Stability conditions on fibred threefolds

TL;DR

This paper proposes a conjectural method to construct Bridgeland stability conditions on fibred threefolds, relying on a Bogomolov-Gieseker type inequality, and proves this inequality for certain cases, establishing stability conditions there.

Contribution

It introduces a conjectural construction of stability conditions on fibred threefolds and proves the key inequality for relative projective planes over curves.

Findings

Proved the Bogomolov-Gieseker type inequality for relative projective planes over curves.

Established the existence of Bridgeland stability conditions on these threefolds.

Provided a framework for a relative version of stability condition construction.

Abstract

We give a conjectural construction of Bridgeland stability conditions on the derived category of fibred threefolds. The construction depends on a conjectural Bogomolov-Gieseker type inequality for certain stable complexes. It can be considered as a relative version of the construction of Bayer, Macr\`i and Toda. We prove the conjectural Bogomolov-Gieseker type inequality in the case of relative projective planes over curves. This gives the the existence of Bridgeland stability conditions on such threefolds.