Stability conditions on fibred threefolds

Hao Max Sun

Published 2022-01-31Version 1

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 inequalities in the case of relative projective planes over curves. This gives the the existence of Bridgeland stability conditions on such threefolds.