The \(n\)-dimensional Cremona group \(\mathrm{Bir}(\mathbb P^n_{\mathbb C})\) is the group of the birational maps of the complex projective space \(\mathbb P^n_\mathbb C\) in itself. For \(n=1\) it coincides with the projective general linear group \(\mathrm{PGL}(2,\mathbb C)\). For \(n=2\) a classical theorem due to Max Noether and Guido Castelnuovo states that it is generated by \(\mathrm{PGL}(3,\mathbb C)\) and by the standard quadratic transformation; modern accounts of this theorem and on the geometry of the plane Cremona group can be found in the books [M. Alberich-Carramiñana, Geometry of the plane Cremona maps. Berlin: Springer (2002; Zbl 0991.14008)] and [I. V. Dolgachev, Classical algebraic geometry. A modern view. Cambridge: Cambridge University Press (2012; Zbl 1252.14001)]. On the other hand, for \(n\geq 3\) the structure of the Cremona group is much more complicated and no analogue of the theorem of Noether-Castelnuovo can be proved (see, in particular, [H. P. Hudson, Cremona transformations in plane and space. Cambridge, University Press (1927; JFM 53.0595.01)].
The purpose of the book under review is to give an account of the many new results on these groups obtained in the last twenty years, in particular concerning their subgroups. It consists of nine chapters, at the beginning of each chapter there is an extended summary of its contents.
Chapter 1 contains the basic definitions and examples, in particular using the language of linear systems and divisors.
Chapter 2 contains the definition of the hyperbolic space of infinite dimension \(\mathbb H^\infty(S)\) associated to a projective surface \(S\); any birational selfmap \(\phi\) of \(S\) induces an isometry of \(\mathbb H^\infty(S)\). Isometries of this space are classified in three types – elliptic, loxodromic, parabolic – and there is a dictionary between this classification and the properties of \(\phi\).
Chapter 3 describes two topologies on \(\mathrm{Bir}(\mathbb P^n_{\mathbb C})\): the Euclidean topology, whose restriction to \(\mathrm{PGL}(n+1,\mathbb C)\) is the classical Euclidean topology, and the Zariski topology. It contains the description, due to J. Blanc, of the maximal algebraic subgroups of the plane Cremona group.
Chapter 4 is devoted to the problem of describing the Cremona groups via generators and relations; it contains a proof of the Noether-Castelnuovo theorem, and then discusses recent results about other descriptions of the plane Cremona group, for instance as amalgamated product of subgroups. It also explains why there is no analogue of Noether-Castelnuovo theorem in higher dimension.
Chapter 5 discusses the existence of linear representations and representations of subgroups of \(SL(n,\mathbb Z)\), \(n\geq 3,\) of the plane Cremona group.
Chapter 6 is devoted to the finite subgroups of \(\mathrm{Bir}(\mathbb P^2_{\mathbb C})\); after recalling the classical results by Bertini and Castelnuovo, it surveys the many recent results on this topic, in particular about finite abelian subgroups and elements of finite order fixing a curve of geometric genus \(\geq 2\).
In Chapter 7 uncountable maximal abelian subgroups of the plane Cremona group are studied and, as an application, a characterization of the group of automorphisms of \(\mathrm{Bir}(\mathbb P^n_{\mathbb C})\) is given.
Chapter 8 is devoted to explain how the study of the hyperbolic space introduced in Chapter 2 can be applied to the study of infinite subgroups of the Cremona groups \(\mathrm{Bir}(\mathbb P^n_{\mathbb C})\).
Finally Chapter 9 focuses on automorphisms of surfaces with positive entropy. The following problem is considered: “ When is a birational selfmap of a complex projective surface birationally conjugate to an automorphism?” Three answers are given.
This book is a very rich and updated source of information on many aspects (even if not all) of the study of Cremona groups, explaining the problems, the techniques and the difficulties, and containing a rich bibliography. It is recommended to anyone interested in this fascinating topic.