The volume of an isolated singularity
By S. Boucksom, Tommaso de Fernex, C. Favre. 2010
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We introduce a notion of volume of a normal isolated singularity that gener- alizes Wahl's characteristic number of surface singularities to arbitrary dimensions. We prove a basic monotonicity property of this volume under finite morphisms. We draw sev- eral consequences regarding the existence of non-invertible finite endomorphisms fixing an isolated singularity. Using a cone construction, we deduce that the anticanonical di- visor 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 notion 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.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/f228b9a7b1f1af90c11894531e3fea84c121eab0
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In Algebra and Number Theory
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Citations in the corpus
- A Guided Tour to Normalized Volume (2018, background): “ log discrepancy, we can rescale such that λ·v ∈Val=1 X,D where Val =1 X,D consists of all valuations v ∈ValX,x with A(X,D)(v) = 1. Remark 2.4. A definition of volume of singularities is also given in [BdFF12]. Their definition is the local analogue of the volume KX whereas our definition is the one of”
- Stability of valuations and Kollár components (2016, background): “hen we modify the birational models. We hope to explore these interesting questions in the future. 2.The volume of klt singularities dened here is dierent with the volume of singularities dened in [BdFF12] (see also [Zha14]). The volume in [BdFF12] is dened using envelops of log discrepancy b-diviso”