The joint project with M.Bolognesi (Rennes) and J.Martens (Edinburgh) on flatness the Oxbury–Pauly abelianization of rank two vector bundles on curves organized into several tasks: a first step must address the fact that the Mumford–Welters connection is only defined on a collection of “pairwise skew subbundles” instead of the whole sum of abelian theta functions, establish the existence of a compatible connection extending them, and inquire uniqueness of it. Once this existence is established, it suffices to check that every summand of the Oxbury–Pauly map (these are uniquely defined) is flat to prove the existence of a unique sum map which is flat. The identification of the respective scalings would certainly be desireable in this step, albeit not strictly speaking necessary to show flatness.

A further facet of this project is a more detailed discussion of these results in genus two, where the non-abelian side reduces to the topic of Kummer surfaces with its rich classical and modern developments. The flat projective connection in this case associates deformations of arbitrary divisors to deformations of the Kummer surface, but the construction is intricate. We aim for a more geometric understanding, at least for the relative anti-canonical sheaf.