Study of algebraic varieties by log Hosge theory

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Generalizing toroidal compactifications of Hermitian symmetric domains by Mumford et al., Kazuya Kato and Usui constructed fine moduli spaces of polarized log Hodge structures (PLH, for short). Moreover, we constructed Borel-Serre compactifications and SL(2)-partial compactifications of Griffiths domains, and also a fundamental diagram of the relationship of all these enlarged spaces. This joint will be published as a book of almost 300 pages in the series of Ann. Math. Studies, Princeton University Press.Assuming the existence of a complete fan, Usui showed that the image of the extended period map, from a compactification of moduli of varieties of general type to the moduli of PLH, is a separated complex algebraic space. This observation shows in particular that, even if the moduli space of PLH has slits in this case, the image of the extended period map does not touch with these slits. This result was published in J. Alg. Geom. in 2006. The restriction of dimension in this paper is now removed and applicable for all dimensions by a recent great advance in minimal model theory.Generalizing SL(2)-orbit theorems of Schmid in pure case in one variable and of Cattani, Kaplan and Schmid in pure case in several variables, Kazuya Kato, Chikara Nakayama and Usui obtained SL(2)-orbit theorem in mixed case in several variables and an estimate of Hodge norm in this situation. This result is submitted.Masanori Asakura and Shuji Saito studied the Jacobian rings of open complete intersections and solved Beilinson's Hodge conjecture for sufficiently general open complete intersections. These results were publishes in Math.Zeit., in Math.Nachr., and in publication of London Math.Soc.