Fukaya category with supports
Abstract
We associate to each compact subset K of a closed symplectic manifold M an A∞-category FukK, called the Fukaya category with support on K. The objects are tautologically unobstructed global Lagrangians endowed with appropriate decorations. The dependence on K is detected by the morphism complexes which can be computed by completed telescopes associated with acceleration data for K. In particular, these vanish for Lagrangians disjoint of K. For K=M this recovers the usual Fukaya category of tautologically unobstructed objects. These categories admit restriction functors under inclusions of compact subsets. We prove a Mayer--Vietoris descent theorem for weakly involutive covers. The construction is carried out over the Novikov ring and is established whenever classical transversality methods apply.
Related
- The Relative Fukaya Category as a Closed-Open Deformation
- Minimal slN Complexes of Rational Tangles via the Fukaya Category
- Symplectomorphism invariance of the wrapped Fukaya category
- Partially wrapped contact Fukaya categories of 1-jet spaces
- The derived Picard group of Fukaya categories of closed surfaces
BibTeX
@misc{abouzaid2026fukaya,
title = {{Fukaya category with supports}},
author = {Mohammed Abouzaid and Yoel Groman and Umut Varolgunes},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35565},
eprint = {2609.35565},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35565},
}
RIS
TY - UNPB
TI - Fukaya category with supports
AU - Mohammed Abouzaid
AU - Yoel Groman
AU - Umut Varolgunes
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35565
UR - https://arxiv.org/abs/2609.35565
AB - We associate to each compact subset K of a closed symplectic manifold M an A_∞-category Fuk_K, called the Fukaya category with support on K. The objects are tautologically unobstructed global Lagrangians endowed with appropriate decorations. The dependence on K is detected by the morphism complexes which can be computed by completed telescopes associated with acceleration data for K. In particular, these vanish for Lagrangians disjoint of K. For K=M this recovers the usual Fukaya category of tautologically unobstructed objects. These categories admit restriction functors under inclusions of compact subsets. We prove a Mayer--Vietoris descent theorem for weakly involutive covers. The construction is carried out over the Novikov ring and is established whenever classical transversality methods apply.
ER -