This compact book is a fabulous contribution to the literature on the gorgeous and important subject of Riemann surfaces, penned by a grandmaster. I guess this effusive phrase requires justification, so here goes. Obviously Donaldson is a grandmaster known for his work on 4-manifolds, and therefore learning about geometric mainstays like Riemann surfaces from him should be a treat on any number of counts. And it is. I’m reminded of the parallel in algebraic geometry as regards the role played by what was once called simply Mumford’s Red Book (i.e. what Springer now markets, in its usual yellow and black, as The Red Book of Varieties and Schemes): you get to find out not only how the subject works — even at its early stages — but you get the scoop from, yes, a grandmaster.
Now, Mumford’s Red Book is famously (or notoriously) informal and written in Mumford’s famous (or notorious) style: he lectures to you from the page, tries to get you to see things from his vantage point, and at the same time proves everything in wonderful detail; in other words, it’s the experience of being, as it were, only one step removed from sitting in his course as a future algebraic geometer.
Does Donaldson see his pedagogical charge the same way? Riemann Surfaces is not as informal as the Red Book, but that difference may be a function of the authors’ personalities. What is true, however, and obvious, is that Donaldson also lectures from the page, and does so wonderfully effectively. For instance, here he is in action as regards the Uniformisation (or Uniformization) Theorem: “In this chapter  we prove the following theorem. … Let X be a connected, simply connected, non-compact Riemann surface. Then X is equivalent to either C or the upper half-plane H. Corollary: Every connected Riemann surface is equivalent to one of the following: … the Riemann sphere S2; C or C/Z … or C/Λ for some lattice Λ; a quotient H/Γ, where Γ … is a discrete subgroup [of PSL(2,R)] acting freely on H. The corollary follows because any Riemann surface is a quotient of its universal cover by an action of its fundamental group, and we have seen that the only simply connected Riemann surface is the Riemann sphere.” This passage very clearly displays the grandmaster’s touch, doesn’t it? The explanation Donaldson gives of the ever-so-important corollary is as elegant and concise as can be: the topology hiding in the background is given just the right amount of airplay, and the hugely important role of quotients of universal covers is featured (a ploy that arises all over mathematics).
After this Donaldson gets to some pedagogy proper: “Our proof … will follow the same general pattern as the one we have already given to classify compact simply connected Riemann surfaces but the non-compactness will require some extra steps …” Yes, the structure of Donaldson’s Riemann Surfaces is, so to speak, exactly as it should be, with the themes arranged in increasing order of complexity, and with the tools and methods laid out so as to take maximum advantage of similarities, parallels, and analogues. In other words, by travelling along with Donaldson you start, to an extent, to think about this kind of geometry like a geometer.
For me it has been many years since I’ve had occasion to deal with Riemann surfaces: the bulk of such activity for me came at the very start of my graduate school education, or even before it, when the late Basil Gordon (may God rest his soul: he passed away in 2012) presented a masterful treatment of modular forms to an audience of budding number theorists (like me). The third case mentioned in the above corollary, that of “a quotient “H/Γ, where Γ … is a discrete subgroup [of PSL(2,R)] acting freely on H,” is of course at the very heart of the entire affair, and I recall vividly how I was struck by the beauty both of the subject itself, and of Gordon’s masterful presentation. It is wonderful to look at this material again — and so much more besides — in Donaldson’s book.
So what are the other things Donaldson does? Well, to continue with the theme of my foregoing reminiscence, his sixth chapter is devoted to elliptic functions and elliptic integrals, which is to say one of the most beautiful and important themes of 19th century mathematics: Abel, Weierstrass, Riemann, and theta functions. Then his seventh chapter hits the Euler characteristic and some of its applications; the chapter finishes with Riemann-Hurwitz and modular curves. Before that it’s all about the “Preliminaries” and the “Basic Theory.” Indeed, chapters 6 and 7 make for something of a transition to what is arguably the heart of the book, the “Deeper Theory,” in which Donaldson presents the material discussed earlier, surrounding the uniformization theorem, and this discussion is set up by a treatment of compact Riemann surfaces (and the Main Theorem: see below). The Riemann-Roch Theorem is encountered in this setting.
After carrying out all this work on said “deeper theory,” Donaldson treats the reader to a smorgasbord of “Further Developments,” including the following: Gauss-Bonnet, Abel-Jacobi (preceded by a discussion of (sheaf) cohomology and line bundles), moduli theory (another Mumford favorite, of course), Dehn twists, and, in the context of ODEs, periods of holomorphic forms, the hypergeometric equation and the Gauss-Manin connection. Phenomenally beautiful material. (By the way, as a number theorist I cannot resist mentioning sections 1.2 and 1.3 of Ch. 11: valuations and “connections with algebraic number theory,” culminating in an excursion into the p-adic numbers).
Now, as indicated above, a few words about the Main Theorem are both irresistible and instructive regarding the Gestalt of what Donaldson provides us with. The statement is this: “Let X be a compact connected Riemann surface and let ρ be a 2-form on X. There is a solution f to …Δf = ρ if and only if the integral of ρ over X is zero, and then the solution is unique up to the addition of a constant.” This immediately discloses the need for quite a bit of differential geometry proper (i.e. differential forms), and in Donaldson (modern) treatment this is brought about with heavy use of cohomologies (Dolbeault and de Rham, and elsewhere sheaf cohomology, of course). I think this is certainly the way to go: these methods are elegant and extremely powerful, and, beyond that, the reader, in using them, becomes exposed to them “in context.” In point of fact, this characterization applies to the entire book: you’ll learn a great deal of differential geometry, as well as topology, as you go through the first part, and then, when the core material rolls around, Donaldson brings you to some rather deep and serious mathematics in relatively short order.
As I said at the start, it’s a grandmaster who’s leading these explorations (and he’s even obliged his readers with exercises throughout). The book is simply wonderful.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.