Sheaves of AQ normal series and supermanifolds

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 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 $(X, T^{\ast}_{X,-})$, where $X$ is a complex manifold and $T^{\ast}_{X,-})$ is a holomorphic vector bundle. We are lead to the notion of an ‘obstruction complex’ associated to a model $(X, T^{\ast}_{X,-})$ 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 $(X, T^{\ast}_{X,-})$ vanishes, then any supermanifold modelled on $(X, T^{\ast}_{X,-})$ will be split.

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Published 30 March 2023