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Holomorphic functions on bundles over annuli

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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.

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Authors and Affiliations

  1. Fudan University, Shanghai, China

    Dan Zaffran

  2. Academia Sinica, Taipei, Taiwan

    Dan Zaffran

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  1. Dan Zaffran
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Correspondence to Dan Zaffran.

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Zaffran, D. Holomorphic functions on bundles over annuli. Math. Ann. 341, 717–733 (2008). https://doi.org/10.1007/s00208-007-0201-4

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  • DOI: https://doi.org/10.1007/s00208-007-0201-4

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