Filtrations and torsion pairs in Abramovich Polishchuk's heart

TL;DR

This paper explores abelian subcategories and torsion pairs in Abramovich Polishchuk's heart, constructs stability conditions on a specific triangulated subcategory, and introduces a generalized notion of stability called $l$-th level stability.

Contribution

It introduces the concept of $l$-th level stability and demonstrates a unique filtration for objects in Abramovich Polishchuk's heart with semistable factors.

Findings

Constructed stability conditions on a subcategory of $D(X\times S)$.

Defined $l$-th level stability as a generalization of slope and Gieseker stability.

Established a unique filtration with decreasing phase vectors for objects in the heart.

Abstract

We study some abelian subcategories and torsion pairs in Abramovich Polishchuk's heart. And we construct stability conditions on a full triangulated subcategory $\mathcal{D}^{\leq 1}_S$ 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.