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The asymptotic behavior of plurigenera for a normal isolated singularity. (English) Zbl 0668.14002

Let (X,x) be a normal isolated singularity on an analytic space X of dimension \(n\geq 2\). In this paper, we study the asymptotic behavior of two kinds of known plurigenera \(\gamma_ m(X,x)\) and \(\delta_ m(X,x)\). First, we introduce the concept of the growth order for certain kinds of number series. Next, we show that the growth order is well-defined for the number series \(\{\gamma_ m(X,x)\}\) and the possible value of the order is either -\(\infty\) or \(n=\dim (X)\). In order to investigate \(\delta_ m(X,x)\), we introduce a third plurigenus \(d_ m(X,x)\), for which the growth order is shown to be well-defined and its possible value to be either -\(\infty\) or an integer between 0 and n-1. The two plurigenera \(\delta_ m(X,x)\) and \(d_ m(X,x)\) turn out to be closely related with each other. The relation yields that the growth order is well-defined also for \(\{\delta_ m(X,x)\}\) and its possible value is either -\(\infty\) or an integer between 0 and n except for n-1. The relation also yields that the two classifications by the growth orders of \(\delta_ m(X,x)\) and \(d_ m(X,x)\) coincide. This classification of isolated singularities is expected to be as reasonable as that of compact varieties by the Kodaira dimension.
We have examples of normal isolated singularities with each possible value as growth order for the three plurigenera. In particular for \(n=2\), we obtain the complete classification by the growth orders of \(\delta_ m(X,x)\) and \(d_ m(X,x)\) (the one by \(\gamma_ m(X,x)\) is well known).
Reviewer: S.Ishii

MSC:

14B05 Singularities in algebraic geometry
32S05 Local complex singularities
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