Holomorphic families of long C^2's

TL;DR

This paper constructs a family of complex surfaces parametrized by a Stein manifold, where each surface is a long C^2 biholomorphic to C^2 for some parameters but not others, illustrating complex deformation phenomena.

Contribution

It introduces a holomorphic family of long C^2 surfaces parametrized by Stein manifolds, demonstrating diverse biholomorphic types within a single family.

Findings

Existence of holomorphic families of long C^2 surfaces parametrized by Stein manifolds.

Variation in biholomorphic type within the family.

Examples of complex surfaces with controlled deformation properties.

Abstract

We construct a holomorphically varying family of complex surfaces X_s, parametrized by the points s in any Stein manifold, such that every X_s is a long C^2 which is biholomorphic to C^2 for some but not all values of s.