Filtrations and Torsion Pairs in Abramovich–Polishchuk’s Heart

Abstract

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}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {D}}_S^{\le 1}$$\end{document}

in

D(X\times S)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$D(X\times S)$$\end{document}

, 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.

Keywords

Bridgeland stability conditions / Product varieties / Filtrations / 14F08 / 14J40 / 18E99

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Funding

National Natural Science Foundation of China(12201011)

RIGHTS & PERMISSIONS

Peking University