Complex Analytic Sets

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.

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 Poincaré-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.