  We study global and local geometry of forms on odd symplectic BV
supermanifolds, constructed from the total space of the bundle of 1-forms on a
base supermanifold. We show that globally 1-forms are an extension of vector
bundles defined on the base supermanifold. In the holomorphic category, we
prove that this extension is split if and only if the super Atiyah class of the
base supermanifold vanishes. This is equivalent to the existence of a
holomorphic superconnection: we show how this condition is related to the
characteristic non-split geometry of complex supermanifolds. From a local point
of view, we prove that the deformed de Rham double complex naturally arises as
a de-quantization of the de Rham/Spencer double complex of the base
supermanifold. Following \v{S}evera, we show that the associated spectral
sequence yields semidensities on the BV supermanifold, together with their
differential in the form of a super BV Laplacian.
