Biran decomposition and relative symplectic cohomology
Abstract
Let (M,ω) be a closed monotone integral symplectic manifold with monotonicity constant τ>0, and let D⊂ M be a symplectic hyperplane section of degree 0<k<τ. By Biran's decomposition theorem, M is the union of a symplectic disk bundle over D and the skeleton of the complement M∖ D equipped with its Liouville structure. In this paper, we compute the relative symplectic cohomology of circle subbundles of this disk bundle over ℤ in terms of the quantum cohomology of D. This relative symplectic cohomology depends on the radius of the circle bundle relative to a certain critical radius. For a broad class of pairs (M,D), we prove that the relative symplectic cohomology over ℤ does not vanish for supercritical radii. Therefore, circle bundles of supercritical radii are heavy subsets, which yields infinite dimensional quasi-flats in the Hamiltonian diffeomorphism group with respect to both the Hofer metric and the spectral metric over ℤ. We also establish that if the degree k of D is greater than one, the skeleton of M∖ D is SH-full over ℤ/kℤ, and hence non-displaceable in M by any symplectomorphism.
Related
- Relative symplectic cohomology in complex projective spaces
- The Fourth Continuous Bounded Cohomology of the Complex Symplectic Group
- Decomposition and framing of F‐bundles and applications to quantum cohomology
- Relative cohomology and base-change
- Symplectic singularities in Relative Langlands duality for products of general linear groups
BibTeX
@misc{jo2026biran,
title = {{Biran decomposition and relative symplectic cohomology}},
author = {Hong-kwon Jo and Jungsoo Kang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01334},
eprint = {2610.01334},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01334},
}
RIS
TY - UNPB
TI - Biran decomposition and relative symplectic cohomology
AU - Hong-kwon Jo
AU - Jungsoo Kang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01334
UR - https://arxiv.org/abs/2610.01334
AB - Let (M,ω) be a closed monotone integral symplectic manifold with monotonicity constant τ>0, and let D⊂ M be a symplectic hyperplane section of degree 0<k<τ. By Biran's decomposition theorem, M is the union of a symplectic disk bundle over D and the skeleton of the complement M∖ D equipped with its Liouville structure. In this paper, we compute the relative symplectic cohomology of circle subbundles of this disk bundle over ℤ in terms of the quantum cohomology of D. This relative symplectic cohomology depends on the radius of the circle bundle relative to a certain critical radius. For a broad class of pairs (M,D), we prove that the relative symplectic cohomology over ℤ does not vanish for supercritical radii. Therefore, circle bundles of supercritical radii are heavy subsets, which yields infinite dimensional quasi-flats in the Hamiltonian diffeomorphism group with respect to both the Hofer metric and the spectral metric over ℤ. We also establish that if the degree k of D is greater than one, the skeleton of M∖ D is SH-full over ℤ/kℤ, and hence non-displaceable in M by any symplectomorphism.
ER -