Holomorphic Functions on Bundles Over Annuli
Abstract
We consider a family Em(D,M) of holomorphic bundles constructed as follows: to any given M in GLn(Z), we associate a "multiplicative automorphism" f of (C*)n. Now let D be a f-invariant Stein Reinhardt domain in (C*)n. Then Em(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 Em(D,M) depends nontrivially on the parameters m, M and D. Our main result is that Em(D,M) is Stein if and only if m log(r(M)) <= 2 π2, 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.