Filtrations on the derived category of twisted K3 surfaces

Abstract

We introduce and study the Shen-Yin-Zhao filtration on derived categories of twisted K3 surfaces. A main contribution is the construction of a twisted Beauville-Voisin class ${\mathfrak{o}}_{\mathcal{X}}\in {\mathrm{CH}}_0(X)$ that extends fundamental results of O'Grady and Shen-Yin-Zhao \cite{OG13, SYZ20} to twisted settings. This class enables: 1. A derived equivalence-invariant filtration ${\mathbf{S}}_{\bullet }({\mathrm{D}}^{(1)}(\mathcal{X}))$ preserved under Fourier-Mukai transforms, 2. A birational invariant filtration ${\mathbf{S}}_{\bullet }^{\mathrm{SYZ}}{\mathrm{CH}}_0$ on Bridgeland moduli spaces. We prove ${\mathbf{S}}_{\bullet }^{\mathrm{SYZ}}{\mathrm{CH}}_0$ coincides with Voisin's filtration ${\mathbf{S}}_{\bullet }^{\mathrm{BV}}{\mathrm{CH}}_0$ (Theorem 1.4), providing a canonical candidate for the conjectural Beauville-Voisin filtration. Applications include Bloch's conjecture for (anti)-symplectic automorphisms and existence of algebraically coisotropic subvarieties.

24 pages