In this paper we look at the following two problems:
Compactification problem:Let X be a smooth complex projective variety, A a divisor on X such that X\A is biregular to an affine homology cell, i.e. a smooth affine scheme with the homology groups of a point. Classify all pairs (X, A)!
Characterisation problem:Let V = Spec(R) be a smooth complex affine scheme. Characterise V by topological or algebraic properties of V itself and at infinity (i.e. of suitable smooth compactifications).
We will present known results about these two problems as well as their relation to other topics like Fano varieties and homogenous spaces.
Our problems are obviously related by the process of compactification which is due to Nagata in this category. Therefore results about one of these problems will give information about the solution of the other.
The compactification problem was first stated by Hirzebruch (see [H]) in the category of complex manifolds: Let X be a complex manifold with an analytic subset A such that X\A is biholomorphic to Cn. Then (X, A) is called a complex analytic compactification of Cn. Problems 26 and 27 in [H] ask for the classification of all pairs (X, A). By the theorem of Hartogs it follows that A has to be a divisor in X, hence this is a special case of our compactification problem in the complex analytic case.
The characterisation problem was attacked by several people, including Ramanujam [R] and [Mi].
In the sequel we give an overview of some of the results, in particular concerning the case of 3-dimensional varieties.