Stronger Bogomolov--Gieseker type inequality on quintic threefold

TL;DR

This paper proves a stronger Bogomolov--Gieseker type inequality for slope-semistable sheaves on quintic threefolds, improving classical bounds and supporting the existence of Bridgeland stability conditions of Gepner type.

Contribution

It introduces a refined inequality combining restriction theorems and Clifford bounds, extending to surfaces and advancing stability condition research.

Findings

Established a piecewise linear inequality on Chern characters

Improved classical Bogomolov--Gieseker bounds for sheaves

Provided evidence for Bridgeland stability conditions of Gepner type

Abstract

We establish a stronger Bogomolov--Gieseker type inequality for slope-semistable sheaves on the smooth quintic threefold. Our approach combines a refined restriction theorem for tilt-stable objects with explicit Clifford-type bounds for semistable bundles on plane quintic curves. As a consequence, we obtain an explicit piecewise linear inequality on the Chern characters of any slope-semistable sheaf improving upon the classical Bogomolov--Gieseker bound and implying Toda's conjectural inequality. The method also yields a stronger Bogomolov--Gieseker type inequality on smooth quintic surfaces. These results provide new evidence toward the existence of a Bridgeland stability condition of Gepner type on the quintic threefold.