Line bundles on super Riemann surfaces

We give the elements of a theory of line bundles, their classification, and their connections on super Riemann surfaces. There are several salient departures from the classical case. For example, the dimension of the Picard group is not constant, and there is no natural hermitian form on Pic. Furthermore, the bundles with vanishing Chern number aren't necessarily flat, nor can every such bundle be represented by an antiholomorphic connection on the trivial bundle. Nevertheless the latter representation is still useful in investigating questions of holomorphic factorization. We also define a subclass of all connections, those which are compatible with the superconformal structure. The compatibility conditions turn out to be constraints on the curvature 2-form.