2002/02/25 by M. Matthias Schmitt, Matthias M. Schmitt, Schmitt, M. Matthias
Mathematics · #32D15 #46E10 (secondary) #46M18 (primary) #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #math.CV #math.FA #msc:32D15 #msc:46E10 #msc:46M18
paper · pdf · doi:10.48550/arxiv.math/0202263
104 pages
arxiv created 2002/02/25 · openalex publication_date 2002/02/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this thesis we solve the coboundary equation δc=d with bounds for cochains with values in a coherent subsheaf of some OpΩ, where Ω is a Stein manifold. In particular the existence of a finite set of global generators is not assumed. Our result applies therefore to the ideal sheaf JV⊂ O\CN of germs of holomorphic functions vanishing on a closed analytic submanifold V⊂\CN. Although we are mainly interested in the estimates for the solutions of δc=d, the techniques used also lead to a proof for the classical Theorem B of Cartan for coherent subsheafs of some OpΩ, avoiding the Mittag-Leffler argument. We derive an extension theorem for holomorphic functions on V to entire functions, with control on growth behaviour. \newline As a corollary we construct a linear tame extension operator H(V)→ H(\CN) under the hypothesis that H(V) is linear tamely isomorphic to the infinite type power series space Λ_∞(k(1)/(n)), n= dim\CV; this condition is also necessary. Here the supnorms on H(V) are taken over intersections of V with polycylinders of polyradii em, m∈ \N. Aytuna asked how much, and what kind of, information about the complex analytic structure of V is carried by the Fréchet space H(V). We prove that H(V) is linear tamely isomorphic to a power series space of infinite type if and only if V is algebraic.