Complex Analytic Sets

1 Fundamentals of the theory of analytic sets.- 1. Zeros of holomorphic functions.- 1.1. Weierstrass' preparation theorem.- 1.2. Dependence of roots on parameters.- 1.3. Discriminant set.- 1.4. Factorization into irreducible factors.- 1.5. Multiplicity of zeros. Divisor of a holomorphic function.- 2. Definition and simplest properties of analytic sets. Sets of codimension 1.- 2.1..- 2.2. Simplest topological properties.- 2.3. Regular and singular points.- 2.4. Dimension.- 2.5. Regularity in ? n and ??n+1.- 2.6. Principal analytic sets.- 2.7. Critical points.- 2.8. Local representation of sets of codimension 1.- 2.9. Minimal defining functions.- 3. Proper projections.- 3.1. Proper maps.- 3.2. Exception of variables.- 3.3. Corollaries.- 3.4. Existence of proper projections.- 3.5. On the dimension.- 3.6. Almost single-sheeted projections.- 3.7. Local representation of analytic sets.- 3.8. Images of analytic sets.- 4. Analytic covers.- 4.1. Definitions.- 4.2. Canonical defining functions.- 4.3. Analytic covers as analytic sets.- 4.4. The theorem of Remmert-Stein-Shiffman.- 4.5. Analyticity of sng A.- 5. Decomposition into irreducible components and its consequences.- 5.1. Connected components of reg A.- 5.2. Decomposition by dimension. Analyticity of sng A and S(A).- 5.3. Irreducibility.- 5.4. Irreducible components.- 5.5. Stratifications.- 5.6. Intersections of analytic sets.- 5.7. The number of defining functions.- 5.8. A theorem on proper maps.- 6. One-dimensional analytic sets.- 6.1. Local parametrization.- 6.2. Normalization and uniformization.- 6.3. Maximum principle.- 7. Algebraic sets.- 7.1. Chow's theorem.- 7.2. Closure of affine algebraic sets.- 7.3. Algebraic sets as analytic covers.- 7.4. Some criteria for being algebraic.- 2 Tangent cones and intersection theory.- 8. Tangent cones.- 8.1. Definitions and simplest properties.- 8.2. The tangent cone and maps.- 8.3. The tangent cone and the ?-process.- 8.4. Analytic description.- 8.5. Tangent vectors and one-dimensional sections.- 8.6. Deviation.- 9. Whitney cones.- 9.1. Definitions and simplest properties.- 9.2. Hierarchy and analyticity.- 9.3. Tangent space.- 9.4. Whitney cones and projections.- 9.5. Singularities of codimension 1. Puiseux normalization.- 10. Multiplicities of holomorphic maps.- 10.1. Multiplicity of projections.- 10.2. Multiplicity of maps.- 10.3. Multiplicities and initial polynomials.- 10.4. Bezout's theorem.- 10.5. Milnor numbers.- 11. Multiplicities of analytic sets.- 11.1. Multiplicity of an analytic set at a point.- 11.2. Multiplicities and the tangent cone.- 11.3. Degree of an algebraic set.- 11.4. Multiplicity sets.- 11.5. Holomorphic chains.- 11.6. The tangent cone as chain.- 11.7. Dependence of the tangent cone on parameters.- 12. Intersection indices.- 12.1. The case of complementary codimensions.- 12.2. Some properties of indices.- 12.3. Intersections of holomorphic chains.- 12.4. Properties of intersection chains.- 12.5. Multiplicities and transversality.- 12.6. Multiplicities of fibers of holomorphic maps.- 3 Metrical properties of analytic sets.- 13. The fundamental form and volume forms.- 13.1. Hermitian manifolds.- 13.2. Volume forms.- 13.3. Wirtinger's inequality.- 13.4. Integration in ?n.- 13.5. Integration over incidence manifolds. Crofton's formula.- 13.6. Relation between projective and affine volumes.- 14. Integration over analytic sets.- 14.1. Lelong's theorem.- 14.2. Properties of integrals over analytic sets.- 14.3. Stokes' theorem.- 14.4. Analytic sets as minimal surfaces.- 14.5. Tangential and normal components of volume.- 14.6. Volumes of analytic subsets of a ball.- 14.7. Volumes of algebraic sets.- 15. Lelong numbers and estimates from below.- 15.1. Lelong numbers.- 15.2. Integral representations.- 15.3. Lower bounds for volumes.- 15.4. Areas of projections.- 15.5. Sequences of analytic sets.- 16. Holomorphic chains.- 16.1. Sequences of holomorphic chains.- 16.2. Intersection chains as currents.- 16.3. Formulas of Poincare-Lelong.- 16.4. Jensen formulas.- 17. Growth estimates of analytic sets.- 17.1. Blaschke's condition.- 17.2. Metrical conditions of algebraicity.- 17.3. Growth estimates of hyperplane sections.- 17.4. Converse estimates.- 17.5. Corollaries and generalizations.- 4 Analytic continuation and boundary properties.- 18. Removable singularities of analytic sets.- 18.1. Singularities of small codimensions.- 18.2. Infectiousness of continuation.- 18.3. Removing pluripolar singularities. Bishop's theorems.- 18.4. Continuation across ?n.- 18.5. Obstructions of small CR-dimensions.- 18.6. "Hartogs' lemma" for analytic sets.- 19. Boundaries of analytic sets.- 19.1. Regularity near the boundary.- 19.2. Boundary uniqueness theorems.- 19.3. Plateau's problem for analytic sets.- 19.4. Preparation lemmas.- 19.5. Boundaries of analytic covers.- 19.6. The Harvey-Lawson theorem.- 19.7. On singularities of analytic films.- 20. Analytic continuation.- 20.1. On continuation of analytic sets.- 20.2. Compact singularities.- 20.3. Continuation across pseudoconcave surfaces.- 20.4. Continuation across an edge.- 20.5. The symmetry principle.- 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.