So, what’s a J-holomorphic curve? Well, as the Preface to the first edition of the book under review states, it goes back to a 1985 paper by Mikhail Gromov, titled “Pseudo-holomorphic curves in symplectic manifolds,” and on p.3 of the book McDuff and Salamon give its definition as a (j,J)-holomorphic mapping from a Riemann surface (with j being — well, what else? — its j-invariant) to an almost complex manifold with its almost complex structure coming from (a matrix) J; thus, J2 = –12n, if we’re playing in 2n dimensions. In this context we have that the target manifold is a 2n-dimensional smooth manifold equipped with a symplectic form ω and it’s part of the game that J is “tamed” by ω, meaning that ω(v,Jv) is positive for all tangent vectors v.
All right: this is rather heady — pretty sophisticated stuff. What does it mean? Well, for one thing, “these curves satisfy a non-linear analogue of the Cauchy-Riemann equations” and their theory and the attendant methods have come to “permeate almost all … aspects” of symplectic geometry and are
of interest in the study of Kähler manifolds, since often when one studies properties of holomorphic curves in such manifolds it is useful to perturb the complex structure to be generic … [but] the perturbed structure may no longer be integrable, and … one is led to … curves that are holomorphic with respect to some nonintegrable complex structure J.
Thus, this second edition AMS Colloquium publication presupposes a lot on the part of the reader, including a good deal of expertise in (or at least some serious familiarity with) Riemann surfaces, complex manifolds, symplectic geometry (and topology), and so on. Regarding the latter, McDuff and Salamon recommend their own book (Introduction to Symplectic Topology) as preparation, or “more elementary treatments such as Berndt [An Introduction to Symplectic Geometry] and Cannas de Silva [Lectures on Symplectic Geometry] as well as classics such as Arnol’d [Mathematical Methods in Classical Mechanics].” I guess that’s really non-negotiable: this book is not aimed at rookies.
In fact the ostensible purpose of the first edition was to provide “an expository account … that explained the main technical steps in the theory of J-holomorphic curves,” but the subject’s rapid growth led to the authors’ subsequent aim “to establish the fundamental theorems in the subject in full and rigorous detail …[as well as to provide] an introduction to current work in symplectic topology.”
In other words, this work is intended to provide a bridge to very hot contemporary mathematics, involving a number of deep geometrical themes. Among these we encounter such things as Gromov-Witten invariants, moduli spaces, quantum cohomology, and Floer homology — to give but a sampling. This is clearly very serious material, requiring a good deal of preparation including algebraic geometry, differential geometry (on anabolic steroids), Riemannian geometry, and symplectic geometry and topology.
McDuff and Salamon note that their second edition is intended to “correct … errors … and update [their] discussion of current work in the field.” So we have in the book under review a high-level and high-quality presentation of a beautiful and important part of modern geometry (in the wake of Gromov, so to speak), fitted into on the order of 700 pages, and this still makes for a compact treatment: the presentation is concise and to-the-point, and the prose is clear even as it is anything but chatty.
This having been said, there is nonetheless a serious pedagogical intent present in the book. For example, there are plenty of remarks in the narrative, aimed at bringing the reader to a certain vantage point with something of a hand up; and there a lot of examples given. Additionally there are exercises that look to be of a reasonable sort, and there are occasional hints given. And, as promised by McDuff and Salamon, their theorems come equipped with full proofs, again written compactly but with great attention to detail.
All in all J-Holomorphic Curves and Symplectic Topology is a scholarly work of great pedagogical value, and an evident sine quo non for entry into the subject of symplectic topology and the hot areas of research it generates and touches. But be forewarned, serious commitment is required from the reader — and rightly so, of course.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.