On the Volume of Isolated Singularities

TL;DR

This paper redefines the local volume of isolated singularities, extends the concept beyond Q-Gorenstein cases, and establishes a positive lower bound for Gorenstein singularities, with examples illustrating the limitations.

Contribution

It provides an equivalent definition of local volume for isolated singularities and generalizes it to non-Q-Gorenstein cases, advancing singularity theory.

Findings

Positive lower bound for local volume in Gorenstein case

Extension of local volume definition to non-Q-Gorenstein singularities

Existence of non-Q-Gorenstein example with zero local volume

Abstract

We give an equivalent definition of the local volume of an isolated singularity Vol_{BdFF}(X,0) given in [BdFF12] in the Q-Gorenstein case and we generalize it to the non-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-Q-Gorenstein example with Vol_{BdFF}(X,0)=0, which does not allow a boundary \Delta such that the pair (X,\Delta) is log canonical.