On associated variety for Lie superalgebras

We define the associated variety $X_{M}$ of a module $M$ over a finite-dimensional superalgebra ${\mathfrak g}$, and show how to extract information about $M$ from these geometric data. $X_{M}$ is a subvariety of the cone $X$ of self-commuting odd elements. For finite-dimensional $M$, $X_{M}$ is invariant under the action of the underlying Lie group $G_{0}$. For simple superalgebra with invariant symmetric form, $X$ has finitely many $G_{0}$-orbits; we associate a number (rank) to each such orbit. One can also associate a number (degree of atypicality) to an irreducible finite-dimensional representation. We prove that if $M$ is an irreducible ${\mathfrak g}$-module of degree of atypicality $k$, then $X_{M}$ lies in the closure of all orbits on $X$ of rank $k$. If ${\mathfrak g}={\mathfrak g}{\mathfrak l}(m|n)$ we prove that $X_{M}$ coincides with this closure.