Associated Varieties of Ordinary Modules over Quasi-Lisse Vertex Algebras

Abstract: We prove that if $V$ is a conical simple self-dual quasi-lisse vertex algebra and $M$ is an ordinary module then $\dim X_M=\dim X_V$. Hence, if moreover $X_V$ is irreducible then $X_M=X_V$. In particular, this applies to quasi-lisse simple affine vertex algebras $L_{k}(\mathfrak{g})$. For admissible $k$ it reproves a result in \cite{A2}, and it further extends it to non-admissible levels.