Functorial Homological Mirror Symmetry for Acyclic Cluster Varieties
Abstract
We introduce the formalism of cluster models, a class of log Calabi-Yau varieties obtained from smooth quasi-affine toric varieties by iterated affine blowups along non-toric codimension 2 centers determined by a really full rank quiver. Cluster models include the truncated cluster varieties of Gross, Hacking, and Keel with respect to really full rank quivers, all really full rank acyclic cluster varieties, and all open toric subvarieties of ℂn×(ℂ*)m. We then construct mirrors to cluster models via iterated Weinstein handle attachments to toric mirrors, so that the corresponding skeleta acquire compact pieces, and we prove homological mirror symmetry for all cluster models, compatibly with freezing, deleting, and contracting vertices of the quiver. This extends toric mirror symmetry and its naturalities beyond the cotangent bundle. In particular, we obtain homological mirror symmetry for all really full rank acyclic cluster varieties. We develop this package with a view towards conjectures of Marsh, Rietsch, and Williams about mirror symmetry for Grassmannians and closed positroid varieties. Motivated by these, we also conjecture that open positroid varieties are self-mirror. Assuming a natural ansatz, we prove this for a class of positroid varieties that includes nontrivial cells in every Grassmannian and all cells in finite-type Grassmannians. Towards this end we introduce a new combinatorial input, the notion of weak k-booleanness for permutations and braids.
Related
- Motivic Mirror Symmetry for Sink-recurrent Cluster Varieties
- Local Homological Mirror Symmetry
- Examples of homological mirror symmetry via Zariski descent
- Acyclic quantum cluster algebras via Hall algebras associated to cluster categories
- Copula Adapted Directed Acyclic Graph for Cluster Representation of Biomedical Data
BibTeX
@misc{hlavinka2026functorial,
title = {{Functorial Homological Mirror Symmetry for Acyclic Cluster Varieties}},
author = {Josephine Hlavinka},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.12751},
eprint = {2611.12751},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.12751},
}
RIS
TY - UNPB
TI - Functorial Homological Mirror Symmetry for Acyclic Cluster Varieties
AU - Josephine Hlavinka
PY - 2026
DA - 2026/11/09
PB - arXiv
DO - 10.48550/arxiv.2611.12751
UR - https://arxiv.org/abs/2611.12751
AB - We introduce the formalism of cluster models, a class of log Calabi-Yau varieties obtained from smooth quasi-affine toric varieties by iterated affine blowups along non-toric codimension 2 centers determined by a really full rank quiver. Cluster models include the truncated cluster varieties of Gross, Hacking, and Keel with respect to really full rank quivers, all really full rank acyclic cluster varieties, and all open toric subvarieties of ℂ^n×(ℂ^*)^m. We then construct mirrors to cluster models via iterated Weinstein handle attachments to toric mirrors, so that the corresponding skeleta acquire compact pieces, and we prove homological mirror symmetry for all cluster models, compatibly with freezing, deleting, and contracting vertices of the quiver. This extends toric mirror symmetry and its naturalities beyond the cotangent bundle. In particular, we obtain homological mirror symmetry for all really full rank acyclic cluster varieties. We develop this package with a view towards conjectures of Marsh, Rietsch, and Williams about mirror symmetry for Grassmannians and closed positroid varieties. Motivated by these, we also conjecture that open positroid varieties are self-mirror. Assuming a natural ansatz, we prove this for a class of positroid varieties that includes nontrivial cells in every Grassmannian and all cells in finite-type Grassmannians. Towards this end we introduce a new combinatorial input, the notion of weak k-booleanness for permutations and braids.
ER -