Periodic Reporting for period 4 - WallCrossAG (Wall-Crossing and Algebraic Geometry)
Reporting period: 2023-12-01 to 2025-02-28
Algebraic geometry is the study of algebraic varieties - geometric shapes arising through polynomials. Algebraic varieties arise naturally whenever points in space (for example, data points in a high-dimensional space of measurements) are related by polynomials. They have found applications in statistics, in phylogenetics, in robotics, and many other fields. Many of the fundamental questions of algebraic geometry- for example, whether we can switch between implicit descriptions of geometric sets via equations, or explicit descriptions via a parametric form - have direct consequences in these applications.
The goal of this project is to answer fundamental questions about the geometry and classification of algebraic varieties. 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, even for concrete and explicit examples of algebraic varieties: how to detect that two varieties are different, or how to detect that they are different in an essential way, how to construct geometric relations between different algebraic varieties.
The methodology of this project is based on wall-crossing and Bridgeland stability conditions, originally developed with a view towards string theory and enumerative geometry. Developments prior to the project had turned them into a potentially widely applicable tool to enhance basic techniques and methods in algebraic geometry. The goal of this project was to bring these methods to their full potential, and apply them to a broad range of questions.
The project was designed to both enhance the underlying methodology, and to develop broader and more systematic applications to the most fundamental questions of algebraic geometry.
The goal of this project is to answer fundamental questions about the geometry and classification of algebraic varieties. 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, even for concrete and explicit examples of algebraic varieties: how to detect that two varieties are different, or how to detect that they are different in an essential way, how to construct geometric relations between different algebraic varieties.
The methodology of this project is based on wall-crossing and Bridgeland stability conditions, originally developed with a view towards string theory and enumerative geometry. Developments prior to the project had turned them into a potentially widely applicable tool to enhance basic techniques and methods in algebraic geometry. The goal of this project was to bring these methods to their full potential, and apply them to a broad range of questions.
The project was designed to both enhance the underlying methodology, and to develop broader and more systematic applications to the most fundamental questions of algebraic geometry.
An ubiquitous technique in algebraic geometry - since its revolution through Grothendieck's work in the 1960s - is to not just consider a single algebraic variety in isolation, but to consider families of algebraic varieties (for example obtained by deforming their defining polynomial equations). Joint work of the PI at the beginning of this project unified this perspective with stability conditions, via his publication "Stability conditions in families". This publication has become a standard reference, and the developed methodology has been as transformative as we originally hoped. It has seen applications in many different contexts, within this ERC project but also by other groups world-wide.
The construction of stability conditions on higher-dimensional varieties has made significant progress, in work by team member Naoki Koseki. This is a key tool that has already been applied by a group at Imperial College to deduce striking consequences in enumerative geometry. 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 situations, and to the first systematic construction of stability conditions in finite characteristic.
In work by various team members, much progress has been made in understanding the geometry of the most explicit type of higher-dimensional varieties, so-called Fano varieties. Work by the PI successfully turned the methods of stability conditions into an effective tool for these varieties, which was subsequently applied by many different team members, but also by groups in Italy, USA and China.
The fundamental questions is that of the classification of algebraic varieties of a certain fixed type: what does the parameters space of all such varieties look like? Which invariants can reliably detect that two such varieties are distinct?
When do two such varieties have many properties in common? On a technical level, results by various team members have greatly clarified the role of certain categories, called "Kuznetsov components", can play in the classification. Thanks to the achievements in the projects, these are now understood to be linked intrinsically to classical Hodge-theoretic classification data, and give much more powerful algebraic and geometric tools for the reconstruction of a given variety from its associated invariants.
Finally, joint work by the PI addressed the fundamental question of the explicit forms (their so-called "Mukai models") of three-dimensional Fano varieties of certain types. Mukai's seminal insight was to described them as lying inside not the standard compactification ("projective space") of linear spaces, but inside different compactifications called "Grassmannians. The equations defining Grassmanninans are well-understood, and Mukai showed that one can determine an additional redunancy-free system of equations defining the Fano threefold insider the Grassmannian.
While Mukai first introduced these models in 1990, the existing proofs The PI's joint work is the first rigorous and complete treatment of this construction. Like Mukai's Ansatz, it is based on the vector bundle method, and on the Brill-Noether theory of one- and two-dimensional varieties (K3 surfaces) inside the three-dimensional Fano variety.
The construction of stability conditions on higher-dimensional varieties has made significant progress, in work by team member Naoki Koseki. This is a key tool that has already been applied by a group at Imperial College to deduce striking consequences in enumerative geometry. 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 situations, and to the first systematic construction of stability conditions in finite characteristic.
In work by various team members, much progress has been made in understanding the geometry of the most explicit type of higher-dimensional varieties, so-called Fano varieties. Work by the PI successfully turned the methods of stability conditions into an effective tool for these varieties, which was subsequently applied by many different team members, but also by groups in Italy, USA and China.
The fundamental questions is that of the classification of algebraic varieties of a certain fixed type: what does the parameters space of all such varieties look like? Which invariants can reliably detect that two such varieties are distinct?
When do two such varieties have many properties in common? On a technical level, results by various team members have greatly clarified the role of certain categories, called "Kuznetsov components", can play in the classification. Thanks to the achievements in the projects, these are now understood to be linked intrinsically to classical Hodge-theoretic classification data, and give much more powerful algebraic and geometric tools for the reconstruction of a given variety from its associated invariants.
Finally, joint work by the PI addressed the fundamental question of the explicit forms (their so-called "Mukai models") of three-dimensional Fano varieties of certain types. Mukai's seminal insight was to described them as lying inside not the standard compactification ("projective space") of linear spaces, but inside different compactifications called "Grassmannians. The equations defining Grassmanninans are well-understood, and Mukai showed that one can determine an additional redunancy-free system of equations defining the Fano threefold insider the Grassmannian.
While Mukai first introduced these models in 1990, the existing proofs The PI's joint work is the first rigorous and complete treatment of this construction. Like Mukai's Ansatz, it is based on the vector bundle method, and on the Brill-Noether theory of one- and two-dimensional varieties (K3 surfaces) inside the three-dimensional Fano variety.
Work by the PI, and by team member Hannah Dell, now allows us to much more systematically take advantage of inherent symmetries. Sometimes this involves explicit symmetries of the geometric shapes involved, and sometimes it instead involves categorical shadows of such symmetries. Combining these symmetries with categorical methods and stability conditions led to a better understanding of the space of stability conditions for many surfaces, and to the classification of Kuznetsov components for Fano threefolds. In each case, this allowed us to answer a prominent open question in the field.
Current joint work in progress by the PI, that will be completed after the formal end of the project, gives an entirely new construction of hyperkaehler varieties of so-called Kummer type. This is based on an entirely novel construction of "Noncommutative abelian surfaces", which are in turn based on derived symmetries of certain K3 surfaces with rich algebraic structures.
Current joint work in progress by the PI, that will be completed after the formal end of the project, gives an entirely new construction of hyperkaehler varieties of so-called Kummer type. This is based on an entirely novel construction of "Noncommutative abelian surfaces", which are in turn based on derived symmetries of certain K3 surfaces with rich algebraic structures.