Abstract
Attached to a vertex algebra \({\mathcal V}\) are two geometric objects. The associated scheme of \({\mathcal V}\) is the spectrum of Zhu’s Poisson algebra \(R_{{\mathcal V}}\). The singular support of \({\mathcal V}\) is the spectrum of the associated graded algebra \(\text{gr}({\mathcal V})\) with respect to Li’s canonical decreasing filtration. There is a closed embedding from the singular support to the arc space of the associated scheme, which is an isomorphism in many interesting cases. In this note we give an example of a non-quasi-lisse vertex algebra whose associated scheme is reduced, for which the isomorphism is not true as schemes but true as varieties.
Dedicated to Professor Anthony Joseph on his seventy-fifth birthday
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Notes
- 1.
Provided that \({\mathcal V}\) is finitely strongly generated.
- 2.
Unless \({\mathcal V}\) is finite-dimensional.
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Acknowledgements
T. A. is supported by JSPS KAKENHI Grants #17H01086 and #17K18724. A. L. is supported by Simons Foundation Grant #318755. We thank Julien Sebag for helpful comments on an earlier draft of this paper.
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Arakawa, T., Linshaw, A.R. (2019). Singular Support of a Vertex Algebra and the Arc Space of Its Associated Scheme. In: Gorelik, M., Hinich, V., Melnikov, A. (eds) Representations and Nilpotent Orbits of Lie Algebraic Systems. Progress in Mathematics, vol 330. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-23531-4_1
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