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Wall-Crossing and Algebraic Geometry

Periodic Reporting for period 1 - WallCrossAG (Wall-Crossing and Algebraic Geometry)

Reporting period: 2019-06-01 to 2020-11-30

The goal of this project is to answer fundamental questions about the geometry of algebraic varieties. Algebraic varieties can be given as the solution set of a system of polynomial equations. Despite fast-moving progress that has completely changed the field several times over the last half century, some of the oldest problems remain among the most challenging: how to detect that two varieties are different, or how to detect that they are different in an essential way (not birational).
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.
Led by Naoki Koseki, the construction of stability conditions on higher-dimensional varieties has made significant progress. The circle of ideas behind this work has also led to much stronger versions of the classical Bogomolov-Gieseker inequality in a wide range of cases, and to the first systematic construction of stability conditions in finite characteristic.
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.
In joint work in progress by the PI, Emanuele Macri (Paris) and Alex Perry (Michigan), a new method has been developed to understand rationality of special cubic fourfolds. This will undoubtedly lead to significant progress; how far this idea will go is still to be seen.
Fei Xie has made excellent progress understanding derived categories of singular Fano threefolds.
Project overview
An algebraic variety