Chow groups of K3 surfaces and spherical objects

TL;DR

This paper proves that a specific subring of the Chow ring of a K3 surface remains invariant under derived equivalences and explores the arithmetic nature of spherical bundles, connecting to deep conjectures.

Contribution

It establishes the invariance of Beauville-Voisin's subring under derived equivalences and links the definability of spherical bundles over number fields to the Bloch-Beilinson conjecture.

Findings

The subring R(X) is preserved under derived equivalences.

Spherical bundles on complex K3 surfaces over number fields are defined over algebraic closures.

Results support the Bloch-Beilinson conjecture in this context.

Abstract

We show that for a K3 surface X the finitely generated subring R(X) of the Chow ring introduced by Beauville and Voisin is preserved under derived equivalences. This is proved by analyzing Chern characters of spherical bundles. As for a K3 surface X defined over a number field all spherical bundles on the associated complex K3 surface are defined over $\bar\QQ$, 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 with Macri and Stellari in arXiv:0710.1645 are central for the discussion.