The Serre problem with Reinhardt fibers

Serre problem for unbounded pseudoconvex Reinhardt domains in $\mathbb C^2$

A characterization of non-hyperbolic pseudoconvex Reinhardt domains in $\mathbb C^2$ for which the answer to the Serre problem is positive is given.

Steinness of bundles with fiber a Reinhardt bounded domain

Let E denote a bundle with fiber D and with basis B. Both D and B are assumed to be Stein. For D a Reinhardt bounded domain of dimension d=2 or 3, we give a necessary and sufficient condition on D for the existence of a non-Stein such E (Theorem 1); for d=2, we give necessary and sufficient criteria for E to be Stein (Theorem 2). For D a Reinhardt bounded domain of any dimension not intersecting any coordinate hyperplane, we give a sufficient criterion for E to be Stein (Theorem 3).

Holomorphic Functions on Bundles Over Annuli

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 π^2, where r(M) denotes the max of the spectral radii of M and its inverse. As corollaries, we: -- …

Cœuré-Loeb 例子作为 Stein 空间的开子集

The Cœuré-Loeb example as an open subset of a Stein space

本文针对塞尔问题，证明了 Cœuré-Loeb 纤维化可作为 Stein 空间的开子集（即 Stein 空间中某解析集的补集），补充了该问题反例的结构性质。