Tilt-stability, vanishing theorems and Bogomolov-Gieseker type inequalities

TL;DR

This paper studies tilt-stability of sheaves on projective varieties, establishing bounds for stability parameters, proving vanishing theorems, and deriving Bogomolov-Gieseker type inequalities for sheaves on projective 3-space.

Contribution

It provides effective bounds for tilt-stability parameters, leading to new vanishing theorems and Bogomolov-Gieseker inequalities for stable sheaves.

Findings

Effective bounds for tilt-stability parameters.

New vanishing theorems for stable sheaves.

Bogomolov-Gieseker type inequalities for sheaves on P^3.

Abstract

We investigate the tilt-stability of stable sheaves on projective varieties with respect to certain tilt-stability conditions depends on two parameters constructed by Bridgeland. For a stable sheaf, we give effective bounds of these parameters such that the stable sheaf is tilt-stable. These allow us to prove new vanishing theorems for stable sheaves and an effective Serre vanishing theorem for torsion free sheaves. Using these results, we also prove Bogomolov-Gieseker type inequalities for the third Chern character of a stable sheaf on $\mathbb{P}^3$.