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The hyper-Kähler derived-equivalence conjecture for Chow motives

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Let XX and YY be projective hyper-Kähler varieties. An exact equivalence is an equivalence between their triangulated bounded derived categories D⁡b(X)\operatorname{D}^b(X) and D⁡b(Y)\operatorname{D}^b(Y). Their Chow motives h(X)\mathfrak{h}(X) and h(Y)\mathfrak{h}(Y) carry Frobenius algebra structures.

Hyper-Kähler motivic conjecture. If D⁡b(X)\operatorname{D}^b(X) and D⁡b(Y)\operatorname{D}^b(Y) are exactly equivalent, then h(X)\mathfrak{h}(X) and h(Y)\mathfrak{h}(Y) are isomorphic as Frobenius algebra objects in the category of Chow motives. In particular, their Chow rings and cohomology rings are isomorphic.

This is the hyper-Kähler specialization of the expected relationship between derived categories and motives. The paper proves the corresponding statement for K3 surfaces and discusses compatibility with known results for Hilbert schemes and birational hyper-Kähler varieties; the general claim remains open.

References

Primary source

Lie Fu and Charles Vial, “A motivic global Torelli theorem for isogenous K3 surfaces”, arXiv:1907.10868 (2021).

derived-categorieshyperkahler-geometrymotives
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Progress summary

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Claimed progress

The general conjecture remains open: derived-equivalent hyper-Kähler varieties are known to have matching cohomology, but a full correspondence of algebraic cycles has not been proved.

The conjecture, stated as Conjecture 4.7 in 2019, predicts that an exact equivalence between projective hyper-Kähler varieties yields an isomorphism of their Chow motives preserving the Frobenius algebra structure. The K3K3 case is proved, while the general statement remains unresolved.

Known results

  • K3K3 surfaces: derived equivalence implies isomorphic Chow motives (2017).
  • Derived-equivalent projective hyper-Kähler varieties have isomorphic cohomology rings; additional motivic results cover birational varieties and Hilbert schemes of K3K3 surfaces (2019).
  • For varieties of K3[n]K3^{[n]}-type, the derived-equivalence conjecture for birational varieties was proved in 2024, but this is not a Chow-motive result.

January 28, 2026 restricted progress

Maulik et al. are reported to claim that homological motives are settled for derived-equivalent varieties of K3[n]K3^{[n]}-type, compatibly with cup products; under Franchetta-type conditions, corresponding Chow-motive isomorphisms are reported. These are restricted or conditional results, not a proof of the full Frobenius-algebra conjecture, and the report is unverified.

Current status (as of August 2026): The K3K3 case and several restricted consequences are known, but the general Chow-motive Frobenius-algebra conjecture remains open; the January 2026 advance is unverified.

Sources

Solutions 0

No solutions have been posted yet.