  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.
