Complex Analytic Sets / Edition 1

by E.M. Chirka

Overview

The theory of complex analytic sets is part of the modern geometrical theory of functions of several complex variables. A wide circle of problems in multidimensional complex analysis, related to holomorphic functions and maps, can be reformulated in terms of analytic sets. In these reformulations additional phenomena may emerge, while for the proofs new methods are necessary. (As an example we can mention the boundary properties of conformal maps of domains in the plane, which may be studied by means of the boundary properties of the graphs of such maps.)

The theory of complex analytic sets is a relatively young branch of complex analysis. Basically, it was developed to fulfill the need of the theory of functions of several complex variables, but for a long time its development was, so to speak, within the framework of algebraic geometry - by analogy with algebraic sets. And although at present the basic methods of the theory of analytic sets are related with analysis and geometry, the foundations of the theory are expounded in the purely algebraic language of ideals in commutative algebras.

In the present book I have tried to eliminate this noncorrespondence and to give a geometric exposition of the foundations of the theory of complex analytic sets, using only classical complex analysis and a minimum of algebra (well-known properties of polynomials of one variable). Moreover, it must of course be taken into consideration that algebraic geometry is one of the most important domains of application of the theory of analytic sets, and hence a lot of attention is given in the present book to algebraic sets.

Product Details

ISBN-13:	9780792302346
Publisher:	Springer Netherlands
Publication date:	07/31/1989
Series:	Mathematics and its Applications , #46
Edition description:	1989
Pages:	372
Product dimensions:	6.10(w) x 9.25(h) x 0.36(d)

Table of Contents

1 Fundamentals of the theory of analytic sets.- 1. Zeros of holomorphic functions.- 2. Definition and simplest properties of analytic sets. Sets of codimension 1.- 3. Proper projections.- 4. Analytic covers.- 5. Decomposition into irreducible components and its consequences.- 6. One-dimensional analytic sets.- 7. Algebraic sets.- 2 Tangent cones and intersection theory.- 8. Tangent cones.- 9. Whitney cones.- 10. Multiplicities of holomorphic maps.- 11. Multiplicities of analytic sets.- 12. Intersection indices.- 3 Metrical properties of analytic sets.- 13. The fundamental form and volume forms.- 14. Integration over analytic sets.- 15. Lelong numbers and estimates from below.- 16. Holomorphic chains.- 17. Growth estimates of analytic sets.- 4 Analytic continuation and boundary properties.- 18. Removable singularities of analytic sets.- 19. Boundaries of analytic sets.- 20. Analytic continuation.- Appendix Elements of multi-dimensional complex analysis.- A1. Removable singularities of holomorphic functions.- A1.2. Plurisubharmonic functions.- A1.3. Holomorphic continuation along sections.- A1.4. Removable singularities of bounded functions.- A1.5. Removable singularities of continuous functions.- A2.1. Holomorphic maps.- A2.2. The implicit function theorem and the rank theorem.- A3. Projective spaces and Grassmannians.- A3.1. Abstract complex manifolds.- A3.5. Incidence manifolds and the σ-process.- A4. Complex differential forms.- A4.1. Exterior algebra.- A4.2. Differential forms.- A4.3. Integration of forms. Stokes’ theorem.- A4.4. Fubini’s theorem.- A4.5. Positive forms.- A5. Currents.- A5.1. Definitions. Positive currents.- A5.3. Regularization.- A5.4. The δ^--problem and the jump theorem.- A6. Hausdorff measures.- A6.1. Definition and simplest properties.- A6.3. The Lemma concerning fibers.- A6.4. Sections and projections.- References.- References added in proof.

Editorial Reviews

Monograph providing advanced graduate students and other researchers with an account of the zero sets of holomorphic functions of several complex variables, which are studied here from a geometric (rather than an algebraic) point of view, with emphasis particularly on the global aspects of the theory. The approach is modern, and many of the results reported are relatively new. (NW) Annotation c. Book News, Inc., Portland, OR (booknews.com)

Booknews