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On isolated Gorenstein singularities. (English) Zbl 0541.14002
Let (X,x) be a germ of an isolated singularity of an n-dimensional analytic space. To investigate a normal isolated singularity (X,x) K. Watanabe introduced pluri-genera \(\{\delta_ m(X,x)\}_{m\in {\mathbb{N}}}\). For a normal isolated Gorenstein singularity (X,x), it is known that either \(\delta_ m(X,x)=0\) for any m, \(\delta_ m(X,x)=1\) for any m or \(\delta_ m(X,x)\) grows in order n as a function in m.
In this article, it is shown that for a normal isolated Gorenstein singularity (X,x), \(\delta_ m(X,x)\leq 1\) holds for every \(m\in {\mathbb{N}}\) if and only if \(H^ i(\tilde X,{\mathcal O}_{\tilde X})\cong H^ i(E,{\mathcal O}_ E)\) for any \(i>0\), where \(f:\tilde X\to X\) is a resolution of the singularity (X,x) with \(E=f^{-1}(x)_{red}\) simple normal crossings. The singularity with the second property is called a Du Bois singularity (Steenbrink). - Let E be as above and decompose it into irreducible components \(E_ i (i=1,2,...,r)\). Then it is also shown that a normal isolated Gorenstein singularity (X,x) is Du Bois if and only if the canonical divisor \(K_{\tilde X}=\sum^{r}_{i=1}m_ iE_ i\) satisfies \(m_ i\geq -1\) for every i. More precisely, \(''\delta_ m(X,x)=0\) for every m” iff \(m_ i\geq 0\) for every i, and \(''\delta_ m(X,x)=1\) for every m” iff \(m_ i\geq -1\) for every i and \(m_ i=-1\) for some i. In the former case, (X,x) is rational. We call (X,x) purely elliptic in the later case. - We classify the set of n-dimensional purely elliptic singularities into n-types by the Hodge structure of the exceptional divisors, and consider the configuration of the exceptional divisor of a certain resolution of a singularity of each type. Next, we construct purely elliptic singularities of each type of any dimension \(n\geq 2\), by means of blowing down.

14B05 Singularities in algebraic geometry
32S05 Local complex singularities
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
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