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 second part of the research project, I focussed on the following additional parts of my ERC project:
3. Investigate the uniqueness of dg enhancements for triangulated categories of geometric nature.
4. Deformation of stability conditions.
5. The study of the homotopy categories of dg and A_\infty categories.
6.The interplay between dg enhancements and the theory of weakly approximable triangulated categories.
As for 3, we essentially extended and improved all the existing results and answered most of the open question on the subject. Regarding 4, we are about to prove a very prowerful result which will allow us to produce examples of stability conditions for very general manifolds with trivial canonical bundle. Concerning 4, we recently posted a preprint where we rpvide a complete comparison between all possible localizations of dg and A_\infty categories and as for 6 we also posted during the last few months a preprint which shows the power of such an interplay and provides a vast generalization of a result by Rickard about Morita theory for schemes.
