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Branched coverings and algebraic functions. (English) Zbl 0706.14017
Pitman Research Notes in Mathematics Series, 161. Harlow: Longman Scientific & Technical; New York: John Wiley & Sons Ltd. 201 p. $ 49.95 (1987).
This book originates from a series of lectures and seminar talks for graduate students at various universities in Japan. It is a research monograph describing research by the author on finite branched coverings of projective complex manifolds in connection with the theory of algebraic functions of several complex variables. The author presents a theory generalizing earlier work in one complex variable, in particular work of A. Weil [J. Math. Pures Appl., IX. Sér. 17, 47-87 (1938; Zbl 0018.06302)].
The book has three chapters: 1. “Branched coverings of complex manifolds”, 2. “Fields of algebraic functions”, 3. “Weil-Tōyama theory”.
The main result in chapter 1 is a theorem providing a suitable sufficient condition for the existence of a finite Galois covering of a projective manifold, branched over a given divisor. In the case of compact Riemann surfaces, this originates in a problem posed by Fenchel. In addition, chapter 1 contains many well chosen examples of finite branched coverings. – Chapter 2 studies finite abelian coverings using the theory of currents developed by G. de Rham and K. Kodaira in “Harmonic integrals”, Lecture Notes Inst. Adv. Study (Princeton 1950). There are results providing necessary and sufficient conditions for the existence of a finite abelian covering onto a projective complex manifold branched over a given divisor. Furthermore, the set of all isomorphism classes of finite abelian branched coverings of a projective complex manifold is described using the notion of rational divisor classes. The results can be considered as higher dimensional generalizations of results of Iwasawa from 1952. – Chapter 3 studies similar questions for finite Galois coverings, but according to the author the results obtained are not completely satisfactory. The book should be a good source for inspiration to further work.
Reviewer: V.L.Hansen

14H30 Coverings of curves, fundamental group
14E20 Coverings in algebraic geometry
14H05 Algebraic functions and function fields in algebraic geometry
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
57M12 Low-dimensional topology of special (e.g., branched) coverings
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
30F10 Compact Riemann surfaces and uniformization