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\({\mathbb A}^{0|1}\)-Torsors, Quotients by Free \({\mathbb A}^{0|1}\)-Actions, and Embeddings into \(\Pi \)-Projective Spaces and Super-Grassmannians G(1|1, n|n)

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Abstract

We study embeddability of superschemes into \(\Pi \)-projective spaces and into supergrassmannians G(1|1, n|n). We give some criteria based on the relation with \({\mathbb A}^{0|1}\)-torsors and \({\mathbb A}^{0|1}\)-fibrations. We also prove the existence of nice quotients for free actions of \({\mathbb A}^{0|1}\) on superschemes.

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Notes

  1. In [8] the group Q(1) is denoted by \(\mathbb {G}_m\) (and what we denote by \(\mathbb {G}_m\) is denoted by \(\mathbb {G}_m^{1|0}\)).

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Acknowledgements

I am grateful to Arkady Vaintrob for helpful discussions and to Emile Bouaziz for suggesting interesting examples of embeddable CY 2|2 supervarieties (see Proposition 5.7). I also thank the anonymous referee for useful comments and for suggesting a simpler proof of Lemma 3.5.

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  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA

    Alexander Polishchuk

  2. National Research University Higher School of Economics, Moscow, Russia

    Alexander Polishchuk

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  1. Alexander Polishchuk
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Correspondence to Alexander Polishchuk.

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Communicated by N. Nekrasov.

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Partially supported by the NSF grant DMS-2001224, and within the framework of the HSE University Basic Research Program and by the Russian Academic Excellence Project ‘5-100’.

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Polishchuk, A. \({\mathbb A}^{0|1}\)-Torsors, Quotients by Free \({\mathbb A}^{0|1}\)-Actions, and Embeddings into \(\Pi \)-Projective Spaces and Super-Grassmannians G(1|1, n|n). Commun. Math. Phys. 402, 2011–2029 (2023). https://doi.org/10.1007/s00220-023-04769-8

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