Strong Bogomolov inequalities for sheaves on threefolds

We present an overview of recent works on the Bogomolov inequality for stable sheaves. The novelty being to use other notions of stability in the derived category to obtain stronger bounds for sheaves. We will discuss mostly the case of Fano 3folds (works by Chunyi Li, and joint work with Bernardara, Schmidt, and Zhao) and the recent paper by Chunyi Li on the quintic 3fold.