Duke Mathematical Journal

The volume of an isolated singularity

Sebastien Boucksom, Tommaso de Fernex, and Charles Favre

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Abstract

We introduce a notion of volume of a normal isolated singularity that generalizes Wahl’s characteristic number of surface singularities to arbitrary dimensions. We prove a basic monotonicity property of this volume under finite morphisms. We draw several consequences regarding the existence of noninvertible finite endomorphisms fixing an isolated singularity. Using a cone construction, we deduce that the anticanonical divisor of any smooth projective variety carrying a noninvertible polarized endomorphism is pseudoeffective.

Our techniques build on Shokurov’s b-divisors. We define the notions of nef Weil b-divisors and of nef envelopes of b-divisors. We relate the latter to the pullback of Weil divisors introduced by de Fernex and Hacon. Using the subadditivity theorem for multiplier ideals with respect to pairs recently obtained by Takagi, we carry over to the isolated singularity case the intersection theory of nef Weil b-divisors formerly developed by Boucksom, Favre, and Jonsson in the smooth case.

Article information

Source
Duke Math. J., Volume 161, Number 8 (2012), 1455-1520.

Dates
First available in Project Euclid: 22 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1337690406

Digital Object Identifier
doi:10.1215/00127094-1593317

Mathematical Reviews number (MathSciNet)
MR2931273

Zentralblatt MATH identifier
1251.14026

Subjects
Primary: 14J17: Singularities [See also 14B05, 14E15]
Secondary: 14C20: Divisors, linear systems, invertible sheaves 14E99: None of the above, but in this section 14F18: Multiplier ideals

Citation

Boucksom, Sebastien; de Fernex, Tommaso; Favre, Charles. The volume of an isolated singularity. Duke Math. J. 161 (2012), no. 8, 1455--1520. doi:10.1215/00127094-1593317. https://projecteuclid.org/euclid.dmj/1337690406


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