Complex Analytic Cycles I: Basic Results on Complex Geometry and Foundations for the Study of Cycles (Grundlehren der mathematischen Wissenschaften)

Preface to the English Edition
	Contents of Volume II
	Acknowledgments
Contents
1 Preliminary Material
	1.1 Holomorphic Mappings
		1.1.1 Definitions
		1.1.2 The Case Where E is Finite-Dimensional
		1.1.3 Holomorphic= Analytic for Arbitrary E
	1.2 Complex Manifolds
		1.2.1 Manifolds
		1.2.2 Vector Bundles
		1.2.3 Projective Space P(E)
		1.2.4 Grassmannians
		1.2.5 Further Examples of Complex Manifolds
		1.2.6 Integration on Oriented Manifolds and Stokes\' Formula
		1.2.7 Differential Forms on a Complex Manifold
	1.3 Symmetric Products of C
		1.3.1 Continuity of Roots
		1.3.2 Weierstrass Preparation Theorem
			Holomorphic Variation of Roots
			A First Application
			Weierstrass Preparation Theorem
			Division Theorem
	1.4 The Symmetric Product of Cp
		1.4.1 Symmetric Products of Topological Spaces
		1.4.2 Vector Symmetric Functions
		1.4.3 Vertical Localization
		1.4.4 Canonical Equations
		1.4.5 Complex Structure on Symk(Cp)
		1.4.6 Stratification
2 Multigraphs and Reduced Complex Spaces
	2.1 Reduced Multigraph
		2.1.1 Proper Mappings
		2.1.2 Analytic Subsets
		2.1.3 Analytic Continuation
		2.1.4 Analytic Étale Coverings
		2.1.5 Reduced Multigraphs
		2.1.6 Local Study of Reduced Multigraphs
		2.1.7 Irreducibility of Reduced Multigraphs
			Irreducible Components
			Path Connectedness
			Trace and Norm: A First Direct Image Theorem
	2.2 Multigraphs
		2.2.1 Basic Definitions
		2.2.2 Classification Map and the Canonical Equation
	2.3 Analytic Subsets
		2.3.1 Local Parameterization Theorem: First Version
		2.3.2 Irreducible Components and Singular Locus
		2.3.3 Maximum Principle
		2.3.4 Ramified Covers
		2.3.5 Local Parameterization Theorem: Final Version
		2.3.6 Analyticity of the Singular Locus
	2.4 Reduced Complex Spaces
		2.4.1 Definitions and Elementary Properties
		2.4.2 Singular Locus and Irreducible Components
		2.4.3 Dimension and Local Irreducibility
		2.4.4 Minimal Embeddings and the Zariski Tangent Space
		2.4.5 Fiber Dimension and Generic Rank
		2.4.6 Symmetric Product of a Reduced Complex Space
		2.4.7 Proper Finite Mappings
		2.4.8 Remmert–Stein Theorem
	2.5 Notes on Chapter 1 and this chapter
		2.5.1 The Preparation and Division Theorems
		2.5.2 The Local Parameterization Theorem
		2.5.3 The Three Definitions of Ramified Covers
		2.5.4 Complex Spaces
		2.5.5 The Theorem of Remmert–Stein
3 Analysis and Geometry on a Reduced Complex Space
	3.1 Transversality and the Zariski Tangent Cone
		3.1.1 Transverse Planes to an Analytic Subset
		3.1.2 Algebraic Cones
		3.1.3 Transversality and the Tangent Cone
		3.1.4 Zariski Tangent Cycle
	3.2 The Theorem of P. Lelong
		3.2.1 Introduction
		3.2.2 Preliminaries
		3.2.3 The Case of a Reduced Multigraph
		3.2.4 Differential Forms on a Reduced Complex Space
			Differentiable Functions on a Reduced Complex Space
			Differential Forms on a Reduced Complex Space
			Natural Topology on Spaces of Differential Forms
		3.2.5 Lelong\'s Theorem : General Case
		3.2.6 Volume
	3.3 Coherent Sheaves
		3.3.1 Coherent Sheaves on a Reduced Complex Space
		3.3.2 Canonical Topology
			An Application
	3.4 Modifications and Blowups
		3.4.1 Modifications
		3.4.2 Blowups
		3.4.3 Meromorphic Mappings
	3.5 Normalization
		3.5.1 Normal Spaces
		3.5.2 Meromorphic Functions
		3.5.3 Locally Bounded Meromorphic Functions
		3.5.4 Universal Denominators
		3.5.5 Additional Material on Normality
		3.5.6 Normalization
		3.5.7 The Weak Normalization
		3.5.8 Complementary Material on Meromorphic Functions
