Sheaves of AQ Normal Series and Supermanifolds

Abstract

On a group $G$, a filtration by normal subgroups is referred to as a normal series. If subsequent quotients are abelian, the filtration is referred to as an \emph{abelian-quotient normal series}, or `AQ normal series' for short. In this article we consider `sheaves of AQ normal series'. From a given AQ normal series satisfying an additional hypothesis we derive a complex whose first cohomology obstructs the resolution of an `integration problem'. These constructs are then applied to the classification of supermanifolds modelled on $(X, T^*_{X, -})$, where $X$ is a complex manifold and $T^*_{X, -}$ is a holomorphic vector bundle. We are lead to the notion of an `obstruction complex' associated to a model $(X, T^*_{X, -})$ whose cohomology is referred to as `obstruction cohomology'. We deduce a number of interesting consequences of a vanishing first obstruction cohomology. Among the more interesting consequences are its relation to projectability of supermanifolds and a `Batchelor-type' theorem: if the obstruction cohomology of a `good' model $(X, T^*_{X, -})$ vanishes, then any supermanifold modelled on $(X, T^*_{X, -})$ will be split.