Complex Analytic Sets - E. M. Chirka

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Résumé :
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 ?-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 ??-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....

Sommaire:
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.)