>>126 もどる 補足

(参考)
https://users.fmf.uni-lj.si/forstneric/datoteke/2016-LongC2.pdf
Long C2 s without holomorphic functions
Franc Forstneric
University of Ljubljana Institute of Mathematics, Physics and Mechanics
CR Geometry and PDEs VII, Levico, June 2016

What is a long Cn?
A complex manifold X of dimension n is said to be a long Cn if it is the union of an increasing sequence of domains
X1 ⊂ X2 ⊂X3 ⊂・・・⊂ ∪j=1〜∞ Xj = X
such that each Xj is biholomorphic to the complex Euclidean space Cn.
Identifying Xj ≃ Cn, each inclusion Xj → Xj+1 is given by a
Fatou-Bieberbach (FB) map φj : Cn → Cn, and X is the direct limit of
the system ( φj)j ∈N. The elements of X are represented by infinite
strings x = (xi,xi+1,...), where i∈ N and
xk+1 = φk(xk), k =i,i +1,....
Another string y = (yj,yj+1,...) determines the same element of X if
and only if one of the following possibilities hold:
略
The rst main theorem Every long C equals C. The situation is very di erent for n > 1. Theorem (L. Boc Thaler and F.F., 2015) For every integer n > 1 there exists a long Cn without any nonconstant holomorphic or plurisubharmonic functions. This theorem gives a strong counterexample to the classical union problem: is an increasing union of Stein manifolds always Stein?

Behnke and Thullen 1933, 1939 Yes for domains in Cn.
J.E. Forn ss 1976 NO: There is an increasing union of complex 3-balls that is not holomorphically convex, hence not Stein.
Key ingredient: a biholomorphic map : 略
on a C3 bounded neighborhood Ω ⊂ C3 of any compact set K ⊂ C3 with ˚K≠∅
such that 略

History continued
Forn ss and Stout 1977 A 3-dimensional increasing union of polydiscs without any nonconstant holomorphic functions.
E.F. Wold 2008 Non-Runge Fatou-Bieberbach domains in C2.
E.F. Wold 2010 Non-Stein (and non-holomorphically convex) long C2
略