Filtrations on the derived category of twisted K3 surfaces

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 $\mathfrak{o}_{\mathscr{X}} \in \operatorname{CH}_0(X)$ 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 $\mathbf{S}_\bullet(\mathrm{D}^{(1)}(\mathscr{X}))$ preserved under Fourier-Mukai transforms, 2. A birational invariant filtration $\mathbf{S}^{\mathrm{SYZ}}_\bullet \operatorname{CH}_0$ on Bridgeland moduli spaces. We prove $\mathbf{S}^{\mathrm{SYZ}}_\bullet \operatorname{CH}_0$ coincides with Voisin's filtration $\mathbf{S}^{\mathrm{BV}}_\bullet\operatorname{CH}_0$ (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.