A note on the Bloch-Beilinson conjecture for K3 surfaces and spherical objects

TL;DR

This paper explores the relationship between spherical objects in the derived category of a K3 surface and the Bloch-Beilinson conjecture, establishing a connection between categorical structures and algebraic cycles.

Contribution

It introduces a dense triangulated subcategory generated by spherical objects and links the existence of a bounded t-structure to the Bloch-Beilinson conjecture for K3 surfaces.

Findings

S^* admits a bounded t-structure if and only if CH^2(X)=Z for K3 surfaces over ar Q.

The subcategory S^* is generated by spherical objects in D^b(Coh(X)).

The work connects derived category structures with deep conjectures in algebraic geometry.

Abstract

For a projective K3 surface X we introduce the dense triangulated subcategory S^* of the bounded derived category D^b(Coh(X)) of coherent sheaves on X that is generated by spherical objects. For a K3 surface X over \bar Q it is shown that S^* admits a bounded t-structure if and only if the Bloch-Beilinson conjecture holds for X, i.e. CH^2(X)=Z.