Bridgeland Stability conditions on threefolds I: Bogomolov-Gieseker type inequalities

TL;DR

This paper develops new stability conditions for derived categories of threefolds, linking them to Bogomolov-Gieseker type inequalities, and proves these inequalities in specific cases.

Contribution

It introduces new t-structures on threefolds' derived categories and establishes their connection to Bogomolov-Gieseker type inequalities for stability conditions.

Findings

Conjecture of Bridgeland stability conditions near large volume limit.

Equivalence of the conjecture to a Bogomolov-Gieseker type inequality.

Proof of a classical Bogomolov-Gieseker inequality for stable complexes.

Abstract

We construct new t-structures on the derived category of coherent sheaves on smooth projective threefolds. We conjecture that they give Bridgeland stability conditions near the large volume limit. We show that this conjecture is equivalent to a Bogomolov-Gieseker type inequality for the third Chern character of certain stable complexes. We also conjecture a stronger inequality, and prove it in the case of projective space, and for various examples. Finally, we prove a version of the classical Bogomolov-Gieseker inequality, not involving the third Chern character, for stable complexes.