  Let $P$ be the image of a period map. We discuss progress towards a
conjectural Hodge theoretic completion $\overline{P}$, an analogue of the
Satake-Baily-Borel compactification in the classical case. The set
$\overline{P}$ is defined and given the structure of a compact Hausdorff
topological space. We conjecture that it admits the structure of a compact
complex analytic variety. We verify this conjecture when $\mathrm{dim} P \le
2$. In general, $\overline{P}$ admits a finite cover $\overline{S}$ (also a
compact Hausdorff space, and constructed from Stein factorizations of period
maps). Assuming that $\overline{S}$ is a compact complex analytic variety, we
show that a lift of the augmented Hodge line bundle $\Lambda$ extends to an
ample line bundle, giving $\overline{P}$ the structure of a projective normal
variety. Our arguments rely on refined positivity properties of Chern forms
associated to various Hodge bundles; properties that might be of independent
interest.
