Singular Support of a Vertex Algebra and the Arc Space of Its Associated Scheme

Abstract

Attached to a vertex algebra V are two geometric objects. The associated scheme of V isthespectrum of Zhu's Poisson algebra Rv.Thesingular support of V is the spectrum of the associated graded algebra gr(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.