On isolated Gorenstein singularities
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.
n-dimensional purely elliptic singularities
Hodge structure of the exceptional divisors
scientific article; zbMATH DE number 3859273
scientific article