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Stability Conditions, Moduli Spaces and Enhancements

Periodic Reporting for period 3 - StabCondEn (Stability Conditions, Moduli Spaces and Enhancements)

Reporting period: 2021-02-01 to 2022-07-31

The project aim at addressing two big open questions in the theory of derived/triangulated categories and their many applications in algebraic geometry.

The first one concerns the theory of {\bf Bridgeland stability conditions}, which provides a notion of stability for complexes in the derived category. The problem of showing that the space parametrizing stability conditions is non-empty is one of the most difficult and challenging ones. Once we know that such stability conditions exist, it remains to prove that the corresponding moduli spaces of stable objects have an interesting geometry (e.g.\ they are projective varieties). This is a deep and intricate problem.

On the more foundational side, the most successful approach to avoid the many problematic aspects of the theory of triangulated categories consisted in considering {\bf higher categorical enhancements} of triangulated categories. On the one side, a big open question concerns the uniqueness and canonicity of these enhancements. On the other side, this approach does not give a solution to the problem of describing all exact functors, leaving this as a completely open question. We need a completely new and comprehensive approach to these fundamental questions.

During the first half of the project, I mainly worked through the following two specific parts of the project:

1. Develop a theory of stability conditions for semiorthogonal decompositions and its applications to moduli problems. The main applications concern cubic fourfolds, Calabi--Yau threefolds and Calabi--Yau categories.

2. Apply these new results to the study of moduli spaces of rational normal curves on cubic fourfolds and their deep relations to hyperk\"ahler geometry.

The theory of stability conditions developed to work on (1) had been very influencial so far and had been applied to several related questions.

In the rest of the research project, I will mainly investigate the following two additional parts of my ERC project:

- Investigate the uniqueness of dg enhancements for the category of perfect complexes and, most prominently, of admissible subcategories of derived categories.

- Develop a new theory for an effective description of exact functors in order to prove some related conjectures.
During the first two years and a half of my ERC project I mainly worked on the construction of stability conditions in families. This new theory has been successfully applied to derived categories of cubic fourfolds in combination with the theory of stability conditions for semiorthogonal decompositions that I developed previously together with my coauthors (Bayer, Lahoz and Macrì).

The same circle of ideas had important effects on the following related questions:

- constructions of new 20-dimensional locally complete families of hyperkaehler manifolds of arbitrary large dimension and degree;

- construction of stability conditions on Gushel-Mukai fourolfds;

- study of the geometry of moduli spaces of rational normal curves of degree 3 and 5 inside cubic fourfolds;

- study uniqueness of enhancements in most cases of interest in algebraic geometry and representation theory;

We got major results in all these areas.
The main achievements so far are based on the following groundbreaking results:

- A relative version of Bridgeland's notion of stability conditions (together with a deformation theorem an results about the existence and the geometry of relative moduli spaces of stable objects);

- A criterion for extending equivalences between semiorthogonal components of derived categories and its interplay with the theory of spherical objects (with applications to Enriques surfaces and a conjecture by Kuznetsov about Artin-Mumford quartic double solids).

By the end of the ERC project we plan to work on the following additional problems:

- Extending results about uniqueness of dg enhancements in geometric contexts;

- New description and characterization of exact functors between derived categories.