Bridgeland stability and Gieseker stability I: Limit stability
Abstract
We study the large volume limit of Bridgeland stability conditions on smooth projective varieties. For each fixed ray in a suitable angular sector, we prove that an object of Db(X) is Bridgeland (semi)stable in the corresponding hearts for all sufficiently large radii if and only if it is a pure H-Gieseker (semi)stable sheaf. In this sector, we identify the category of coherent sheaves as the limiting heart in the sense of eventual membership. We also identify a perverse coherent limiting heart in the opposite angular sector. The key ingredients are uniform HN-phase estimates for bounded families of pure sheaves, coefficient bounds for the coherent images of potential destabilizers, and a criterion forcing narrow exact sequences in the varying hearts to be bounded saturated coherent sequences.
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BibTeX
@misc{xu2026bridgeland,
title = {{Bridgeland stability and Gieseker stability I: Limit stability}},
author = {Chunkai Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37032},
eprint = {2610.37032},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37032},
}
RIS
TY - UNPB
TI - Bridgeland stability and Gieseker stability I: Limit stability
AU - Chunkai Xu
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37032
UR - https://arxiv.org/abs/2610.37032
AB - We study the large volume limit of Bridgeland stability conditions on smooth projective varieties. For each fixed ray in a suitable angular sector, we prove that an object of D^b(X) is Bridgeland (semi)stable in the corresponding hearts for all sufficiently large radii if and only if it is a pure H-Gieseker (semi)stable sheaf. In this sector, we identify the category of coherent sheaves as the limiting heart in the sense of eventual membership. We also identify a perverse coherent limiting heart in the opposite angular sector. The key ingredients are uniform HN-phase estimates for bounded families of pure sheaves, coefficient bounds for the coherent images of potential destabilizers, and a criterion forcing narrow exact sequences in the varying hearts to be bounded saturated coherent sequences.
ER -