Holomorphic Functions on Bundles Over Annuli

TL;DR

This paper investigates the conditions under which certain holomorphic bundles over annuli are Stein, revealing how their complex structure depends on parameters and providing classifications and insights into longstanding conjectures.

Contribution

It establishes a precise criterion for when these bundles are Stein, linking geometric parameters to spectral properties, and offers new classifications and perspectives on existing conjectures.

Findings

E_m(D,M) is Stein iff m log(r(M)) <= 2π^2

Provides a classification of Reinhardt domains in all dimensions

Offers insights into counterexamples to Serre's question and reformulates a disproved conjecture

Abstract

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.