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Mathematische Annalen

, Volume 341, Issue 4, pp 717–733 | Cite as

Holomorphic functions on bundles over annuli

  • Dan Zaffran
Article

Abstract

We consider a family \(\big\{ E_m(D,M) \big\}\) of holomorphic bundles constructed as follows:from any given \(M\in GL_n({\mathbb{Z}})\) , we associate a “multiplicative automorphism” \(\varphi\) of \(({\mathbb{C}}^*)^n\) . Now let \(D\subseteq ({\mathbb{C}}^*)^n\) be a \(\varphi\) -invariant Stein Reinhardt domain. Then E m (D, M) is defined as the flat bundle over the annulus of modulus m > 0, with fiber D, and monodromy \(\varphi\) . We show that the function theory on E m (D, M) depends nontrivially on the parameters m, M and D. Our main result is that
$$E_m(D,M) \text{\ is Stein if and only if\ } m \log \rho (M) \leq 2 \pi^2,$$
where ρ(M) denotes the max of the spectral radii of M and M −1. As corollaries, we: (1) obtain a classification result for Reinhardt domains in all dimensions; (2) establish a similarity between two known counterexamples to a question of J.-P. Serre; and (3) suggest a potential reformulation of a disproved conjecture of Siu Y.-T.

Keywords

Stein Bounded Domain Holomorphic Function Laurent Series Stein Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Fudan UniversityShanghaiChina
  2. 2.Academia SinicaTaipeiTaiwan

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