  This work is part of a project to construct completions of period mappings. A
proper topological SBB-esque completion is constructed. The fibres of are
projective varieties, and the image is a union of quasi-projective varieties;
one wants to endow the topological completion with a compatible algebraic
structure. This raises questions about: (i) the global geometry of the fibres;
and (ii) the existence of period matrix representations on neighborhoods of
such fibres over which the restricted extension is still proper. The purpose of
this paper is to investigate these questions.
