Tilt-stability on singular schemes and Bogomolov-Gieseker-type inequalities

TL;DR

This paper extends tilt-stability and Bogomolov-Gieseker inequalities to singular schemes, verifying conjectures for certain threefolds and constructing stability conditions on relative Kuznetsov components.

Contribution

It generalizes tilt-stability and Bogomolov-Gieseker inequalities to singular schemes and verifies related conjectures for specific classes of threefolds.

Findings

Verified the conjecture for all Fano threefolds with certain singularities.

Established Bogomolov-Gieseker-type inequalities for semistable sheaves on any projective scheme.

Constructed stability conditions on relative Kuznetsov components for singular Fano threefolds.

Abstract

We generalize the framework of tilt-stability to singular schemes and formulate the generalized Bogomolov-Gieseker inequality conjecture of Bayer-Macr\`i-Toda for singular threefolds. We also develop relative versions of these constructions, generalizing corresponding results in [BLM+21]. Along the way, we establish Bogomolov-Gieseker-type inequalities for semistable sheaves on any projective scheme. By extending previous techniques, we verify the conjecture for all Fano threefolds with canonical Gorenstein $\mathbb{Q}$-factorial singularities and a series of singular Calabi-Yau threefolds. Furthermore, we construct stability conditions on the relative Kuznetsov components associated with families of singular Fano threefolds, thereby proving a singular analogue of a conjecture of Kuznetsov-Shinder.