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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.

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