The volume of an isolated singularity. (English) Zbl 1251.14026

Boucksom, Sebastien; De Fernex, Tommaso; Favre, Charles

Duke Math. J. 161, No. 8, 1455-1520 (2012).

In the paper under review, the authors introduce a new notion of the volume of a normal isolated singularity \( (X,0)\) that, in dimension \(2\) agrees with Wahl’s characteristic number of surface singularities. They show that the volume \(\text{Vol} (X,0)\in \mathbb R _{\geq 0}\) satisfies the following properties:
– If \(X\) is \(\mathbb Q\)-Gorenstein, then \(X\) has log canonical singularities if and only if \(\text{Vol} (X,0)=0\).
– If \(\phi :(X,0)\to (Y,0)\) is a finite morphism of degree \(e(\phi)\), then \(\text{Vol} (X,0)\geq e(\phi )\text{Vol} (Y,0)\).
Two interesting consequences are: 1) If \(\phi :(X,0)\to (X,0)\) is a finite non-invertible morphism, then \(\text{Vol} (X,0)=0\) so that if \(X\) is \(\mathbb Q\)-Gorenstein then \(X\) is log canonical, and 2) If \(V\) is a smooth projective variety with a non-invertible polarized endomorphism \(\phi\), then \(-K_V\) is pseudo-effective.

Reviewer: Christopher Hacon (Salt Lake City)

MSC:

14J17	Singularities of surfaces or higher-dimensional varieties
14C20	Divisors, linear systems, invertible sheaves
14F18	Multiplier ideals
14B05	Singularities in algebraic geometry
14E99	Birational geometry

Keywords:

isolated singularities

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