Did you know that the Čech-approach to calculating the cohomology of a manifold with coefficients in the associated sheaf of germs of holomorphic functions is due to none other that Jean Leray himself? — see p.187 of this book. In my own mathematical travels I first came across the Čech approach in Serre’s gorgeous Annals paper “Faisceaux algébriques cohérents,” dating back to 1955 but still one of the best sources for almost all things sheaf, factoring in that Grothendieck hadn’t gotten off the ground yet: his first salvo, the Tôhoku paper, “Sur quelques points d’algèbre homologique,” was still two years away, and his reformulation of sheaf cohomology in strict categorical terms appeared only later.
The book under review, whose birth year is 1965, but whose contents go back to the authors’ Princeton lectures of the early 1960s, certainly sports a lot of cohomology in the style of Serre (and Leray, and H. Cartan, and Kodaira, and Hirzebruch, and so on), but despite the mention of Grothendieck’s seminal (but unfinished) EGA (i.e., his Eléments de Géométrie Algébrique), none of his functorial characterizations are in evidence anywhere: it’s all more in the way of old-fashioned sheaf theory, to coin a phrase.
Nonetheless sheaf cohomology is a wonderful thing, even if it’s still not everyone’s cup of tea. It’s true, I think, that a path toward sheaf theory featuring Čech-style methodology, familiar to every one who knows even a modicum of algebraic topology, makes for a considerably more comfortable experience even for the otherwise unwilling. On p.118 of the book under review, we accordingly encounter a definition of “sheaf of abelian groups” that fits with Serre’s FAC, and what follows is indeed very much along the same lines. (Interestingly we also find on p.120, a phrase in the handwriting of a previous owner of the hard-copy source which was evidently reproduced to yield the present AMS Chelsea edition: “Called stack in Swan, ‘The Theory of Sheaves,’ p.25,” with the corresponding phrase used by Gunning-Rossi being “presheaf”. Piquant!)
Well, enough about my love-affair with sheaves: the book is about several complex variables, not sheaves. However, it is fair to say that in the hands of Henri Cartan, Leray’s sheaves found an application par excellence to “several CV,” with a major feature being the reformulation and extension of the earlier results of Oka which, in turn, mark the first truly radical advance in the subject since the 1930s. Gunning-Rossi note that
a foundation for the subject was laid late in the nineteenth century by Weierstrass, and around the turn of the century by Cousin, Hartogs and Poincaré … Significant work in many directions was achieved by Bergman, Behnke, Bochner and others, in papers appearing from about the mid-1920s until the present time … but the main problems were still there. Then Oka brought into the subject a brilliant collection of new ideas based primarily in the earlier work of H. Cartan, and in a series of papers written between 1936 and 1953, systematically eliminated these problems…
They then go on to say that “Oka’s work had a far wider scope, and it was H. Cartan who realized this and developed an algebraic basis for the theory.” Bingo: Bring in the sheaves.
Now that the cat is out of the bag, then, let the authors speak for themselves some more: “This book has been written with the prospective student of several complex variables in mind” and “The prerequisites for reading this book are, essentially, a good undergraduate training in analysis (principally the classical theory of functions of one complex variable), algebra, and topology.” Plus ça change, plus c’est la meme chose.
So this is really a terrific book. It teaches you a lot of very beautiful and important material that spills over well beyond the ostensible borders of several CV: Gunning-Rossi include discussions of varieties, analytic sheaves, cohomology galore (and yes, the other approach they feature for getting at the the cohomology of a manifold with coefficients in its sheaf of germs of holomorphic functions is due to Dolbeault: see in this connection e.g. Griffiths-Harris, Principles of Algebraic Geometry). There is also coverage of Stein spaces (both geometrically and sheaf-theoretically) and, finally, pseudoconvexity. This is how the book ends, of course; it naturally begins with a very thorough treatment of the basic material on holomorphic functions with Cousin and Weierstrass, for example, making their appearances.
There are good reasons why this book is an AMS Chelsea player. One of these is that it is a pure pleasure to read: the prose is crystal clear and anything but prolix. At the same time it evinces the authors’ experience with this material in the classroom: it’s a pedagogical marvel, very different from many of today’s efforts which err either on the side of too little motivation or on the side of overwhelming density. To wit, on p.17 Gunning-Rossi state and prove the inverse mapping (or function) theorem, while on p.284, closing the book proper, they prove Kodaira’s theorem on projective varieties: they are happy to get into relatively elementary material in some detail, and at the same time they don’t hold back at all when it comes to rather profound and more sophisticated stuff. And the i’s are dotted and the t’s are crossed.
It’s a wonderful book!
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.