Rigid and obstructed tangent bundles on Calabi--Yau threefolds
Abstract
We address two questions of Huybrechts concerning rigidity and singular deformation spaces of tangent bundles on Calabi--Yau threefolds, without assuming simple connectedness. Our first example is a free (ℤ/2)3 quotient of a smooth intersection of four quadrics in ℙ7. Its tangent bundle is stable and infinitesimally rigid, and it also provides a three-dimensional counterexample to the proposed classification in Peternell's Question~1.6. For a classical Igusa quotient of a product of three elliptic curves, we determine the analytic semiuniversal deformation germ of its polystable tangent bundle, keeping the underlying threefold fixed. This germ is isomorphic to the product of two copies of the union of the three coordinate axes in ℂ3. It is therefore reduced and singular, with nine two-dimensional irreducible components.
Related
- Chern bounds and tangent geometry of polarized Calabi-Yau threefolds
- Stable atomic vector bundles are 1-obstructed
- The Tate Conjecture for Ordinary Calabi-Yau Threefolds
- 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{wan2026rigid,
title = {{Rigid and obstructed tangent bundles on Calabi--Yau threefolds}},
author = {Xueyuan Wan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27271},
eprint = {2609.27271},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27271},
}
RIS
TY - UNPB
TI - Rigid and obstructed tangent bundles on Calabi--Yau threefolds
AU - Xueyuan Wan
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27271
UR - https://arxiv.org/abs/2609.27271
AB - We address two questions of Huybrechts concerning rigidity and singular deformation spaces of tangent bundles on Calabi--Yau threefolds, without assuming simple connectedness. Our first example is a free (ℤ/2)^3 quotient of a smooth intersection of four quadrics in ℙ^7. Its tangent bundle is stable and infinitesimally rigid, and it also provides a three-dimensional counterexample to the proposed classification in Peternell's Question~1.6. For a classical Igusa quotient of a product of three elliptic curves, we determine the analytic semiuniversal deformation germ of its polystable tangent bundle, keeping the underlying threefold fixed. This germ is isomorphic to the product of two copies of the union of the three coordinate axes in ℂ^3. It is therefore reduced and singular, with nine two-dimensional irreducible components.
ER -