On stability conditions for the quintic threefold

TL;DR

This paper establishes new stability conditions for sheaves on smooth quintic threefolds, leveraging Clifford inequalities and Bogomolov-Gieseker type inequalities to advance understanding of derived categories in algebraic geometry.

Contribution

It introduces stronger Bogomolov-Gieseker inequalities for stable sheaves and tilt-stable objects on quintic threefolds, and constructs an open subset of stability conditions.

Findings

Proved stronger Bogomolov-Gieseker inequalities.

Constructed an open subset of stability conditions.

Extended the framework of Bayer, Bertram, Macr extquotesingle i, Stellari, and Toda.

Abstract

We study the Clifford type inequality for a particular type of curves $C_{2,2,5}$, which are contained in smooth quintic threefolds. This allows us to prove some stronger Bogomolov-Gieseker type inequalities for Chern characters of stable sheaves and tilt-stable objects on smooth quintic threefolds. Employing the previous framework by Bayer, Bertram, Macr\`i, Stellari and Toda, we construct an open subset of stability conditions on every smooth quintic threefold in $\mathbf{P}^4_{\mathbb C}$.