The geometry of algebraic curves over an algebraically closed field, has been well understood since the late 19th century, with foundational details taken care in the 20th century. In dimension two we have surfaces. Over the complex numbers, beginning in the mid 19th century, surfaces of low degree on complex projective space \({\mathbb P}^3\) had already caught the attention of geometers. The names of Cayley, Steiner, Clebsch, Noether, and Kummer are associated to this pioneering work, and we still cherish the beautiful theorem that on a smooth cubic surface there are exactly 27 lines.

At the beginning of the 20th century, Castelnuovo, Enriques, Severi, and several other geometers belonging to the classical Italian school of algebraic geometry, attempted a classification of algebraic surfaces, looking for something similar to the classification of algebraic curves. The main tools used in this attempt were birational invariants such as the Betti numbers or the genus.

Nowadays this classification is formulated in terms of the Kodaira dimension of the surface. According to the Enriques-Kodaira classification, there are four birational classes of algebraic surfaces, depending on the value of the Kodaira dimension: \(-\infty\), \(0\), \(1\) or \(2\). There is a finer classification that subdivides each of these four classes in different types according to the values of some of their invariants.

A modern exposition of this classification can be found, first in Shafarevich’s seminar *Algebraic Surfaces* (Proc. Steklov Mat. Inst. 75, 1965, translated by the AMS in 1967), in A. Beauville’s concise *Surfaces algébriques complexes *(Astérisque 54, Soc. Math. France, 1978), in Barth et al., *Compact Complex Surfaces* (Springer, 1984, Second Enlarged Edition, 2004), or in Badescu *Algebraic Surfaces* (Springer, 2001). It should be emphasized that all these references consider only the case when the ground field is the complex numbers, since a classification in positive characteristic, by D. Mumford and E. Bombieri, was only obtained relatively recently, in the late 1970s.

The finer geometry of surfaces in each type or class has kept geometers occupied, either as a beautiful subject of itself or as a testing ground and reality check for the classification of higher dimensional varieties. This brings us to the book under review: K3 *surfaces*, so-named by André Weil in a felicitous moment honoring the pioneering work of Kummer, the fact that every complex algebraic K3 surface is Kähler, the Kodaira classification, and the beautiful K2 mountain in Kashmir, the second highest mountain in Earth. (A *caveat* is in order, since the fact that a complex K3 surface is Kähler was not proved until 1983, by Siu, verifying a conjecture of Piateshki-Shapiro and Shafarevich. Moreover, in the third page of the preface of the book under review, I just learned that there is a mountain in the Himalayas, Broad Peak, apparently also named K3, but I agree with Weil that K2 is stunningly beautiful.

This outline only touches upon some *geometric* aspects of the theory of K3 surfaces, but there is also a rich *arithmetic* side, and for this it is enough to mention that, as part of the general Weil conjectures for algebraic varieties over finite fields, the analogue of the Riemann hypothesis for K3 surfaces was proved by Deligne a few years before his general theorem.

Assuming a basic background in algebraic (or complex) geometry, in the first seven chapters the book under review introduces K3 surfaces, from their definition as algebraic varieties or complex manifolds and their Hodge structure, to moduli spaces of polarized K3 surfaces, surjectivity of the period map and the global Torelli theorem. Along the way, proofs of the Weil and Tate conjectures for K3 surfaces are obtained, via an auxiliary abelian variety.

Chapters eight to ten introduce or review constructions and techniques on K3 surfaces (ample, nef, effective and Kähler cones, vector bundles, and the symplectic structure of the moduli spaces of sheaves), while chapter 12 recalls some facts about the Chow and Grothendieck groups, and then surveys some recent specific results on K3 surfaces, including a discussion of a conjecture of Bloch and Beilinson for K3 surfaces over a number field. Chapters eleven and thirteen discuss particular facts on elliptic K3 surfaces and rational points on projective K3 surfaces. Chapters fourteen and fifteen are devoted to some fundamental results about lattices and symplectic automorphisms of K3 surfaces. In chapters seventeen and eighteen the author treats the Picard and Brauer group of a K3 surface over a field of arbitrary characteristic, which allows him to discuss Tate’s conjecture again. Lastly, in chapter sixteen, using derived categories, the author discusses some derived variations of some results on K3 surfaces.

As can be gathered from this account, the book allows different readings. Some chapters are independent of others, a few of them are very detailed, almost self-contained, while some others chapters can be read as surveys, with precise references in the rich extensive bibliography of 655 items, an extra bonus for the interested reader, giving an up-to-date treatment of the subject. The book leaves out only a few borderline topics; the only exclusion that I regretted was an overview of Enriques surfaces and their relations to K3 surfaces.

The book is a welcome addition to the literature, especially since its scope ranges from a very good introduction to K3 surfaces to the more recent advances on these surfaces and related topics. Before the publication of this book, K3 surfaces were treated in some of the general references on algebraic surfaces mentioned before, for example in the last chapter of the book by Barth et al, and also in Beauville et al *Géometrie des surfaces* K3: *modules et périods* (Astérisque 126, Soc. Math. France, 1985).

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is [email protected].