Stability conditions on Fano threefolds of Picard number one

TL;DR

This paper proves a conjectural inequality for stable objects on Fano threefolds with Picard number one, leading to new stability conditions and bounds for Chern characters, advancing the understanding of derived categories in algebraic geometry.

Contribution

It establishes the Bogomolov-Gieseker type inequality for tilt slope stable objects on these Fano threefolds, enabling the construction of geometric stability conditions.

Findings

Proved the conjectural inequality for tilt slope stable objects.

Constructed an open subset of geometric stability conditions on D^b(X).

Derived a new bound for Chern characters of slope semistable sheaves.

Abstract

We prove the conjectural Bogomolov-Gieseker type inequality for tilt slope stable objects on each Fano threefold X of Picard number one. Based on the previous works 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 the Chern characters of slope semistable sheaves on X.