Geometry of moduli spaces of sheaves via wall-crossing

Abstract

We plan to study geometric properties of moduli spaces of sheaves on surfaces and threefolds such as their birational geometry, by studying the wall-crossing behavior of certain families of Bridgeland stability conditions. This techniques have been applied successfully on several families of surfaces and very few threefolds, mainly due to the lack of examples of Bridgeland stability conditions in dimensions higher than two. More precisely, the constructions of Bridgeland stability conditions on threefolds rely on the existence of a conjectural inequality on the Chern classes of a stable object (playing the role of the Bogomolov-Gieseker inequality for the case of surfaces). This inequality does not have a closed form and varies from threefold to threefold. One of the goals of this project is to provide new examples of this type of inequalities on Calabi-Yau threefolds of Picard rank greater than one, where very few examples are known. (AU)