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 $\mathfrak{o}{\mathscr{X}} \in \operatorname{CH}_0(X)$ 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.
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