Derived Picard reconstruction for graded surfaces with stops
Abstract
We study reconstruction of graded surfaces with stops (S,M,η) from their mapping class groups and the derived Picard groups of their partially wrapped Fukaya categories. In genus at least two, we prove that the ungraded mapping class group determines the surface with stops (S,M), while the derived Picard group with the shift distinguished determines the underlying graded surface (S,η). Nevertheless, the derived Picard group with both the shift and the Serre functor distinguished has countably infinite fibres. We also construct arbitrarily large finite families of pairwise non-equivalent categories with the same invariant and Grothendieck rank. For line fields induced by nowhere-vanishing vector fields, we show that the rational Serre representation determines the surface with stops (S,M) in every genus. For arbitrary line fields in genus at least two, the full integral derived Picard representation, with the shift distinguished, has fibres of cardinality at most four. It completely determines the graded surface with stops (S,M,η) when S has one or at least five boundary components, while non-singleton fibres occur for two, three, and four boundary components.
Related
- The derived Picard group of Fukaya categories of closed surfaces
- Reconstruction of graphs and surfaces from derived categories
- Residual finiteness and cuspidal cohomology of Picard modular surfaces
- Densities of Picard Rank Jumps for K3 Surfaces Over Global Fields
- Sharp Volume Bounds in Picard Rank One and Polynomial Bounds for Log Calabi-Yau Surfaces
BibTeX
@misc{chang2026derived,
title = {{Derived Picard reconstruction for graded surfaces with stops}},
author = {Wen Chang and Dongjian Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09319},
eprint = {2610.09319},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09319},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Derived Picard reconstruction for graded surfaces with stops
AU - Wen Chang
AU - Dongjian Wu
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09319
UR - https://arxiv.org/abs/2610.09319
AB - We study reconstruction of graded surfaces with stops (S,M,η) from their mapping class groups and the derived Picard groups of their partially wrapped Fukaya categories. In genus at least two, we prove that the ungraded mapping class group determines the surface with stops (S,M), while the derived Picard group with the shift distinguished determines the underlying graded surface (S,η). Nevertheless, the derived Picard group with both the shift and the Serre functor distinguished has countably infinite fibres. We also construct arbitrarily large finite families of pairwise non-equivalent categories with the same invariant and Grothendieck rank. For line fields induced by nowhere-vanishing vector fields, we show that the rational Serre representation determines the surface with stops (S,M) in every genus. For arbitrary line fields in genus at least two, the full integral derived Picard representation, with the shift distinguished, has fibres of cardinality at most four. It completely determines the graded surface with stops (S,M,η) when S has one or at least five boundary components, while non-singleton fibres occur for two, three, and four boundary components.
ER -