A homotopy theorem for Oka theory
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Mathematische Annalen
A homotopy theorem for Oka theory Luca Studer1 Received: 5 December 2018 / Revised: 4 July 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We prove a homotopy theorem for sheaves. Its application shortens and simplifies the proof of many Oka principles such as Gromov’s Oka principle for elliptic submersions.
1 Introduction 1.1 Motivation Oka theory is the art of reducing proofs in complex geometry to purely topological statements. Its applications reach beyond complex geometry; for example to the study of minimal surfaces [1]. The power of the theory lies in the fact that there are many problems—some of them almost a century old—for which Oka theory provides the only known approaches. Examples include the following theorems. Theorem (Grauert [10]) Two complex analytic vector bundles over a Stein base which are isomorphic as complex topological vector bundles are complex analytically isomorphic. Theorem (Forster, Ramspott [5]) The ideal sheaf of a smooth complex analytic curve in a Stein manifold X of dimension n ≥ 3 is generated by n − 1 holomorphic functions X → C. Theorem (Gromov [12]) Every continuous map from a Stein manifold X to Cn \Y is homotopic to a holomorphic map given that Y ⊂ Cn is an algebraic subvariety of codimension at least 2. Theorem (Leiterer [16]) For holomorphic maps a, b : X → Mat(n × n, C) defined on a Stein manifold X the equation f (x)a(x) f (x)−1 = b(x), x ∈ X has a holomorphic solution f : X → GLn (C) if there is a smooth solution of the same equation.
Communicated by Ngaiming Mok.
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Luca Studer [email protected] Universität Bern, Bern, Switzerland
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L. Studer
Theorem (Kutzschebauch, Lárusson, Schwarz [15]) A holomorphic action of a complex reductive Lie group G on Cn is linearizable if there is a smooth G-diffeomorphism from Cn to a G-module which induces a biholomorphism on the corresponding categorical quotients, and on every reduced fiber of the quotient map. All known proofs of these results depend on a specific Oka principle. That is, roughly speaking, on a theorem which states that there are only topological obstructions to a complex analytic solution of an associated problem. In concrete terms, Grauert’s result is proved using the Oka principle for principal G-bundles [3,10]. All others depend on extensions of Grauert’s work, namely on the Oka principle for admissible pairs of sheaves [4] in the case of Forster and Ramspott’s and Leiterer’s results, on the Oka principle for elliptic submersions [9,12] in the case of Gromov’s result, and on the Oka principle for equivariant isomorphisms [14] in the case of the result due to Kutzschebauch, Lárusson and Schwarz. The first three theorems can be proved alternatively with Forstneriˇc’s Oka principles for stratified fiber bundles [7], a generalization of Gromov’s work to stratified settings including more general fibers and possibly non-smooth base spaces. Complete proofs of these powerful tools fill a book. However, a careful study of the literature reveals that all
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