The volume of an isolated singularity

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.