The volume of an isolated singularity

TL;DR

This paper introduces a new volume concept for isolated singularities, generalizing previous notions, and explores its properties and implications for endomorphisms and divisor theory using advanced algebraic geometry techniques.

Contribution

It defines a volume for isolated singularities, proves its monotonicity under finite morphisms, and applies it to study endomorphisms and divisor properties in algebraic geometry.

Findings

Volume of isolated singularities generalizes Wahl's characteristic number.

Monotonicity of volume under finite morphisms is established.

Implications for the existence of non-invertible endomorphisms and divisor pseudoeffectiveness are derived.

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 non-invertible finite endomorphisms fixing an isolated singularity. Using a cone construction, we deduce that the anticanonical divisor of any smooth projective variety carrying a non-invertible polarized endomorphism is pseudoeffective. Our techniques build on Shokurov's b-divisors. We define the notion of nef Weil b-divisors, and of nef envelopes of b-divisors. We relate the latter to the pull-back 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.