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Cattani, Kaplan and Schmid (1986) established the SL(2)-orbit theorem in several variables for the degeneration of polarized Hodge structure. The aim of the present paper is to generalize it for the degeneration of mixed Hodge structure whose graded quotients by the weight filtration are polarized. As an application, we obtain a mixed Hodge version of an estimate of the Hodge metric for the degeneration of polarized Hodge structure.

For a linear algebraic group G over Q, we consider the period domains D classifying G-mixed Hodge structures, and construct the extended period domains D BS , D SL(2) , and Γ \ D Σ . In particular, we give toroidal partial compactifications of mixed Mumford-Tate domains. Contents §0. Introduction §1. The period domain D 1.1 Definition of the period domain D 1.2 Relation to [11] Section 5 and [12] 1.5-1.8 of Deligne 1.3 The real analytic structure of D 1.4 The complex analytic structure of D 1.5 Polarizability 1.6 Relations with usual period domains and Mumford-Tate domains §2. The space of Borel-Serre orbits 2.1 Real analytic manifolds with corners 2.2 Borel-Serre liftings 2.3 Review of Borel-Serre theory 2.4 The set D BS 2.5 The real analytic structure of D BS 2.6 Global properties of D BS §3. The space of SL(2)-orbits 3.1 The set D SL(2) when G is reductive

We construct toroidal partial compactifications of the moduli spaces of mixed Hodge structures with polarized graded quotients. They are moduli spaces of log mixed Hodge structures with polarized graded quotients. We construct them as the spaces of nilpotent orbits.

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