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Pairwise non-isomorphic twisted K-theories for the orbi-Fanos Vk

Construct gerbes with connection ℒk over each Fano orbifold Vk arising from the S^1-quotients of Brieskorn-type hypersurface links, such that the twisted K-theory groups {}^{ℒk}Kgrp(Vk) are pairwise non-isomorphic for distinct k in {1,…,28}.

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Background

Building on connections between exotic spheres, orbifold bases of S1-bundles, and gerbes classified by Deligne cohomology, the authors suggest using twisted K-theory to distinguish birational types of these orbifolds.

They formulate a conjecture asserting the existence of twists yielding non-isomorphic K-theory groups across the 28 cases.

References

Conjecture For each $k\in {1,\ldots,28}$ there exist gerbes with connections $\cal L_k$ over the Fano orbifolds $V_k$ have non-such that, for each $k\neq k'\in {1,\ldots,28}$ it holds that ${}{\mathcal L_k}K_{\mathrm{grp}(V_k)\not\cong {}{\mathcal L_{k'}K_{\mathrm{grp}(V_{k'}).$

— A Gromov-Witten approach to $G$-equivariant birational invariants (2405.07322 - Cavenaghi et al., 12 May 2024) in Section 6.3, “Birational invariants for orbifolds via twisted K-theory”