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On the volume of isolated singularities
Published online by Cambridge University Press: 23 June 2014
Abstract
We give an equivalent definition of the local volume of an isolated singularity $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\rm Vol}_{\text {BdFF}}(X,0)$ given in [S. Boucksom, T. de Fernex, C. Favre, The volume of an isolated singularity. Duke Math. J. 161 (2012), 1455–1520] in the
$\mathbb{Q}$-Gorenstein case and we generalize it to the non-
$\mathbb{Q}$-Gorenstein case. We prove that there is a positive lower bound depending only on the dimension for the non-zero local volume of an isolated singularity if
$X$ is Gorenstein. We also give a non-
$\mathbb{Q}$-Gorenstein example with
${\rm Vol}_{\text {BdFF}}(X,0)=0$, which does not allow a boundary
$\Delta $ such that the pair
$(X,\Delta )$ is log canonical.
MSC classification
- Type
- Research Article
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- © The Author 2014
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