Bogomolov-Gieseker type inequalities on ruled threefolds

TL;DR

This paper proves a strengthened Bogomolov-Gieseker type inequality for ruled threefolds, which supports the existence of explicit Bridgeland stability conditions and advances understanding of stability in algebraic geometry.

Contribution

The paper extends a conjecture on tilt-stable complexes to arbitrary tilt-slope and proves it for ruled threefolds, improving previous results.

Findings

Proves the conjecture for ruled threefolds.

Shows the conjecture implies the support property of Bridgeland stability.

Establishes the existence of explicit stability conditions.

Abstract

We strengthen a conjecture by the author. This conjecture is a Bogomolov-Gieseker type inequality involving the third Chern character of mixed tilt-stable complexes on fibred threefolds. We extend it from complexes of mixed tilt-slope zero to arbitrary relative tilt-slope. We show that this stronger conjecture implies the support property of Bridgeland stability conditions, and the existence of explicit stability conditions. We prove our conjecture for ruled threefolds, hence improving a previous result by the author.