Translator Disclaimer
1 June 2012 The volume of an isolated singularity
Sebastien Boucksom, Tommaso de Fernex, Charles Favre
Duke Math. J. 161(8): 1455-1520 (1 June 2012). DOI: 10.1215/00127094-1593317


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.


Download Citation

Sebastien Boucksom. Tommaso de Fernex. Charles Favre. "The volume of an isolated singularity." Duke Math. J. 161 (8) 1455 - 1520, 1 June 2012.


Published: 1 June 2012
First available in Project Euclid: 22 May 2012

zbMATH: 1251.14026
MathSciNet: MR2931273
Digital Object Identifier: 10.1215/00127094-1593317

Primary: 14J17
Secondary: 14C20 , 14E99 , 14F18

Rights: Copyright © 2012 Duke University Press


This article is only available to subscribers.
It is not available for individual sale.

Vol.161 • No. 8 • 1 June 2012
Back to Top