Complex Analytic Sets

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.