Chow groups of K3 surfaces and spherical objects

Abstract

We show that for a K3 surface $X$ the finitely generated subring $R(X)\subset {\mathrm{CH}}^{\ast }(X)$ 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 $\bar{Q}$, this is compatible with the Bloch–Beilinson conjecture. Besides the work of Beauville and Voisin [5], Lazarfeld’s result on Brill–Noether theory for curves in K3 surfaces [15] and the deformation theory developed in [12] are central for the discussion.