Slope stability of tangent bundles of smooth toric Fano varieties
Abstract
We construct smooth toric Fano n-folds of Picard number n+2 whose tangent bundles are slope-stable with respect to the anticanonical polarization, for every n≥4. The construction gives toric fibrations with fibre the del Pezzo surface of degree six over products of projective lines, parametrized by multisets of roots of A2. Every nonempty multiset of nonzero roots with vanishing sum yields a stable tangent bundle. Among these multisets, the resulting variety is Kähler-Einstein if and only if the multiset is invariant under negation or under the order-three rotation of the root hexagon. We also construct stable, non-Kähler-Einstein examples with nonzero twist sum in every even dimension at least four. The proofs reduce slope inequalities and barycenter computations to integrals over the moment hexagon. For a smooth toric Fano variety with strictly semistable tangent bundle, we prove that polystability is equivalent to decomposition as a nontrivial product of smooth toric Fano varieties with anticanonically stable tangent bundles. Exact evaluation of Klyachko's criterion extends the stability classification to dimensions five and six, recovering the classifications of Steffens and Reynolds in dimensions three and four. Together with the product criterion, this determines polystability for all 8,630 varieties in dimensions three through six, and hence the existence of Hermitian-Einstein metrics on their tangent bundles with respect to an anticanonical Kähler form.
Related
- On the existence of smooth varieties of any dimension carrying no Ulrich bundles for some very ample line bundle
- Tangent bundles of Quot schemes of vector bundles on curves
- Non-null ruled surfaces and pseudo hyper-dual tangent sphere bundles
- Uniform RC-positivity of tangent bundles
- Positive mass and rigidity for asymptotically flat tangent bundles
BibTeX
@misc{wuebben2026slope,
title = {{Slope stability of tangent bundles of smooth toric Fano varieties}},
author = {Bernd Johannes Wuebben},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.20411},
eprint = {2608.20411},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.20411},
}
RIS
TY - UNPB
TI - Slope stability of tangent bundles of smooth toric Fano varieties
AU - Bernd Johannes Wuebben
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2608.20411
UR - https://arxiv.org/abs/2608.20411
AB - We construct smooth toric Fano n-folds of Picard number n+2 whose tangent bundles are slope-stable with respect to the anticanonical polarization, for every n≥4. The construction gives toric fibrations with fibre the del Pezzo surface of degree six over products of projective lines, parametrized by multisets of roots of A_2. Every nonempty multiset of nonzero roots with vanishing sum yields a stable tangent bundle. Among these multisets, the resulting variety is Kähler-Einstein if and only if the multiset is invariant under negation or under the order-three rotation of the root hexagon. We also construct stable, non-Kähler-Einstein examples with nonzero twist sum in every even dimension at least four. The proofs reduce slope inequalities and barycenter computations to integrals over the moment hexagon. For a smooth toric Fano variety with strictly semistable tangent bundle, we prove that polystability is equivalent to decomposition as a nontrivial product of smooth toric Fano varieties with anticanonically stable tangent bundles. Exact evaluation of Klyachko's criterion extends the stability classification to dimensions five and six, recovering the classifications of Steffens and Reynolds in dimensions three and four. Together with the product criterion, this determines polystability for all 8,630 varieties in dimensions three through six, and hence the existence of Hermitian-Einstein metrics on their tangent bundles with respect to an anticanonical Kähler form.
ER -