Complexity and Module Varieties for Classical Lie Superalgebras

Let
$\mathfrak g=\mathfrak g_{\bar 0}\oplus \mathfrak g_{\bar 1}$
be a classical Lie superalgebra and
$\mathcal F$
be the category of finite-dimensional
$\mathfrak g$
-supermodules which are semisimple over
$\mathfrak g_{\bar 0}$
. In this paper we investigate the homological properties of the category
$\mathcal F$
. In particular, we prove that
$\mathcal F$
is self-injective in the sense that all projective supermodules are injective. We also show that all supermodules in
$\mathcal F$
admit a projective resolution with polynomial rate of growth, and, hence, one can study complexity in ℱ. If
$\mathfrak g$
is a type I Lie superalgebra, we introduce support varieties which detect projectivity and are related to the associated varieties of Duflo and Serganova. With this new approach, we prove that the conditions of being tilting or projective are equivalent for type I Lie superalgebras.