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An old conjecture of Bondal–Orlov–Kawamata predicts that K-equivalent varieties are D-equivalent, see Kawamata's paper D-equivalence and K-equivalence for definitions. In particular this applies to birational Calabi–Yau varieties. See Progress on Bondal–Orlov derived equivalence conjecture for the Bondal–Orlov formulation of the conjecture and the known cases. These conjectures are mostly open, and considered to be an important bridge between derived categories and birational geometry.

Derived categories and Chow motives play the role of universal cohomology theories, in noncommutative, and commutative worlds respectively. Do we expect that K-equivalence implies isomorphism of rational, or even integral Chow motives?

Example 1. Integral Chow motives of varieties related by a standard flop are isomorphic: Q. Jiang - On the Chow theory of projectivization. This was the motivation for the question.

Example 2 (from Nico Berger's comment). Birational irreducible holomorphic symplectic varieties have isomorphic integral Chow motives: U. Riess - On the Chow ring of birational irreducible symplectic varieties.

Remarks. In many cases, D-equivalence is known to imply isomorphisms for rational Chow motives, but not for integral ones: examples can be found among K3 surfaces. It is also known that D-equivalence does not imply K-equivalence, even for rational surfaces: Uehara - An example of Fourier–Mukai partners of minimal elliptic surfaces.

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    $\begingroup$ @abx: Read carefully; Huybrechts considers rational Chow motives. $\endgroup$ Jan 22, 2021 at 7:46
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    $\begingroup$ Riess proved that K-equivalent (i.e. birational) irreducible holomorphic symplectic varieties have isomorphic integral Chow motives, see arxiv.org/abs/1304.4404. $\endgroup$ Jan 22, 2021 at 8:34
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    $\begingroup$ @crystalline: Yes indeed, thanks — I was misled by the statement of Theorem 1.1 in Huybrechts' paper. Doesn't "Chow motive" usually refer to the integral ones? $\endgroup$
    – abx
    Jan 22, 2021 at 8:36
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    $\begingroup$ There is an upgrade of Huybrechts' result by Fu-Vial stating that the rational Chow motives of derived equivalent K3 surfaces are isomorphic as "Frobenius algebra objects", see arxiv.org/abs/1907.10868. Their Question 1 asks for the case of ihs manifolds. $\endgroup$ Jan 22, 2021 at 9:19
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    $\begingroup$ @abx: I agree with that, but my confidence in Evgeny's expertise on this topic led me to double check Huybrechts' statement more carefully. $\endgroup$ Jan 22, 2021 at 15:02

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K-equivalence $\implies$ isomorphism of (rational) Chow motives is a conjecture going back to this 2002 paper of Wang (Conjecture 2.2); see this overview paper of his for what else K-equivalence should imply on the cohomological level.

Some important special cases, such as ordinary flops and Mukai flops are proven in this 2010 paper by Lee, Lin and Wang. In 2016 the case of semismall K-equivalences has been proved by Liu: "Motivic equivalence under semismall flops".

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