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Orlov's motivic conjecture for derived-equivalent varieties

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Let XX and YY be smooth projective varieties. Orlov's motivic conjecture predicts

DbCoh⁡(X)≃DbCoh⁡(Y)⟹h(X)Q≃h(Y)Q,D^b\operatorname{Coh}(X)\simeq D^b\operatorname{Coh}(Y) \quad\Longrightarrow\quad h(X)_{\mathbb Q}\simeq h(Y)_{\mathbb Q},

where h(X)Qh(X)_{\mathbb Q} denotes the rational Chow motive.

A Fourier-Mukai equivalence produces correspondences between XX and YY, but they generally mix codimensions and Tate twists. The difficulty is to extract mutually inverse correspondences of pure codimension dim⁡X\dim X.

The conjecture is known for K3 surfaces and some other special cases, but remains open in higher dimensions, notably for important hyperkähler examples. A related noncommutative Hodge conjecture asks whether rational Hodge classes of a smooth proper dg category, or an admissible component of Db(X)D^b(X), come from Chern characters of rational KK-theory classes.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Orlov's motivic derived invariance conjecture

    Let XX and YY be smooth projective varieties over a field kk, and let Db(X)D^b(X) and Db(Y)D^b(Y) denote their bounded derived categories of coherent sheaves. Assume that there is an equivalence of triangulated categories

    Db(X)≃Db(Y).D^b(X) \simeq D^b(Y).

    Orlov's motivic conjecture. The rational Chow motives of XX and YY are isomorphic.

    This is the stronger motivic form of Orlov's conjecture mentioned in the source. It would imply agreement of a much richer invariant than cohomology, but the source gives no evidence of a resolution.

    source: Dion Leijnse, “Varieties admitting a holomorphic symplectic form: LLV algebras and derived equivalences”, arXiv:2605.26398 (2026).

References

References

Source: https://arxiv.org/html/2601.20289v1 Source: https://ems.press/content/serial-article-files/31056

algebraic-geometryderived-categoriesmotives
Posted by Vihaan Dheer Posted

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