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Stability of vector bundles and reduction mod p

  • Adrian Langer
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Abstract

Let E be a vector bundle on a normal complex projective variety. We prove that slope stability of mathopSym2E and Zariski density of the reductions of E modulo p that are not strongly geometrically stable imply that the bundle E is projectively unitary flat. This generalizes Shepherd--Barron's result from rank $2$ to arbitrary rank. We also show that if E has rank ≤ 3 we can replace stability of mathopSym2E by stability of E but we cannot do so if E has rank $4$. We also generalize these results to reflexive sheaves on some mildly singular normal complex projective varieties.

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Cite

BibTeXRIS
BibTeX
@misc{langer2026stability,
  title = {{Stability of vector bundles and reduction mod $p$}},
  author = {Adrian Langer},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14276},
  eprint = {2610.14276},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14276},
}
RIS
TY  - UNPB
TI  - Stability of vector bundles and reduction mod p
AU  - Adrian Langer
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.14276
UR  - https://arxiv.org/abs/2610.14276
AB  - Let E be a vector bundle on a normal complex projective variety. We prove that slope stability of mathopSym^2E and Zariski density of the reductions of E modulo p that are not strongly geometrically stable imply that the bundle E is projectively unitary flat. This generalizes Shepherd--Barron's result from rank $2$ to arbitrary rank. We also show that if E has rank ≤ 3 we can replace stability of mathopSym^2E by stability of E but we cannot do so if E has rank $4$. We also generalize these results to reflexive sheaves on some mildly singular normal complex projective varieties.
ER  -