The original edition of this textbook for graduate students in complex geometry, in which a comprehensive exposition of the modern theory of period mappings in transcendental algebraic geometry and Hermitean differential geometry is provided, appeared about 15 years ago. Written under the same title, and by the same three authors, this book back then gave a systematic introduction to a subject originally developed by Ph. A. Griffiths in the late 1960s, and continuously further advanced by him, and many others until today.
As for a rather detailed and appraising analysis of the first edition, we may refer to our review (Zbl 1030.14004) published in 2003.
The present book is the thoroughly revised, reworked, and significantly expanded second edition of the original. As the authors point out, their ample teaching experience, based on using the text over the past fourteen years, made them realize that several of the core aspects of period domains needed much more explanation, on the one hand, and an appropriate taking up of additional, more recent topics due to newer developments on the other. Especially the Lie group aspects of period domains have been elaborated in more depth than before. In this context, Section 4.3 on variations of Hodge structures as well as Chapter 12 (on the structure of period domains) and Chapter 13 (on curvature estimates for period domains) have been completely rewritten, and a new “Part four” of about 80 pages has been added to the former three main parts of the text. Also, some new material has been interwoven into Chapters 5 and 6, and a more detailed discussion of Higgs bundles can now be found in the reworked Chapter 13.
The new Part 4 is titled “Additional topics” and contains the three further Chapters 15 to 17. The group-theoretic formulation of the concept of a Hodge structure as well as Mumford-Tate groups, together with their associated Mumford-Tate subdomains and period maps, are discussed in Chapter 15. In the sequel, in Chapter 16, Mumford-Tate domains and their quotients by certain discrete subgroups, the so-called Mumford-Tate varieties, are treated in a more abstract, purely axiomatic fashion, and in this context, Shimura varieties are put in their place. Finally, in Chapter 17, various particular subvarieties of Mumford-Tate varieties are described, with a special view toward Hodge loci, the moduli space of cubic surfaces as a Shimura variety, Shimura curves and their embeddings, and other modular subvarieties.
Also, the three appendices of the first edition have been complemented by a new Appendix D, in which the basics on Lie groups and algebraic groups are briefly collected.
Of course, both the bibliography and the index have been suitably updated.
Apart from the reported various improvements and additions, the well-tried text of the original edition has been left largely intact, thereby maintaining, even enhancing the outstanding rôle of this unique, modern textbook as a standard source in this central area of complex geometry.