Associated Varieties of Finite-Group Orbifolds

Abstract

Let $V$ be a simple, strongly finitely generated vertex algebra with a connected nonnegative integer grading, and let $G$ be a finite group of grading-preserving automorphisms. In this paper, we prove that $X_{V^G}\cong X_V/G$ as reduced Poisson varieties. More precisely, the canonical surjection $R_{V^G}\longrightarrow (R_V)^G $ has nilpotent kernel. The proof uses a uniform derivative bound in a relative $C_2$-quotient, which also implies that $V^G$ is strongly finitely generated. Consequently, quasi-lisseness and $C_2$-cofiniteness descend to finite-group orbifolds, and hence, by existing results, so does strong rationality.