Filtrations and torsion pairs in Abramovich Polishchuk's heart

We study some abelian subcategories and torsion pairs in Abramovich Polishchuk’s heart. And we construct stability conditions on a full triangulated subcategory ${\mathcal{D}}_S^{\le 1}$ in $D(X\times S)$, 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.