Categorical measures for finite group actions

TL;DR

This paper compares two categorical measures associated with finite group actions on varieties, demonstrating their equality in many cases and providing counterexamples, thereby advancing understanding of orbifold invariants and related conjectures.

Contribution

It establishes conditions under which the equivariant and extended quotient categorical measures coincide, confirming a conjecture and clarifying necessary assumptions.

Findings

Equivalence of measures in many cases using orbifold factorization

Counterexamples where measures differ

Validation of a conjecture on the necessity of certain conditions

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.