Bridgeland stability conditions on surfaces

Treballs finals del Màster en Matemàtica Avançada, Facultat de Matemàtiques, Universitat de Barcelona: Curs: 2022-2023. Director: Martí Lahoz Vilalta[en] The main goal of this Master Thesis is to study the stability manifold on complex projective surfaces. In this case, the stability manifold is the complex manifold that parametrizes Bridgeland stability conditions on the derived category of coherent sheaves on the surface. First of all, we present the classical study of stable sheaves on curves as the elementary model of Bridgeland stability to provide some intuition. Then we explain some basic definitions and results on triangulated categories and we construct the derived category of an abelian category. Next, we introduce the concept of Bridgeland stability conditions and prove their existence on surfaces. The key result to prove the existence is the Bogomolov-Gieseker inequality. The last part of this memoir explains the method that Feyzbakhsh, Li and Liu have developed to improve the Bogomolov-Gieseker inequality to enlarge the known region of the stability manifold for some specific surfaces. We have explored the possibility to apply this method to other surfaces