Categorical measures for finite group actions

Bergh, Daniel; Gorchinskiy, Sergey; Larsen, Michael; Lunts, Valery

doi:10.1090/jag/768

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Categorical measures for finite group actions

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Bergh, Daniel
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1090/jag/768
(Publisher version)

https://doi.org/10.48550/arXiv.1709.00620
(Preprint)

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Citation

Bergh, D., Gorchinskiy, S., Larsen, M., & Lunts, V. (2021). Categorical measures for finite group actions. Journal of Algebraic Geometry, 30(4), 685-757. doi:10.1090/jag/768.

Cite as: https://hdl.handle.net/21.11116/0000-0009-4908-D

Abstract

Given a variety with a finite group action, we compare its equivariant categorical measure, that is, the categorical measure of the corresponding quotient stack, and the categorical measure of the extended quotient. Using weak factorization for orbifolds, we show that for a wide range of cases, these two measures coincide. This implies, in particular, a conjecture of Galkin and Shinder on categorical and motivic zeta-functions of varieties. We provide examples showing that, in general, these two measures are not equal. We also give an example related to a conjecture of Polishchuk and Van den Bergh, showing that a certain condition in this conjecture is indeed necessary.