Hans Grauert

Several complex variables

I Holomorphic Functions.- 1 Power Series.- 2 Complex Differentiable Functions.- 3 The Cauchy Integral.- 4 Identity Theorems.- 5 Expansion in Reinhardt Domains.- 6 Real and Complex Differentiability.- 7 Holomorphic Mappings.- II Domains of Holomorphy.- 1 The Continuity Theorem.- 2 Pseudoconvexity.- 3 Holomorphic Convexity.- 4 The Thullen Theorem.- 5 Holomorphically Convex Domains.- 6 Examples.- 7 Riemann Domains over ?n.- 8 Holomorphic Hulls.- III The Weierstrass Preparation Theorem.- 1 The Algebra of Power Series.- 2 The Weierstrass Formula.- 3 Convergent Power Series.- 4 Prime Factorization.- 5 Further Consequences (Hensel Rings, Noetherian Rings).- 6 Analytic Sets.- IV Sheaf Theory.- 1 Sheaves of Sets.- 2 Sheaves with Algebraic Structure.- 3 Analytic Sheaf Morphisms.- 4 Coherent Sheaves.- V Complex Manifolds.- 1 Complex Ringed Spaces.- 2 Function Theory on Complex Manifolds.- 3 Examples of Complex Manifolds.- 4 Closures of ?n.- VI Cohomology Theory.- 1 Flabby Cohomology.- 2 The ?ech Cohomology.- 3 Double Complexes.- 4 The Cohomology Sequence.- 5 Main Theorem on Stein Manifolds.- VII Real Methods.- 1 Tangential Vectors.- 2 Differential Forms on Complex Manifolds.- 3 Cauchy Integrals.- 4 Dolbeaults Lemma.- 5 Fine Sheaves (Theorems of Dolbeault and de Rham).- List of symbols.

Theorie der Steinschen Räume

1. Der klassische Satz von Mittag-LeIDer, nach dem in jedem Gebiete der GauB schen Zahlenebene ce meromorphe Funktionen mit vorgegebenen Hauptteilen konstruiert werden konnen, wurde bereits 1895 von P. Cousin auf den Fall von mehreren komplexen Veranderlichen iibertragen. Allerdings konnten Cousin und nachfolgende Autoren den analogen Satz nur fUr spezielle Gebiete, namlich Zylindergebiete des m-dimensionalen komplexen Zahlenraumes cern, beweisen. m Es zeigte sich, daB keineswegs in allen Gebieten des ce, 2 S; m

Vektorbündel vom rang 2 über dem n-dimensionalen komplex-projektiven raum

In this paper we construct holomorphic vector-bundles of rank 2 on complex-projective spaces to canonical data. More detailed results for the projective plane are given.

Theory of q-Convexity and q-Concavity

Section 1 deals with the cohomology with coefficients in the sheaf of local holomorphic functions O near to q-convex and q-concave boundary points of domains in the complex number space ℂ n . Section 2 carries over the results to complex spaces and to arbitrary coherent analytic sheaves, proves extension theorems and introduces the Frechet topology in the set of local cross sections. In § 3 the finiteness of the dimension of cohomology vector spaces is proved in certain cases. Section 4 gives some applications: When can a hole in a complex space be filled. When do hulls for cohomology classes exist and when does the cohomology vanish?

On Meromorphic Equivalence Relations

We denote by X a weakly normal (see § 2.3.) complex space with countable topology and by R ⊂ X × X an analytic set with the following two properties: 1) R contains the diagonal D ⊂ X × X 2) R is mapped by the reflexion (x1,x2) →, (x2,x1) : \( X \times X\tilde \to X \times X\) onto itself.

Hyperbolicity of the complement of plane curves

In this paper it is proved that the complement in ℙ2 of a holomorphic curve D of genus g≧2 is a hermitian hyperbolic complex manifold provided that certain conditions on the singularities of the dual D* of D are satisfied and that every tangent at D* intersects D* in at least two distinct points.

Statistical geometry and space-time

In this paper I try to construct a mathematical tool by which the full structure of Lorentz geometry to space time can be given, but beyond that the background — to speak pictorially — the subsoil for electromagnetic and matter waves, too. The tool could be useful to describe the connections between various particles, electromagnetism and gravity and to compute observables which were not theoretically related, up to now. Moreover, the tool is simpler than the Riemann tensor: it consists just of a setS of line segments in space time, briefly speaking.

Extension of Analytic Objects

Section 1 deals with the extension of coherent quotient sheaves into q-concave boundaries. The old notion of gap sheaf is essential here. The end of § 1 and the beginning of § 2 is devoted to the theorems of Remmert and Stein for the extension of analytic sets into other analytic sets. It is of importance that the boundedness of the area gives a necessary and sufficient condition (Stoll-Bishop theorem). If the boundary is q-concave but has corners stronger conditions are necessary. Of especial interest is the extension into the full complex projective space (see § 3). The last section gives the extension of absolute (i.e. not quotient) sheaves into q-concave boundaries. This also gives the „Kontinuitatssatz“ for meromorphic maps; in the case of meromorphic functions the theorem first was proved by Hellmut Kneser.

Analytic Sets. Coherence of Ideal Sheaves

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.

Der Weierstraßsche Vorbereitungssatz

In diesem Kapitel wollen wir uns eingehender als bisher mit Potenzreihen im Cn befassen. Ziel unserer Bemuhungen wird es sein, eine Art Divisionsalgorithmus fur Potenzreihen zu finden, durch den die Untersuchung der Nullstellen von holomorphen Funktionen erleichtert wird.