Compactifications of moduli spaces of mixed Hodge structure and log geometry

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We constructed the Borel-Serre compactification and the SL(2)-compactification of the moduli space of the mixed Hodge structures by the method of log geometry. The Borel-Serre compactification is constructed by the Borel-Serre orbits and has nice properties such as local compactness. The SL(2)-compactification is constructed by the SL(2)-orbits and closely related to the compactification by the nilpotent oribits (the mixed version of Kato-Usui space) via the SL(2)-orbit theorem, which associates an SL(2)-orbit to a nilpotent orbit.