Holomorphic Functions on Bundles Over Annuli

Details Holomorphic Functions on Bundles Over Annuli Holomorphic Functions on Bundles Over Annuli

2008-10-10

arxiv.org/abs/0810.1817v1

Math. Ann. 341 (2008), no. 4, 717--733

Mathematics

Complex Variables

Scientific paper

10.1007/s00208-007-0201-4

We consider a family E_m(D,M) of holomorphic bundles constructed as follows: to any given M in GL_n(Z), we associate a "multiplicative automorphism" f of (C*)^n. Now let D be a f-invariant Stein Reinhardt domain in (C*)^n. Then E_m(D,M) is defined as the flat bundle over the annulus of modulus m>0, with fiber D, and monodromy f. 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(r(M)) <= 2 \pi^2, where r(M) denotes the max of the spectral radii of M and its inverse. As corollaries, we: -- obtain a classification result for Reinhardt domains in all dimensions; -- establish a similarity between two known counterexamples to a question of J.-P. Serre; -- suggest a potential reformulation of a disproved conjecture of Siu Y.-T.

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