  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.
