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 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.

48 pages. v4: Appendix is new, plus several minor changes made throughout the paper following the referee's suggestions. Final version, to appear in Duke Math J

References in corpus (2)

• Singularities on normal varieties
• Polarized endomorphisms of uniruled varieties (with Appendix by Y. Fujimoto and N. Nakayama)

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• Polarized endomorphisms of normal projective threefolds in arbitrary characteristic
• On the Volume of Isolated Singularities
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• Vanishing theorems and adjoint linear systems on normal surfaces in positive characteristic
• The jumping coefficients of non-Q-Gorenstein multiplier ideals
• On the geometry of spaces of filtrations on local rings