Filtrations and Torsion Pairs in Abramovich–Polishchuk’s Heart

We study abelian subcategories and torsion pairs in Abramovich–Polishchuk’s heart. And we apply the construction from Liu (J Reine Angew Math 770:135–157, 2021) on a full triangulated subcategory \({\mathcal {D}}_S^{\le 1}\) ${\mathcal{D}}_S^{\le 1}$ in \(D(X\times S)\) $D(X\times S)$ , for an arbitrary smooth projective variety S. We also define a notion of l-th level stability, which is a generalization of the slope stability and the Gieseker stability. We show that for any object E in Abramovich–Polishchuk’s heart, there is a unique filtration whose factors are l-th level semistable, and the phase vectors are decreasing in a lexicographic order.