2015/01/27 by Mohammad Reza Rahmati, Rahmati, Mohammad Reza
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1501.06886
openalex publication_date 2015/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 log--toric Hodge stack \cDlog\MT,Σ := [D\MT,Σ/Γ], obtained from a Mumford--Tate domain \DM and a fan Σ of nilpotent cones by forming the quotient of the Kato--Usui partial compactification D\MT,Σ by a neat arithmetic group Γ⊂ \MT(\Q). We show that \cDlog\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 σ∈Σ it admits a canonical analytic log--étale chart of the form ([Fσ/Gσ]× \cTσ)^∘, where Fσ is the space of nilpotent orbits modulo unipotent actions, Gσ is a finite symmetry group of the associated limiting mixed Hodge structures, and \cTσ is a toric Deligne--Mumford stack refining the toric variety Dσ attached to σ. This decomposition cleanly separates Hodge-theoretic information from the combinatorial and stacky boundary data.