Stable G-birationality via twisted orbifold arc-space motivic measures

Investigate whether the discrete-torsion–twisted orbifold motivic measure μorb,α(L(X)) is (i) a motivic measure and (ii) a stable G-birational invariant; in particular, determine whether inequality μorb,α(L(X×P^N))≠μorb,α′(L(X′×P^N)) implies that Y×P^N is not G-birational to Y′×P^N.

Background

Extending the motivic framework to products with projective space, the authors pose whether their twisted measures detect stable G-birational inequivalence.

They explicitly ask for both the measure property and the stability property under products, with a concrete inequality criterion.

References

Question Let $X=Y/G$ be a global quotient orbifold. Is the quantity $\mu_{\mathrm{orb},\alpha}(\mathrm{L}(X))=\sum_{(g)\in T_1}\left[\left(Y{g}/C{\alpha}(g)\right)/X\right]\bb L{\iota_{(g{-1})}$ (1) a motivic measure (2) a stable $G$-birational invariant 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 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 for some $N\geq 1$ there exist $\alpha\in \mathrm{H}2(\pi_1{\text{orb}(X\times \bb PN);\bb C*)$, $\alpha'\in \mathrm{H}2(\pi_1{\text{orb}(X'\times \bb PN);\bb C*)$ such that $\mu_{\mathrm{orb},{\alpha}(\mathrm{L}(X\times \bb PN))\neq \mu_{\mathrm{orb},{\alpha'}(\mathrm{L}(X'\times \bb PN))$ then $Y\times \bb PN\not\sim_GY'\times \bb PN?$

— A Gromov-Witten approach to $G$-equivariant birational invariants (2405.07322 - Cavenaghi et al., 12 May 2024) in Section 6.4, “Stable G-rationality”