A generalized Bogomolov-Gieseker inequality for the three-dimensional projective space

TL;DR

This paper proves a generalized Bogomolov-Gieseker inequality for tilt-stable complexes on threefolds, confirming the conjecture for the three-dimensional projective space by reducing the problem to small polarizations.

Contribution

It establishes the inequality for the three-dimensional projective space, advancing the understanding of stability conditions on threefolds.

Findings

Inequality holds for sufficiently small polarizations.

Confirmed the conjecture for the three-dimensional projective space.

Provides a method to verify the inequality in general cases.

Abstract

A generalized Bogomolov-Gieseker inequality for tilt-stable complexes on a smooth projective threefold was conjectured by Bayer, Toda, and the author. We show that such inequality holds true in general, if it holds true when the polarization is sufficiently small. As an application, we prove it for the three-dimensional projective space.