Singular support of a vertex algebra and the arc space of its associated scheme

TL;DR

This paper explores the relationship between the singular support of a vertex algebra and the arc space of its associated scheme, providing examples where the natural embedding is not an isomorphism of schemes.

Contribution

It offers a specific example of a vertex algebra where the singular support and arc space are not scheme-isomorphic, clarifying the nuanced relationship between these geometric objects.

Findings

The associated scheme of the vertex algebra can be reduced.

The embedding from singular support to arc space can fail to be an isomorphism of schemes.

In some cases, the isomorphism holds as varieties but not as schemes.

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.