The atypicality conjecture for support varieties of basic classical Lie superalgebras

Let ${\mathfrak{g}}={\mathfrak{g}}_{\0}\oplus {\mathfrak{g}}_{\1}$ be a basic classical Lie superalgebra over the complex numbers, let $\mathcal{F}$ be the category of finite-dimensional integrable ${\mathfrak{g}}$-supermodules, and let $\mathcal{V}_{({\mathfrak{g}},{\mathfrak{g}}_{\0})}(M)$ denote the support variety of a supermodule $M$. For a simple supermodule $L(\lambda)$ of highest weight $\lambda$, let $\operatorname{atyp}(\lambda)$ denote its atypicality.

The atypicality conjecture. For every basic classical Lie superalgebra ${\mathfrak{g}}$ and every simple supermodule $L(\lambda)$ in $\mathcal{F}$,

$$\dim \mathcal{V}_{({\mathfrak{g}},{\mathfrak{g}}_{\0})}(L(\lambda))=\operatorname{atyp}(\lambda).$$

This conjecture gives a geometric interpretation of atypicality through support varieties. It was proved for ${\mathfrak{gl}}(m|n)$, and this paper proves it for ${\mathfrak{osp}}(m|2n)$; the general formulation beyond these cases is not asserted as solved here.