Cousin problems

Abstract: The main purpose of this paper is to generalize the celebrated L${}^2$ extension theorem of Ohsawa-Takegoshi in several directions : the holomorphic sections to extend are taken in a possibly singular hermitian line bundle, the subvariety from which the extension is performed may be non reduced, the ambient manifold is K{a}hler and holomorphically convex, but not necessarily compact.

Abstract: Historical work on the emergence of sheaf theory has mainly concentrated on the topological origins of sheaf cohomology in the period from 1945 to 1950 and on subsequent developments. However, a shift of emphasis both in time-scale and disciplinary context can help gain new insight into the emergence of the sheaf concept. This paper concentrates on Henri Cartan’s work in the theory of analytic functions of several complex variables and the strikingly different roles it played at two stages of the emergence of sheaf theory: the definition of a new structure and formulation of a new research programme in 1940–1944; the unexpected integration into sheaf cohomology in 1951–1952. In order to bring this two-stage structural transition into perspective, we will concentrate more specifically on a family of problems, the so-called Cousin problems, from Poincare (1883) to Cartan. This medium-term narrative provides insight into two more general issues in the history of contemporary mathematics. First, we will focus on the use of problems in theory-making. Second, the history of the design of structures in geometrically flavoured contexts—such as for the sheaf and fibre-bundle structures—which will help provide a more comprehensive view of the structuralist moment, a moment whose algebraic component has so far been the main focus for historical work.

Abstract: The purpose of this paper is to give a historical survey of the development of methods in the theory of partial differential equations for the study of the Levi and Cousin problems in complex analysis. Success was achieved by the mid 1960's but we begin further back, with the background in Hodge theory and with early unsuccessful attempts to exploit the Bergman kernel. Some examples of later date illustrating the usefulness of such methods are also given.

Abstract: Introduction Pages 1. Enoncé des résultats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 2. Faisceaux souples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3. Problème de Cousin à croissance. . . . . . . . . . . . . . . . . . . . . . . 111 4. Fonctions à croissance lente dans un t u b o i ' d e . . . . . . . . . . . . . 114 5. Démonstration du théorème 1.2 . . . . . . . . . . . . . . . . . . . . . . . 116 6. Démonstration du théorème 1.8 . . . . . . . . . . . . . . . . . . . . . . . 119