Periodic Reporting for period 1 - WallCrossAG (Wall-Crossing and Algebraic Geometry)
Reporting period: 2019-06-01 to 2020-11-30
The methodology of this project, however, is based on wall-crossing and stability conditions, originally developed with a view towards string theory and counting invariants. Recent developments have turned them into a potentially widely applicable tool to enhance basic techniques and methods in algebraic geometry. The goal of this project is to bring these methods to their full potential, and apply them to a broad range of questions.
The project will enhance the underlying methodology - e.g. by the construction of stability conditions in higher dimensions. But it will also apply them to yield tangible progress on algebraic geometry questions of quite classical flavour, for example in Brill-Noether theory or the geometry of higher-dimensional Fano varieties.
Shizhou Zhang and Augustinas Jacovskis are making fast progress on understanding many of the questions raised in the original proposal for Fano threefolds. Recently, Zhang has disproved a conjecture by Kuznetsov using the technique of moduli spaces and Bridgeland stability conditions.
Led by the PI, a recent preprint led to a new and very geometric proof of the classical Torelli theorem for cubic threefolds.
Fei Xie has made excellent progress understanding derived categories of singular Fano threefolds.