There is a legend that Oscar Zariski and Pierre Samuel had planned to write a textbook on algebraic geometry. Realizing that such a book would require many algebraic prerequisites, they decided to begin by setting out the commutative algebra they needed. Thus, says the legend, was born their famous two-volume Commutative Algebra.
I don’t know if that story is true, but si non è vero, è bene trovato: algebraic geometry does tend to require background from many different parts of mathematics, and it is hard to decide what to do about that when one is to write an introduction. And so it comes about that Günter Harder opens his preface to these Lectures by saying “I want to begin with a defense or apology for the title of this book,” because four of the five chapters are dedicated to subjects other than algebraic geometry. Only the fifth chapter actually moves toward the official subject of the Lectures, and then mostly in an analytic vein with only the first hints of what a fully algebraic approach might look like.
The first two chapters introduce category theory and homological algebra, the latter focusing mostly on the theory of derived functors. Chapters 3 and 4 are about sheaves and sheaf cohomology. Chapter 5 deals with compact Riemann surfaces, complex tori, abelian varieties. Only on page 271 does one get a section called “Towards the Algebraic Theory.”
While the reader looking specifically for the algebraic point of view might be surprised, there is no disappointment here, since this volume includes all sorts of beautiful and useful mathematics. The first three chapters do feel preliminary, but there is real meat to the treatment of sheaf cohomology, including quite a bit of algebraic topology and geometry of complex manifolds, including Poincaré duality, the Lefschetz fixed point formula, de Rham and Dolbeault cohomology, and Hodge theory. The same is true of chapter 5: the techniques are mostly “transcendental”, but the material is classical algebraic geometry of curves and their Jacobian varieties.
My only criticism is on the production side: someone decided to fit as much text as possible on the pages, making the margins very narrow and the lines very long. This makes the text very hard to read. It reminded me of a comment by Robert Bringhurst in his Elements of Typographic Style: long lines are, he says, “a sign, generally speaking, that the emphasis is on the writing instead of the reading…” Indeed, the impression is that there is more concern with putting the text onto paper than with making it readable. Given the quality of the paper and the binding, this decision strikes a dissonant note.
But maybe it’s just me and my tired old eyes. Harder is a master of the subject, and so this volume is full of insights. It will be a particularly valuable resource for those who want to learn the sheaf-theoretic approach to complex algebraic varieties.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.