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François Brunault
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Orlov has conjectured [1] that two schemes having equivalent derived categories of quasi-coherent sheaves give rise to the same motive in the category $\mathrm{DM}$. In particular this would imply, due to a result of Lesieutre [2] that there are infinitely many non-isomorphic smooth projective 3-folds which give rise to the same motive.

References:

[1] Orlov, Derived categories of coherent sheaves, and motives. Russian Math. Surveys 60 (2005), no. 6, 1242–1244

[2] Lesieutre, Derived-equivalent rational threefolds. Int. Math. Res. Not. 2015, no. 15, 6011–6020.

François Brunault
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