Discrete-torsion–twisted orbifold motivic measure and G-birational invariance

Establish that the discrete-torsion–twisted orbifold motivic measure μorb,α(L(X))=∑_{(g)∈T1} [(Y^g/C^α(g))/X]·L^{ι_{(g^{-1})}} is a motivic measure and a G-birational invariant.

Background

Using inner local systems from discrete torsion α∈H2(π1orb(X);C*), the authors define twisted Chen–Ruan cohomology that modifies centralizer actions.

They conjecture an analogous motivic measure incorporating the twisted centralizer Cα(g) and ask to prove its motivic and G-birational invariance properties.

References

Conjecture The quantity given by Equation eq:motivicmeasuretorsioned below $\mu_{\mathrm{orb},\alpha}(\mathrm{L}(X))=\sum_{(g)\in T_1}\left[\left(Y{g}/C{\alpha}(g)\right)/X\right]\bb L{\iota_{(g{-1})}$ is (1) a motivic measure (2) a $G$-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 $\mathrm{Spec}\bb C[[z]]\rightarrow X$.

— A Gromov-Witten approach to $G$-equivariant birational invariants (2405.07322 - Cavenaghi et al., 12 May 2024) in Section 6.2, “Twisted K-theory”