Postnikov-Stability versus Semistability of Sheaves

TL;DR

This paper introduces a new stability concept in triangulated categories, which encompasses semistable sheaves and aids in compactifying moduli spaces of stable bundles through Fourier-Mukai transforms.

Contribution

It proposes Postnikov-stability, a novel stability notion preserved under equivalences, and applies it to compactify moduli spaces of stable bundles with complexes.

Findings

Postnikov-stability generalizes semistability for sheaves.

The new stability notion is preserved by equivalences.

Application to compactify moduli spaces of stable bundles.

Abstract

We present a novel notion of stable objects in a triangulated category. This Postnikov-stability is preserved by equivalences. We show that for the derived category of a projective variety this notion includes the case of semistable sheaves. As one application we compactify a moduli space of stable bundles using genuine complexes via Fourier-Mukai transforms.