Holomorphic functions on bundles over annuli

We consider a family of holomorphic bundles constructed as follows:from any given , we associate a “multiplicative automorphism” of . Now let be a -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 . 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 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.