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Generalized Bogomolov--Gieseker inequalities via Frobenius

  • Chunyi Li
  • Zhiyu Liu
  • Alexander Perry
  • Xiaolei Zhao
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Abstract

We establish a Bogomolov--Gieseker-type inequality for tilt-semistable objects on smooth projective varieties of dimension ≥ 3, in the spirit of the Bayer--Macrì--Toda conjecture. A key ingredient is an analysis of the action of the Frobenius morphism on tilt-semistable objects. As an application, we prove the Castelnuovo Bound Conjecture, up to an explicit constant term. Combining our results with work of Feyzbakhsh and Thomas, we further show that on a Calabi--Yau threefold, arbitrary rank Donaldson--Thomas invariants of semistable sheaves are determined by rank one invariants, and hence are symplectic invariants.

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Cite

BibTeXRIS
BibTeX
@misc{li2026generalized,
  title = {{Generalized Bogomolov--Gieseker inequalities via Frobenius}},
  author = {Chunyi Li and Zhiyu Liu and Alexander Perry and Xiaolei Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18595},
  eprint = {2610.18595},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18595},
}
RIS
TY  - UNPB
TI  - Generalized Bogomolov--Gieseker inequalities via Frobenius
AU  - Chunyi Li
AU  - Zhiyu Liu
AU  - Alexander Perry
AU  - Xiaolei Zhao
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18595
UR  - https://arxiv.org/abs/2610.18595
AB  - We establish a Bogomolov--Gieseker-type inequality for tilt-semistable objects on smooth projective varieties of dimension ≥ 3, in the spirit of the Bayer--Macrì--Toda conjecture. A key ingredient is an analysis of the action of the Frobenius morphism on tilt-semistable objects. As an application, we prove the Castelnuovo Bound Conjecture, up to an explicit constant term. Combining our results with work of Feyzbakhsh and Thomas, we further show that on a Calabi--Yau threefold, arbitrary rank Donaldson--Thomas invariants of semistable sheaves are determined by rank one invariants, and hence are symplectic invariants.
ER  -