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Inhaltsverzeichnis

Frontmatter

Chapter 1. Complex Spaces

Abstract
In this chapter we are dealing with complex spaces and analytic sheaves. The theory here is widely formal. We only discuss such topics which are really needed later in this book. Our presentation is partly rigorous and partly expository.
Hans Grauert, Reinhold Remmert

Chapter 2. Local WEIERSTRASS Theory

Abstract
All local function theory originates from the famous Weierstrass Preparation Theorem which “prepares us so well” for all further discussions. WEIERSTRASS himself called his theorem “Vorbereitungssatz” (cf. Math. Werke 2, p.135), he writes there in a footnote: “Diesen Satz habe ich seit dem Jahre 1860 wiederholt in meinen Universitäts-Vorlesungen vorgetragen.” The Preparation Theorem expresses the fundamental fact that the zero set of a holomorphic function g displays, at least locally in suitable coordinates, an “algebraic” and hence “finite” character. This is the reason why finite holomorphic maps nowadays are the most important tool in local function theory.
Hans Grauert, Reinhold Remmert

Chapter 3. Finite Holomorphic Maps

Abstract
In this chapter we systematically exploit the Weierstrass techniques developed so far. Our main results are the Finite Mapping Theorem, Rückert Nullstellensatz for coherent sheaves, an Open Mapping Lemma for finite holomorphic maps and, last not least, a Local Description Lemma for closed complex subspaces of domains in number spaces. All this is in the center of Local Analytic Geometry and fundamental for further investigations.
Hans Grauert, Reinhold Remmert

Chapter 4. Analytic Sets. Coherence of Ideal Sheaves

Abstract
Analytic sets are zero sets of holomorphic functions. Such sets were already considered in the nineteenth century — long before the notion of a complex space was coined — as the natural generalization of algebraic sets which are zero sets of polynomials. One reason to study analytic sets and not just systems of the type w1= f1(z1, ..., z m ), ..., w n = f n (z1, ..., z n ) is that quite often the implicit function theorem cannot be applied to solve a given set of holomorphic equations.
Hans Grauert, Reinhold Remmert

Chapter 5. Dimension Theory

Abstract
There are basically three possibilities to introduce the notion of dimension at a point x of a complex space X: topologically as the dimension of the topological space X at x; analytically as the smallest integer n such that n holomorphic functions have x as an isolated zero; and algebraically as the smallest number d of germs f1x ,...,fdx∈m x , such that the ring \({\mathcal{O}_x}/({f_{1x}}, \ldots ,{f_{dx}}){\mathcal{O}_x}\) is artinian.
Hans Grauert, Reinhold Remmert

Chapter 6. Analyticity of the Singular Locus. Normalization of the Structure Sheaf

Abstract
In this chapter we first prove in §§1 and 2 that a reduced complex space X is smooth outside of a nowhere dense analytic set. The notion of embedding dimension is basic. In points where a complex space is not smooth the difference of embedding dimension and dimension is a good measure for the deviation from smoothness.
Hans Grauert, Reinhold Remmert

Chapter 7. RIEMANN Extension Theorem and Analytic Coverings

Abstract
If p is a point in a domain D ⊂ ℂ, then each bounded holomorphic function f in D\p has a unique holomorphic extension to D. The same is true for domains D in ℂn,1 ≤ n < ∞, if p is replaced by a nowhere dense analytic set A in D. If A has dimension ≤n −2 everywhere it is no longer necessary to assume that f is bounded in D\A. These two statements are known as the Riemann Extension Theorems; for the convenience of the reader we reproduce proofs in Section 1.
Hans Grauert, Reinhold Remmert

Chapter 8. Normalization of Complex Spaces

Abstract
The main goal of this chapter is to show that every reduced complex space X has, up to a canonical isomorphism, a normalization \(\hat X\). This space \(\hat X\)will turn out to be a one-sheeted analytic covering of X. For this reason we first develop a general theory of such coverings and prove a Local Existence Theorem. This theorem easily implies OKA’s theorem that the normalization sheaf \({{\hat{\mathcal{O}}}_{X}}\) of the structure sheaf \({{\mathcal{O}}_{X}}\) is \({{\mathcal{O}}_{X}}\)-coherent.
Hans Grauert, Reinhold Remmert

Chapter 9. Irreducibility and Connectivity. Extension of Analytic Sets

Abstract
In this chapter we are concerned with questions of connectivity of reduced complex spaces X and of residue spaces X \ A where A is thin in X. If X is connected such sets A may disconnect X as is shown by the standard example of a space consisting of two complex lines intersecting in a single point. Spaces for which this phenomenon cannot occur are called irreducible, we give different characterizations for such spaces (cf. Theorem 1.2). A basic role is played by the Global Decomposition Theorem 2.2. We demonstrate the power of this theorem by various applications in Sections 2 and 3.
Hans Grauert, Reinhold Remmert

Chapter 10. Direct Image Theorem

Abstract
If f: X → Y is a holomorphic map between complex spaces X and Y, the direct images f (q)(S) of a coherent O X -sheaf S in general are not coherent O Y -sheaves. For example if Y is just a single (reduced) point, coherence of the q-th image sheaf f(q)(S) means that f (q) (S)=H q (X, S) is a finite dimensional complex vector space. There are many examples of non-compact spaces X where this is not the case.
Hans Grauert, Reinhold Remmert

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