Categorical measures for finite group actions

Bergh, D.; Gorchinskiy, S.; Larsen, M.; Lunts, V.

doi:10.1090/jag/768

Authors: D. Bergh, S. Gorchinskiy, M. Larsen and V. Lunts
Journal: J. Algebraic Geom. 30 (2021), 685-757
DOI: https://doi.org/10.1090/jag/768
Published electronically: April 28, 2021
Uncorrected version: Original version posted March 16, 2021
Corrected version: A correction was made to the funding information.
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Abstract | References | Additional Information

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 that 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.

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Additional Information

D. Bergh
Affiliation: Department of Mathematical Sciences, University of Copenhagen, 1172 Copenhagen, Denmark
MR Author ID: 1147933
Email: dbergh@math.ku.dk

S. Gorchinskiy
Affiliation: Steklov Mathematical Institute of Russian Academy of Sciences, Russian Federation
MR Author ID: 786536
Email: gorchins@mi.ras.ru

M. Larsen
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
MR Author ID: 293592
Email: mjlarsen@indiana.edu

V. Lunts
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405; and National Research University, Higher School of Economics, Russian Federation
MR Author ID: 265625
Email: vlunts@indiana.edu

Received by editor(s): June 12, 2019
Received by editor(s) in revised form: April 1, 2020
Published electronically: April 28, 2021
Additional Notes: The first named author was partially supported by the Danish National Research Foundation through the Niels Bohr Professorship of Lars Hesselholt, by the Max Planck Institute for Mathematics in Bonn, and by the DFG through SFB/TR 45.
The work of the second named author was performed at the Steklov International Mathematical Center and was supported by the Ministry or Science and Higher Education of the Russian Federation (agreement no. 075-15-2019-1614). He is also very grateful for excellent working conditions to FIM ETH Zürich, where part of this work was done.
The third named author was partially supported by NSF grant DMS-1702152.
The fourth named author was partially supported by the NSA grant H98230-15-1-0255 and by the Laboratory of Mirror Symmetry NRU HSE, RG Government grant ag. no. 14.641.31.0001.
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