The asymptotic behavior of plurigenera for a normal isolated singularity
type: Item
instance of: scholarly article
title: The asymptotic behavior of plurigenera for a normal isolated singularity
zbMATH Open document ID: 0668.14002
DOI: 10.1007/BF01453603
author: Shihoko Ishii
published in: Mathematische Annalen
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164665
review text: 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).
reviewed by: Shihoko Ishii
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 32S05
zbMATH DE Number: 4093621
zbMATH Keywords: normal isolated singularity on an analytic space
zbMATH Keywords: plurigenus
zbMATH Keywords: classification of isolated singularities
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4093621