Holomorphic functions on bundles over annuli

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We consider a family {E-m( D, M)} of holomorphic bundles constructed as follows: from any given M is an element of GL(n)(Z), we associate a "multiplicative automorphism" phi of (C*)(n). Now let D subset of (C*)(n) be phi-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 phi. 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) is Stein if and only if m log.( M) <= 2 pi(2), where rho(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.
Publisher
SPRINGER
Issue Date
2008-08
Language
English
Article Type
Article
Keywords

FIBER-BUNDLES; SERRE PROBLEM; BOUNDED DOMAIN; BASE

Citation

MATHEMATISCHE ANNALEN, v.341, no.4, pp.717 - 733

ISSN
0025-5831
DOI
10.1007/s00208-007-0201-4
URI
http://hdl.handle.net/10203/88627
Appears in Collection
MA-Journal Papers(저널논문)
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