	3.6 Local Bound of Volume of General Fibers
		3.6.1 Local Blowing Up
		3.6.2 The Theorem
	3.7 Direct Image and Enclosure
		3.7.1 Holomorphic Mappings with Values in a TVS
		3.7.2 Reduced Multigraphs in a Sequentially Complete TVS
		3.7.3 The Direct Image Theorem
		3.7.4 Theorem on Encloseability
			The Symmetric Algebra of a Topological Vector Space
			Proof of the Encloseability Theorem
	3.8 Holomorphic Convexity: The Quotient Theorem
		3.8.1 Dirac Mapping
		3.8.2 Holomorphically Convex Spaces
		3.8.3 Stein Spaces and the Remmert Reduction
		3.8.4 The Quotient Theorem of H. Cartan
	3.9 Notes on This Chapter
		3.9.1 Transversality and the Zariski Tangent Cone
		3.9.2 Algebraic Cones
		3.9.3 The Theorem of P. Lelong, Canonical Topology, Modifications and Blowups
		3.9.4 Normalization
		3.9.5 Weak Normalization
		3.9.6 Bound of Volume of General Fibers
		3.9.7 Direct Image and Encloseability: Holomorphic Convexity
		3.9.8 Quotient Theorem
4 Families of Cycles in Complex Geometry
	4.1 Families of Cycles
		4.1.1 Cycles
			Why Cycles?
			Basic Definitions
		4.1.2 Elementary Operations on Cycles
			Addition and Order
			The Universal Family of Cycles
		4.1.3 Functorial Properties
			Restriction to Open Subsets
			The Direct Image of a Cycle
			Direct Image: The Case of a Proper Morphism
			The Case of a Closed Embedding
			Direct Image: The General Case
			Cartesian Product
	4.2 Continuous Families of Cycles
		4.2.1 Scales
		4.2.2 Topology of Cnloc(M) and of Cn(M)
			Topology of Cn(M)
			The Natural Topology on  H(, `3́9`42`\"̇613A``45`47`\"603ASymk(B))
			The Natural Map μ: Ωk(E)→H(, `3́9`42`\"̇613A``45`47`\"603ASymk(B))
		4.2.3 Functions Defined by Integration
			Local Character of the Topology of Cnloc(M)
		4.2.4 Cnloc(M) and Cn(M) are Second Countable
		4.2.5 Continuity of Direct Image Maps
			Direct Image with Parameters
		4.2.6 Integration of Cohomology Classes: Topological Case
		4.2.7 Compactness and the Theorem of E. Bishop
			Topology and Hausdorff Metric
			The Topology on Cnloc(M) and the Hausdorff Topology on K(M)*
			Compactness in Cnloc(M)
			Compactness in Cn(M)
		4.2.8 Cycles as Currents
			Preface
			Direct Image of Cycles and Direct Image of Currents
			Wirtinger\'s Inequality and the Weak Topology
	4.3 Analytic Families of Cycles
		4.3.1 Basic Definitions
		4.3.2 Multiplicity of a Point in a Cycle
		4.3.3 Graph of an Analytic Family of Cycles
		4.3.4 The Case of a Normal Parameter Space
		4.3.5 Stability of Analytic Families by Direct Images
	4.4 Fundamental Counterexample
		4.4.1 What Does Not Work!
		4.4.2 Example
	4.5 Characterization of Isotropic Morphisms: Applications
		4.5.1 Isotropic Morphisms
		4.5.2 Integration of Cohomology Classes
			Complex Découpage
	4.6 Finiteness of the Space of Cycles: Applications
		4.6.1 Finiteness Theorem
		4.6.2 Some Consequences
	4.7 Theorem on Connectedness
		4.7.1 Number of Irreducible Components
		4.7.2 Connected Cycles
	4.8 Relative Cycles
		4.8.1 Preliminaries
		4.8.2 The Space of Cycles Relative to a Morphism
	4.9 Fibers of a Proper Meromorphic Mapping
		4.9.1 The Case of a Proper Holomorphic Map
		4.9.2 The Case of a Proper Meromorphic Map
		4.9.3 Almost Holomorphic Mappings
	4.10 Analytic Families of Holomorphic Mappings
	4.11 Appendix I: Complexification
		4.11.1 Conjugation on a Complex Vector Space
		4.11.2 Complexification of a Real Vector Space
		4.11.3 The Complex Case
		4.11.4 Orientation of a Complex Vector Space
		4.11.5 The Space p,pR(E)
		4.11.6 Positivity in the Sense of P. Lelong
	4.12 Appendix II: Locally Convex Topological Vector Spaces
Bibliography
Index