Beauville-Voisin filtrations on zero cycles of moduli space of stable sheaves on K3 surfaces

TL;DR

This paper demonstrates that Voisin's filtration on zero cycles of moduli spaces of stable sheaves on K3 surfaces aligns with other proposed filtrations, confirming a conjecture and advancing understanding of hyper-K"ahler geometry.

Contribution

It proves the equivalence of Voisin's filtration with other filtrations on zero cycles in moduli spaces of stable sheaves on K3 surfaces, confirming a key conjecture.

Findings

Voisin's filtration is equivalent to other filtrations.

Confirmed a conjecture in Barros-Flapan-Marian-Silversmith.

Enhanced understanding of zero cycle filtrations on hyper-K"ahler manifolds.

Abstract

The Beauville-Voisin conjecture predicts the existence of a filtration on projective hyper-K\"ahler manifolds opposite to the conjecture Bloch-Beilinson filtration, called the Beauivlle-Voisin filtration. Voisin has introduced a filtration on zero cycles of an arbitrary projective hyper-K\"ahler manifold. On moduli space of stable objects of a projective K3 surface, there are other candidates constructed by Shen-Yin-Zhao, Barros-Flapan-Marian-Silversmith and more recently by Vial from different point of views. According to the work of Vial, all of them are proved to be equivalent except Voisin's filtration. In this paper, we show that Voisin's filtration is the same as the other filtrations. As an application, we prove a conjecture in Barros-Flapan-Marian-Silversmith's paper.