Toroidal Compactifications and Stacky Cohomology of Mumford--Tate Domains

Abstract

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} \[ \cD^{\log}_{\MT,Σ} := [D_{\MT,Σ}/Γ], \] obtained from a Mumford--Tate domain $\DM$ and a fan $\Sigma$ of nilpotent cones by forming the quotient of the Kato--Usui partial compactification $D_{\MT,Σ}$ by a neat arithmetic group $Γ\subset \MT(\Q)$. We show that $\cD^{\log}_{\MT,Σ}$ 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--étale chart of the form \[ \bigl([F_σ/G_σ]\times \cT_σ\bigr)^\circ, \] 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 $\cT_σ$ 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.