Of course everyone knows Abel's exhortation that we should seek out "the masters, not their pupils," if we are to learn mathematics well and effectively. But do we take Abel's advice to heart in this day and age when the sheer size of the mathematical community and the mathematical enterprise brings such huge numbers of books and papers before our eyes? A topic like that of complex manifolds not only sports dozens of sources available to a student or an outside researcher, but there are different perspectives to be considered, different objectives to be met, and so forth. Moreover, now that this particular area of mathematics has demonstrated its staying-power and fecundity, not to mention its ubiquity, works by the trailblazers are all too often regarded as passé, having been supplemented by more modern texts, addressing more up to date concerns.
What, then, of the book under review, Complex Manifolds, by James Morrow and Kunihiko Kodaira? Being an AMS Chelsea publication, it is, like all Chelsea books, a classic, or a standard text from a different age and time, and is accordingly distinguished on historical grounds. But much more is true: the material covered in these nearly two-hundred pages comes directly from Kodaira's lectures given at Stanford forty years ago, in 1965–1966. This, then, is a course on complex manifolds given by an undoubted master at an exceptionally important time in the history of mathematics, particularly for the kind of algebraic geometry (and differential geometry) that saw its full flowering in the systematic and widespread application of sheaf theoretic methods. Therefore, given the incomparable opportunity of learning how to look at complex manifolds in the style of Kodaira, which engenders a strong recommendation for Complex Manifolds in its own right, we reap the additional benefit of learning a good deal of (old-fashioned) sheaf theory set in a phenomenally important context. It is worth mentioning in this connection that special explicit care is given to relevant aspects of vector bundles and to infinitesimal deformations: a somewhat rare and therefore extremely useful thing to do these days. Caveat: Kodaira is pretty terse, even if his mathematics is gorgeous and elegant (of course terseness and elegance are anything but antithetical properties, but the reader should gird his loins.)
As the Preface of the book conveys, one of the primary goals of Kodaira's lectures was to address vanishing and embedding theorems, particularly as regards Kähler manifolds: it was Kodaira who showed that if a Kähler manifold is Hodge then it is algebraic. But while this orientation is certainly of great and more than historical importance, it is appropriate to draw attention, too, to the last part of the book, treating elliptic partial differential equations. The appearance of this important theme underscores the fact that the methods dealt with in this connection, including sheaf cohomology in particular, constitute a large proportion of the contemporary arsenal for dealing with an impressive collection of themes, from algebraic geometry proper to, e.g., the microlocal analysis of Sato and Kashiwara.
Truly, Kodaira qualifies as one of the masters responsible for revolutionizing the indicated areas of, for lack of a more descriptive word, algebraic geometry, together with Spencer, Hirzebruch, and so on: is it right to think of a school complementary to the French school of Cartan, Serre, and Grothendieck? There is no question that this beautifully constructed book, full of elegant (and very economical) arguments underscores Abel's aforementioned dictum. Perhaps especially today, when so much is asked of the student of this material in the way of prerequisites, one can do no better than to turn to a master.
Michael Berg is Professor of Mathematics at Loyola Marymount University in California.