## Theory of Stein Spaces1. The classical theorem of Mittag-Leffler was generalized to the case of several complex variables by Cousin in 1895. In its one variable version this says that, if one prescribes the principal parts of a merom orphic function on a domain in the complex plane e, then there exists a meromorphic function defined on that domain having exactly those principal parts. Cousin and subsequent authors could only prove the analogous theorem in several variables for certain types of domains (e. g. product domains where each factor is a domain in the complex plane). In fact it turned out that this problem can not be solved on an arbitrary domain in em, m ~ 2. The best known example for this is a "notched" bicylinder in 2 2 e . This is obtained by removing the set { (z , z ) E e 11 z I ~ !, I z 1 ~ !}, from 1 2 1 2 2 the unit bicylinder, ~ :={(z , z ) E e llz1 |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

Soft and Flabby Sheaves | 4 |

Coherent Sheaves and Coherent Functors | 10 |

Chapter B Cohomology Theory | 28 |

Copyright | |

22 other sections not shown

### Other editions - View all

### Common terms and phrases

algebra analytic applications arbitrary associated block bounded C-algebra called canonical Chapter choose clear closed coherent cohomology commutative compact complex space condition consequence consider construct contained continuous converges coordinates Cousin cover defined Definition denote determines differentiable direct divisor domain equation equivalent exact sequence example exhaustion exists fact finite flabby Fréchet function functor germs given Hence Hº(X holomorphic functions holomorphically convex ideal immediately implies important induces injection isomorphic Lemma locally free sheaf meromorphic module natural neighborhood open set Paragraph particular presheaf problem Proof properties prove quotient R-modules rank reduced Remark resolution respect restriction result rings satisfies sequence sheaf sheaves stalk Stein space structure subset supp Suppose surjective Theorem theory tion topology uniquely vanish vector zero