Period mappings and properties of the augmented Hodge line bundle

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.