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Compact Riemann surfaces. (English) Zbl 0758.30002
Lectures in Mathematics, ETH Zürich. Basel: Birkhäuser. 120 p. (1992).
This remarkable book distinguishes from other books treating the subject by methods of algebraic topology and several complex variables, by the contents which besides the usual results includes special theorems not presented in other monographs and by the choice of the proofs, as well as by the conciseness and elegance of the style.
It also presents a brillant survey of the present interest of Riemann’s ideas and methods, which “expressed in modern language, differ very little (if at all) from the work of modern authors” (preface). Thus the Jacobian and the singularities of the theta divisor are presented from Riemann’s own point of view. Moreover, Serre’s duality is exposed in the spirit of Serre’s original paper, an interesting example of a Riemann surface is presented quoting a paper of de Rham. In the same time the book opens the way to most of the recent research topics and selective references stimulate the reader to pursue the study in these directions.
The preliminary part of the book contains: §1. Algebraic functions, proper maps, coverings; §2. Riemann surfaces; §3. The sheaf of germs of holomorphic functions on Riemann surfaces, the Riemann surfaces of an analytic — and in §4 an algebraic — function; §5. Sheaves, cohomology, Leray’s and Mittag-Leffler’s theorems; §6. Vector and line bundles, divisors.
A second cycles of paragraphs culminates in a Riemann-Roch theorem: §7. Finiteness theorems (the new proof avoids Schwartz’s theorem), consequences for meromorphic sections in holomorphic line bundles and meromorphic functions; §8. Dolbeault’s isomorphism, the canonical bundle; §9. Weyl’s lemma, Serre’s duality for holomorphic vector bundles; §10. The Riemann-Roch theorem and applications: vanishing theorems for cohomological groups, duality pairing, residue version of Serre’s duality; §11. Riemann-Hurwitz formula, topological invariance of the genus, Weierstraß points, gap theorems; §12. Hyperelliptic surfaces. The connections to algebraic geometry constitute another quality of the book: Imbedding theorems in $$P^ N$$ and the canonical map lead to the interpretation of the compact Riemann surfaces as a projective curve and §13 deals with the geometry of these curves: theorems by Bertini, Castelnuovo, Clifford, M.Noether, a geometric form of the Riemann-Roch theorem.
The final part of the book is dedicated to the Jacobian of a compact Riemann surface.: §14. Riemann’s bilinear relations; §15. The Jacobian, the Abel-Jacobi map, Abel’s theorem; §16. Automorphy factors, Riemann’s theta function, Lefschetz’s embedding theorem; §17. The theta divisor $$\theta$$, the Jacobi inverse problem, Riemann’s factorization theorem, functions with prescribed poles or essential singularities; §18. Torelli’s theorem (H. Martens’ proof); §19. Riemann’s theorem on the singularities of $$\theta$$, other results on meromorphic functions and on quadrics.
Some familiarity with concepts and methods utilized in the book especially with the algebraic geometry, facilitate its study, but the reader’s effort is rewarded by the deep acquainstance with this beautiful field of mathematics.

##### MSC:
 30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable 30F10 Compact Riemann surfaces and uniformization 30F99 Riemann surfaces