Notes on minimal normal compactifications of \(\mathbb{C}^2/G\) (Q1769941)

The author considers compact complex surfaces \((X,C)\) such that \(X\setminus C\) is biholomorphic to the affine surface \(\mathbb C^2/G\) for a finite subgroup \(G\) of \(\text{GL}(2,\mathbb C)\). \(X\) is assumed to be smooth along the compactifying curve \(C\) and \(C\) is a minimal normal crossing divisor in the usual sense. Two main results are as follows: 1) A homology plane \(S\) is isomorphic to \(\mathbb C^2\) if and only if it has at least two non-isomorphic minimal normal algebraic compactifications \((X, C)\). 2) If \(G\) is non-cyclic then any minimal normal compactification \((X, C)\) of \(\mathbb C^2/G\) is algebraic and \(C\) has the expected dual graph. Earlier \textit{M. Abe, M. Furushima} and \textit{M. Yamasaki} [Kyushu J. Math. 54, 87--101 (2000; Zbl 1045.32020)] studied this problem for almost all \(G\), using the theory of cluster sets of holomorphic mappings due to \textit{T. Nishino} and \textit{M. Suzuki} [Publ. Res. Inst. Math. Sci. 16, No. 2, 461--497 (1980; Zbl 0506.32007)]. The present proof uses standard algebraic surface theory.