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Smooth compactifications of locally symmetric varieties. With the collaboration of Peter Scholze. 2nd ed. (English) Zbl 1209.14001

Cambridge Mathematical Library. Cambridge: Cambridge University Press (ISBN 978-0-521-73955-9/pbk). x, 230 p. (2010).
The book under review is a new edition of the authors’ celebrated research monograph [Smooth compactification of locally symmetric varieties. Lie Groups: History, Frontiers and Applications. Vol. IV. Brookline, Mass.: Math Sci Press. IV, 335 p. (1975; Zbl 0334.14007)], which must be seen as one of the milestones in contemporary algebraic and complex-analytic geometry. Thirty-five years ago, the original text provided a universal method for constructing desingularizations of a class of quotient spaces, which has proven itself to be crucial in the study of moduli spaces in algebraic geometry ever since. Alas, despite its fundamental significance, this standard reference has been unavailable in the last couple of decades, and the request for a second edition has become more and more urgent in the course of time.
Corresponding to the proposal of a second edition of their book, the authors fortunately agreed to undertake such a rewarding project, and the result is the present new edition of this classic. As they point out in the preface to the second edition of the book, the authors have decided to leave the text of the original basically intact, thereby preserving its content, structure, and style of presentation likewise.
However, in this new edition, the text has been completely re-typeset (in TEX), several recognized errors have been corrected, the whole presentation has been polished and streamlined, the notation has been made consistent and uniform, an index has been added, and the further developments in the field in the last three decades are reflected by a supplementary bibliography serving as a very detailed, rather complete guide to the more recent literature in the subject.
Now as before, the text consists of four chapters of varying authorship. As for their respective contents, we may refer to the review of the original edition (loc. cit.) from 1975 by Yu. G. Zarhin, as the basic material has been left essentially unaltered. However, let us briefly recall both the goal and the general structure of this classic monograph.
The objects of study are quotient spaces \(D/\Gamma\), where \(D\) is a bounded symmetric domain and \(\Gamma\) a neat arithmetic subgroup of \(\operatorname{Aut}(D)^0\). In this context, the authors’ goal is the construction of a family of non-singular compactifications \(\overline{D/\Gamma}\) of the space \(D/\Gamma\), generalizing earlier approaches and results by Baily-Borel, Igusa, Hirzebruch, and Satake, respectively. The authors’ approach builds heavily on the theory of toroidal embeddings as developed by G. Kempf, F. Knudsen, D. Numford and B. Saint-Donat [Lecture Notes in Mathematics. 339. Berlin-Heidelberg-New York: Springer-Verlag. VIII, 209 p. (1973; Zbl 0271.14017)].
Chapter 1 of the present monograph provides a quick review of the theory of toroidal embeddings, together with some classical examples illustrating this framework. Chapter 2 develops the basic ingredients from the polyhedral reduction theory of self-adjoint cones, whereas Chapter 3 takes up the explicit construction of the smooth compactifications \(\overline{D/\Gamma}\) of \(D/\Gamma\). Chapter 4 provides two important applications of the theory developed so far. More precisely, it is shown that the quotient \(D/\Gamma\) is an algebraic variety of general type (in Kodaira’s classification) when the arithmetic group \(P\) is sufficiently small, and that the (a priori) analytic smooth compactification \(\overline {D/\Gamma}\) of \(D/\Gamma\) is indeed a projective variety in many concrete cases. In particular, these geometric applications make the significance of the authors’ general approach in moduli theory strikingly evident.
No doubt, this classic will maintain its outstanding role in algebraic geometry, Hermitian differential geometry, group representation theory, and arithmetic geometry also in the future, especially for active researchers and graduate students in these related areas of contemporary pure mathematics. In this regard, the present new edition of it is certainly more than welcome.

MSC:

14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
22E99 Lie groups
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
14M27 Compactifications; symmetric and spherical varieties
01A75 Collected or selected works; reprintings or translations of classics
22E40 Discrete subgroups of Lie groups
53C35 Differential geometry of symmetric spaces
57S30 Discontinuous groups of transformations
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