cone vertices, any smoothing forces a relation among link classes bounding in the local Milnor fibers. The relation is nonzero at every node and degree-seven point, and at each degree-six point smoothed through the one-dimensional component. The proof uses vanishing periods and applies to smoothing arcs of arbitrary contact order, extending the necessity direction of Friedman's criterion. We apply the obstruction to a Batyrev hypersurface whose thirty singularities are individually smoothable but which admits no smoothing, even formally. On its mirror partner, two nodes cannot be smoothed by any analytic deformation, independently of a locally nonsmoothable cone point.
ER  -