Compactification of \({\mathbb{C}}^ 3\) with reducible boundary divisor
A compactification of \({\mathbb{C}}^ n\) is a pair (X,A), where X is a compact complex manifold and A an analytic subset, such that \(X\setminus A\) is biholomorphic to \({\mathbb{C}}^ n\). The author classifies projective compactifications of \({\mathbb{C}}^ 3\) with two normal boundary divisors at infinity intersecting transversally. They are divided in three classes: \({\mathfrak P}^ 2\)-bundles over \({\mathfrak P}^ 1\), \({\mathfrak P}^ 1\)-bundles over \({\mathfrak P}^ 2\) and blow ups of the four Fano-3-folds \({\mathfrak P}^ 3\), \({\mathfrak Q}^ 3\), \({\mathfrak V}_ 5\) and \({\mathfrak A}_{22}\), which are exactly the irreducible compactifications of \({\mathbb{C}}^ 3\). The boundary surfaces and their singularities are also classified. Especially it follows, that such a compactification is rational.
projective compactifications of \({\mathbb{C}}^ 3\)
scientific article; zbMATH DE number 4114997
scientific article