Stable G-birational invariants via discrete torsion and gerbes

Show that for global quotient orbifolds X=Y/G, (i) the image of H^2(G;C*) in H^2(π1^orb(X);C*) is a stable birational invariant of X; and (ii) if Y is rationally connected, then G-birational types of Y are classified by gerbes with connection over X (via H^3(BX;Z)), while G-stable birational types are classified by the image of H^2(G;C*) in H^2(π1^orb(X);C*).

Background

Leveraging the analogy with Brauer groups and unramified cohomology in stable rationality, the authors propose a cohomological classification of G-(stable) birational types via discrete torsion and gerbes.

They formulate a two-part conjecture specifying the invariants and the classification framework.

References

Conjecture Let $X=Y/G$ be a global quotient orbifold. Then (1) The images $\mathrm{H}2(G;\bb C*)\rightarrow \mathrm{H}2(\pi_1{\text{orb}(X);\bb C*)$ is a stable birational invariant for $X$ just as the Brauer group $\mathrm{Br}(Y)$ is a stable birational invariant of $Y$. (2) If $Y$ is rationally connected, then the $G$-rational types of $Y$ are classified by the gerbes with connections over $X$, thus, can be ready from $\mathrm{H}3(BX;\bb Z)$. For the $G$-stable birational types of $Y$, these can be classified by the images $\mathrm{H}2(G;\bb C*)\rightarrow \mathrm{H}2(\pi_1{\text{orb}(X);\bb C*)$.

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