Stability Conditions on $\mathbb P^3$

TL;DR

This paper refines stability conditions on projective threefolds, specifically on $P^3$, establishing a subset satisfying a Bogomolov-Gieseker inequality and analyzing the global dimension of these stability conditions.

Contribution

It constructs a refined subset of stability conditions on $P^3$ and proves the global dimension is 3 for these conditions, proposing a conjecture on the contractibility of a key component.

Findings

Constructed a subset of stability conditions satisfying a Bogomolov-Gieseker inequality.

Proved the global dimension is 3 for these stability conditions on $P^3$.

Formulated a conjecture on the contractibility of a principal component.

Abstract

We construct a subset of the space of stability conditions for any projective threefold with an ample polarization that satisfies a certain Bogomolov-Gieseker inequality to refine the result in arXiv:1410.1585. Then, we demonstrate that the global dimension, as defined in arXiv:2008.00282 and arXiv:1807.00469, is 3 for any stability condition on $\mathbb P^3$ constructed in arXiv:1410.1585. Finally, we formulate a conjecture concerning the contractibility of a principal connected component of $\mathrm{Stab}(\mathbb P^3)$.