On boundedness of semistable sheaves

TL;DR

This paper presents a new simple proof of the boundedness of semistable sheaves with fixed invariants on smooth projective varieties, also providing a quick proof of Bogomolov's inequality in characteristic zero.

Contribution

It introduces a novel, simplified proof technique for boundedness and Bogomolov's inequality that avoids complex restriction theorems.

Findings

Established boundedness of semistable sheaves with fixed invariants.

Provided a quick proof of Bogomolov's inequality in characteristic zero.

Simplified the proof process for key stability results.

Abstract

We give a new simple proof of boundedness of the family of semistable sheaves with fixed numerical invariants on a fixed smooth projective variety. In characteristic zero our method gives a quick proof of Bogomolov's inequality for semistable sheaves on a smooth projective variety of any dimension $\ge 2$ without using any restriction theorems.