Superschemes and supermanifolds with free 𝔸^0|n-action

Abstract

We prove that given an analytic supermanifold X over a complete field on which the additive supergroup 𝔸^0|n of super-dimension 0|n acts freely (in an appropriate sense), the quotient X/𝔸^0|n exists as an analytic supermanifold, and the canonical morphism X → X/𝔸^0|n is an 𝔸^0|n-principal bundle. A stronger version of this result is proven for C^∞-supermanifolds, whereby X → X/𝔸^0|n is a trivial 𝔸^0|n-principal bundle. Previously, Polishchuk [18] established an algebro-geometric analogue, which proves that a superscheme X with free 𝔸^0|n-action gives rise to a locally trivial 𝔸^0|n-torsor X → X/𝔸^0|n. This result is proven here in an alternative, elementary way. To all of these, a Hopf-algebraic approach is effectively employed.

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BibTeX RIS

BibTeX

@misc{masuoka2026superschemes,
  title = {{Superschemes and supermanifolds with free $\mathbbA^0|n$-action}},
  author = {Akira Masuoka and Takuya Oe},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25564},
  eprint = {2610.25564},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25564},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Superschemes and supermanifolds with free 𝔸^0|n-action
AU  - Akira Masuoka
AU  - Takuya Oe
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25564
UR  - https://arxiv.org/abs/2610.25564
AB  - We prove that given an analytic supermanifold X over a complete field on which the additive supergroup 𝔸^0|n of super-dimension 0|n acts freely (in an appropriate sense), the quotient X/𝔸^0|n exists as an analytic supermanifold, and the canonical morphism X → X/𝔸^0|n is an 𝔸^0|n-principal bundle. A stronger version of this result is proven for C^∞-supermanifolds, whereby X → X/𝔸^0|n is a trivial 𝔸^0|n-principal bundle. Previously, Polishchuk [18] established an algebro-geometric analogue, which proves that a superscheme X with free 𝔸^0|n-action gives rise to a locally trivial 𝔸^0|n-torsor X → X/𝔸^0|n. This result is proven here in an alternative, elementary way. To all of these, a Hopf-algebraic approach is effectively employed.
ER  -

Abstract

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