Classics in Mathematics. Berlin: Springer-Verlag. ix, 234 p. DM 59.00; öS 460.20; sFr. 57.00 (1995).
In the early 1950s, both complex-analytic geometry and algebraic geometry began to undergo a radically new development, which finally led to the present shape of these two disciplines. It was basically the theory of sheaves and their cohomology which, under the influence of H. Cartan, K. Kodaira, J.-P. Serre, A. Grothendieck, and others, provided the appropriate framework for a subtantially new foundation of these two areas in mathematics and, simultaneously, for the great progress achieved since then.
One of the most spectacular and far-reaching results, obtained at the very beginning of this development, was due to the author. As early as in 1954, he succeeded in generalizing the classical Riemann-Roch theorem for algebraic curves and algebraic surfaces to general nonsingular complex projective varieties. His ingenious proof, which involved the entire power of the new sheaf- and cohomology-theoretic techniques, represented a milestone in transcendental algebraic geometry and made the ubiquitous significance (as well as the great perspectives) of the new methods strikingly evident. The author’s epoch-making proof of a more general Riemann-Roch theorem was the decisive beginning of a further extensive development in this direction, culminating in more and more general Riemann-Roch-type theorems in the theory of algebraic schemes and, nowadays, even in arithmetical algebraic geometry.
The first edition of the author’s book (under review) appeared in 1956; see Zbl 0070.16302. At that time, it was intended as being a research report, including an introduction to the theory of sheaves, vector bundles, and their cohomology machinery (cohomology groups, characteristic classes, cobordism theory, index theory, etc.) as used in the course of the treatise. This very layout made the book also into a textbook which, for the first time, offered a systematic and coherent account of the new techniques (and concepts) in algebraic geometry. Very quickly Hirzebruch’s book became a standard text in transcendental algebraic geometry, and in the sequel it saw many new editions and translations.
The third edition of the book, which appeared in 1978 as a translation into English, was enriched and updated by two large appendices. These appendices, written by R. L. E. Schwarzenberger and A. Borel, explained the new developments in Riemann-Roch theory and index theory since the first edition of the book. – The present edition is an unchanged reprint of the second, corrected printing of that third, enlarged edition. Over the years, the author’s outstanding classic text has maintained its strong attractivity to any generation of mathematicians. The unique character of this book is based upon its high degree of clarity, rigor and comprehension, not to speak of the depth and beauty of its contents. There is certainly no doubt that the book under review is (and will remain) among the most important classics in transcendental algebraic geometry.