When P. A. Griffiths (Amer. J. Math., 90 (1968), 568-626) introduced the classifying space D of polarised Hodge structures, he was already aware of the fact that it was quite desirable to add to this space points at infinity. This book is a realisation of Griffiths’ idea. Technically, it is not at all easy and the authors describe a whole series of enlargements of the classifying space. The reader is advised to look at the fundamental diagram, where they can find the enlargements and mappings. To accomplish the enlargement idea, Kato and Usui develop a logarithmic Hodge theory. Here they use the logarithmic structures created by J. M. Fontaine and L. Illusie. They divide the exposition into two topics. The first one consists of “Toroidal partial compactifications and moduli of polarised logarithmic Hodge structures”; the second one is “The eight enlargements of D and the fundamentals diagram“. The book is a highly specialised monograph, which will be appreciated first of all by specialists in the field. But it is well and carefully written and even a beginner can learn a lot from it. It is recommended for the beginner to skip the overview on a first reading and start directly with chapter 1. At the end, there is list of symbols and an index. Both of them are very good and help substantially with orientation in the book. It is a nice book and can be strongly recommended.

Reviewer:

jiva