In 1970, Phillip Griffiths envisioned that points at infinity
could be added to the classifying space D of polarized Hodge
structures. In this book, Kazuya Kato and Sampei Usui realize this
dream by creating a logarithmic Hodge theory. They use the
logarithmic structures begun by Fontaine-Illusie to revive
nilpotent orbits as a logarithmic Hodge structure.
The book focuses on two principal topics. First, Kato and Usui
construct the fine moduli space of polarized logarithmic Hodge
structures with additional structures. Even for a Hermitian
symmetric domain D, the present theory is a refinement of the
toroidal compactifications by Mumford et al. For general D, fine
moduli spaces may have slits caused by Griffiths transversality at
the boundary and be no longer locally compact. Second, Kato and
Usui construct eight enlargements of D and describe their relations
by a fundamental diagram, where four of these enlargements live in
the Hodge theoretic area and the other four live in the
algebra-group theoretic area. These two areas are connected by a
continuous map given by the SL(2)-orbit theorem of
Cattani-Kaplan-Schmid. This diagram is used for the construction in
the first topic.
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