We show that for a K3 surface X the finitely generated subring R (X) ⊂ CH* (X) introduced by Beauville and Voisin is preserved under derived equivalences. This is proved by analyzing Chern characters of spherical bundles (and complexes). As for a K3 surface X defined over a number field all spherical bundles on the complex K3 surface _X_ℂ are defined over ℚ, this is compatible with the Bloch–Beilinson conjecture. Besides the work of Beauville and Voisin , Lazarfeld’s result on Brill–Noether theory for curves in K3 surfaces  and the deformation theory developed in  are central for the discussion.
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Daniel Huybrechts, Chow groups of K3 surfaces and spherical objects. J. Eur. Math. Soc. 12 (2010), no. 6, pp. 1533–1551DOI 10.4171/JEMS/240