Orbifold arc-space motivic measure as a G-birational invariant

Show that the motivic measure μorb(L(X)) defined by μorb(L(X))=∑_{(g)∈T1} [X_{(g)}/X]·L^{ι_{(g^{-1})}} is a G-birational invariant of global quotient orbifolds X=Y/G; in particular, demonstrate that if μorb(L(X))≠μorb(L(X′)) then Y is not G-birational to Y′.

Background

Inspired by the McKay–Reid motivic framework, the authors propose an orbifold motivic measure on the arc space that sums contributions from twisted sectors with degree shift.

They conjecture that this measure detects G-birational inequivalence of the underlying smooth G-varieties producing the orbifold quotients.

References

We conjecture the following. The motivic measure below is a $G$-birational invariant $\mu_{\mathrm{orb}(\mathrm{L}(X))=\sum_{(g)\in T_1}\left[X_{(g)}/X\right]\bb L{\iota_{(g{-1})},$ where $\bb L=[\bb A1]$ is the Tate motive and $\mathrm{L}(X)$ is the space of arcs in $X$ whose $\bb C$-points correspond to formal arcs $\mathrm{Spec}\bb C[[z]]\rightarrow X$. In other words, let a finite group $G$ act regularly generically free on smooth projective irreducible varieties $Y,~Y'$ of dimension $d\geq 2$. Let $X,~X'$ be the resulting global quotient orbifolds (obtained as orbit spaces). If $\mu_{\mathrm{orb}(\mathrm{L}(X))\neq \mu_{\mathrm{orb}(\mathrm{L}(X'))$ then $Y\not\sim_GY'$.

— A Gromov-Witten approach to $G$-equivariant birational invariants (2405.07322 - Cavenaghi et al., 12 May 2024) in Section 6.1, “A motivic interpretation of G-birationality”