Toroidal Compactifications and Stacky Cohomology of Mumford--Tate Domains

Mumford--Tate domains parametrize polarized Hodge structures with fixed Mumford--Tate group and play a central role in the geometry of period maps. Their degenerations are governed by nilpotent orbits and limiting mixed Hodge structures, whose asymptotics are encoded in the logarithmic compactifications of Kato--Usui. In this paper we construct and study the \emph{log--toric Hodge stack} $$\cD^{\log}_{\MT,\Sigma} := [D_{\MT,\Sigma}/\Gamma],$$ obtained from a Mumford--Tate domain $\DM$ and a fan $\Sigma$ of nilpotent cones by forming the quotient of the Kato--Usui partial compactification $D_{\MT,\Sigma}$ by a neat arithmetic group $\Gamma \subset \MT(\Q)$. We show that $\cD^{\log}_{\MT,\Sigma}$ is a global quotient Deligne--Mumford stack, that it admits a natural logarithmic structure extending the period domain, and that near every boundary stratum associated to a cone $\sigma\in\Sigma$ it admits a canonical analytic log--\'etale chart of the form $$\bigl([F_\sigma/G_\sigma]\times \cT_\sigma\bigr)^\circ,$$ where $F_\sigma$ is the space of nilpotent orbits modulo unipotent actions, $G_\sigma$ is a finite symmetry group of the associated limiting mixed Hodge structures, and $\cT_\sigma$ is a toric Deligne--Mumford stack refining the toric variety $D_\sigma$ attached to $\sigma$. This decomposition cleanly separates Hodge-theoretic information from the combinatorial and stacky boundary data.