Arithmetic Smith equivalence in characteristic not 2

Establish an A-infinity quasi-equivalence F: Tw^πFuk(X)_0 ≅ Tw^πFuk(Σ_2) over any field K of characteristic not 2, where X is the intersection of two quadrics in CP^5 and Σ_2 is a genus-2 curve.

Background

Smith proved over \mathbb{C} that a component of the Fukaya category of the intersection of two quadrics in CP5 is equivalent to the Fukaya category of a genus-2 curve. The conjecture asks for an arithmetic strengthening: the same quasi-equivalence over any characteristic-≠2 field.

However, Proposition 5.2 in the paper shows a necessary condition: such a quasi-equivalence forces the ground field to contain a square root of −1, so the conjecture cannot hold in full generality as originally phrased. The authors then focus on fields containing √−1 and explore implications for quantum cohomology and quantum Steenrod operations if the equivalence holds.

References

Conjecture 5.1 Over a field K of characteristic not 2, there is an A_{\infty}-quasi-equivalence \begin{equation} F: Tw{\pi}\mathrm{Fuk}(X)_0\simeq Tw{\pi}\mathrm{Fuk}(\Sigma_2). \end{equation}

— Quantum Steenrod operations and Fukaya categories  (2405.05242 - Chen, 2024) in Section 5, Conjecture 5.1