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.
