PreprintarXiv licencearXiv
Algebraic Vector Bundles on Cluster Varieties
Abstract
We study algebraic vector bundles on cluster varieties. Over a perfect field of characteristic different from two, every algebraic vector bundle is trivial on a cluster variety with principal coefficients at an acyclic seed of irreducible finite Dynkin type. The proof is based on integral contractions of cellular 𝔸1-chain complexes and successive freezing. We also construct smooth cluster varieties admitting indecomposable rank-two vector bundles, whose nontriviality is detected by quadratic Euler classes. The cellular calculation also gives a criterion for stable freeness and computes integral real cohomology and the images of cycle class maps.
Related
- Degrees and directional defects of embedded algebraic vector bundles
- On the existence of smooth varieties of any dimension carrying no Ulrich bundles for some very ample line bundle
- Ampleness of non-symmetric line bundles on degenerations of abelian varieties
- Existence of Ulrich bundles on polarized surfaces and weakly Ulrich bundles on polarized varieties in positive characteristic
- Abundance of affine algebraic structures on stabilized cotangent bundles of surfaces
BibTeX
@misc{liu2026algebraic,
title = {{Algebraic Vector Bundles on Cluster Varieties}},
author = {Haoyang Liu and Tianle Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12748},
eprint = {2610.12748},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12748},
}
RIS
TY - UNPB
TI - Algebraic Vector Bundles on Cluster Varieties
AU - Haoyang Liu
AU - Tianle Liu
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12748
UR - https://arxiv.org/abs/2610.12748
AB - We study algebraic vector bundles on cluster varieties. Over a perfect field of characteristic different from two, every algebraic vector bundle is trivial on a cluster variety with principal coefficients at an acyclic seed of irreducible finite Dynkin type. The proof is based on integral contractions of cellular 𝔸^1-chain complexes and successive freezing. We also construct smooth cluster varieties admitting indecomposable rank-two vector bundles, whose nontriviality is detected by quadratic Euler classes. The cellular calculation also gives a criterion for stable freeness and computes integral real cohomology and the images of cycle class maps.
ER -