Indeterminacy of period map and Griffiths heights for hypersurfaces

Source abstract

We prove that the period map $\overline {\mathcal P}$, from the GIT moduli space of degree $d$ hypersurface $X$ in $\mathbb P^n$ to the Baily-Borel compactification of the period image associated to the middle cohomology $H^{n-1}_{\rm prim}(X,\mathbb Z)$, extends regularly to $X$ if and only if $X$ has at worst nodal singularities (resp. suspensions of slc surface singularities) when $n$ is even (resp. $n$ is odd). This verifies and generalizes Laza-O'Grady's prediction on the indeterminacy locus of the period map for quartic K3s to arbitrary hypersurfaces. For Calabi-Yau hypersurfaces, we prove that the birational map from the GIT moduli space to the BBH compactification extends regularly to $X$ if and only if $X$ has semi-log canonical singularities. The results apply to distinguish the Baily-Borel and BBH compactifications. In the process, we answer two questions of Mordant on the Griffiths height for families of hypersurfaces over curves, which are crucial to the proofs of the main results.