Holomorphic families of long \(\mathbb{C}^{2}\)'s (Q2845053)

This paper in several complex variables adds to the long line of results on ``unusual'' domains and complex manifolds. It constructs holomorphic families of complex manifolds parametrized by a Stein manifold whose fibers are all smoothly diffeomorphic to \({\mathbb C}^n\), \(n\geq2\), but only some of them are biholomorphic to \({\mathbb C}^n\), and some of them are not (even Stein). More precisely, here is a corollary of his Theorem~1.1 that the author points out. For any two countable sets \(A,B\subset{\mathbb C}\) with \(A\cap B=\emptyset\) there is a holomorphic family \(X_s\), \(s\in{\mathbb C}\), of long \({\mathbb C}^2\)'s such that \(X_s\), \(s\in A\), is biholomorphic to \({\mathbb C}^2\), but \(X_s\), \(s\in B\), is not biholomorphic to \({\mathbb C}^2\) (not even Stein). Here a long \({\mathbb C}^2\) is a complex manifold \(X\) of dimension \(2\) that can be written as the increasing union \(X=\bigcup_{j=1}^\infty X^j\) of an exhaustion by open subsets \(X^j\subset X\) with each \(X^j\) biholomorphic to \({\mathbb C}^2\). One of the first type of unusual domains in \({\mathbb C}^n\), \(n\geq2\), is the class of Fatou-Bieberbach domains from the early 1900s having to do with iteration and holomorphic hulls. The author and others have developed a thorough understanding of approximation and iteration of automorphisms and biholomorphisms of domains in complex manifolds, beginning with \({\mathbb C}^n\) and its subdomains, referred to as Andersén-Lempert theory. He uses a (non-Runge) Fatou-Bieberbach domain of Wold to start up a complicated iteration scheme that is constructed with the help of automorphisms (and approximations of automorphisms on subdomains by automorphisms on the whole space) to move around a pair of Stolzenberg discs to create or destroy holomorphic convexity (in the limit manifold) at the required parameter values \(s\). NEWLINENEWLINEThe paper is well written but is suitable only to those familiar with the references [\textit{E. Andersén} and \textit{L. Lempert}, Invent. Math. 110, No. 2, 371--388 (1992; Zbl 0770.32015); the author and \textit{J.-P. Rosay}, Invent. Math. 112, No. 2, 323--349 (1993; Zbl 0792.32011); \textit{G. Stolzenberg}, Math. Ann. 164, 286--290 (1966; Zbl 0141.27303); \textit{D. Varolin}, J. Geom. Anal. 11, No. 1, 135--160 (2001; Zbl 0994.32019); \textit{E. F. Wold}, Math. Ann. 340, No. 4, 775--780 (2008; Zbl 1137.32002); \textit{E. F. Wold}, Ark. Mat. 48, No. 1, 207--210 (2010; Zbl 1189.32004)].