Stability conditions on Fano threefolds of Picard number 1

Abstract

We prove the conjectural Bogomolov–Gieseker type inequality for tilt-stable objects on each Fano threefold $X$ of Picard number 1. In view of the previous works [1], [2] and [3] on Bridgeland stability conditions, this induces an open subset of geometric stability conditions on D${}^b(X)$. We also get a new stronger bound for Chern characters of slope semistable sheaves on $X$.