On the structure of a log smooth pair in the equality case of the Bogomolov-Gieseker inequality

TL;DR

This paper investigates the geometric structure of log smooth pairs under the condition of equality in the Bogomolov-Gieseker inequality, focusing on the semistability of the logarithmic tangent bundle and the canonical extension sheaf.

Contribution

It characterizes the structure of log smooth pairs when the logarithmic tangent bundle attains equality in the Bogomolov-Gieseker inequality, extending understanding of their geometric properties.

Findings

Logarithmic tangent bundle is semistable under the equality condition.

The structure of the pair is explicitly described in this equality case.

Analysis includes the behavior of the canonical extension sheaf.

Abstract

We study the structure of a log smooth pair when the equality holds in the Bogomolov-Gieseker inequality for the logarithmic tangent bundle and this bundle is semistable with respect to some ample divisor. We also study the case of the canonical extension sheaf.