Holomorphic functions on bundles over annuli

Abstract

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.