On singular 4-manifolds of the homology type of \(CP^ 2\)
We investigate the relations between certain global topological properties of compact complex surfaces and the analytic structure of their singular points, with these results among others: 1. Let X be a normal Gorenstein surface of the integral homology type of \({\mathbb{C}}{\mathbb{P}}^ 2\), with vanishing geometric genus and with \(H^ 2(X,{\mathbb{Z}})\) generated, say, by g. Then the singular point(s) of X is (are) precisely the rational double point(s) associated to the Dynkin diagram \(''E_ k''\), where \(k=9-g^ 2\) and where \(''E_ k''\) is the graph with k vertices, \(k=0,1,...,9\), connecting the pairs (1,4), \((j,j+1)\), \(j=2,...,k-1.\)- 2. Conversely, if X is any normal complex surface with only rational double points as singularities and with \(H^ i(X,{\mathbb{Q}})\cong H^ i({\mathbb{C}}{\mathbb{P}}^ 2,{\mathbb{Q}})\) for all i, then either X is \({\mathbb{C}}{\mathbb{P}}^ 2\), the complex quadric surface \({\mathbb{Q}}^ 2_ 0\), or a maximally degenerate Del Pezzo surface; or else X is derived from a minimal non-singular Enriques surface \(\tilde X\), or from a minimal non-singular projective surface \(\tilde X\) of general type with q(\~X)\(=p_ g(\tilde X)=0\), by blowing down non-singular rational curves of grade -2.
scientific article; zbMATH DE number 3861330
scientific article