The aim of this project is to investigate some instances of the theory of derived categories of coherent sheaves. In particular, the P.I. has three main objectives: (1) To study the geometry of moduli spaces of complexes for the projective plane, K3 surfaces and cubic 4-folds. (2) To realize ``classical'' questions and conjectures in enumerative geometry as wall-crossing phenomena in derived category, the starting example being the total space of the canonical bundle over the projective plane. (3) To give a complete description of equivalences between the derived categories of K3 surfaces and of their deformations.
The broader context of this project is the area of algebraic geometry. Starting as a pure mathematical subject, in the recent years algebraic geometry has seen many interactions with other areas of science. The present project is motivated and inspired precisely by connections, envisioned by the Fields Medal winner M. Kontsevich almost twenty years ago, with theoretical physics and string theory. In particular, by bringing intuition and techniques from physics to tackle classical problems in pure mathematics and, vice versa, to provide mathematical rigorous foundations to physics constructions.
The aim of the project was to investigate some instances of the theory of derived categories of coherent sheaves. Although the notion of derived category goes back to the sixties, in the work of Verdier and Grothendieck, the idea of using the derived category to study the birational geometry of algebraic varieties is relatively new. It arose from Theoretical Physics, and it has been developped first in the work of Kontsevich, Bondal, Orlov, Bridgeland, Katzarkov and Kuznetsov, among others. The main idea in the project was to use a new notion of stability for complexes (called Bridgeland stability; this also arose in Theoretical Physics, in the work of Douglas and Aspinwall, among others) to study questions related to moduli of sheaves on certain classes of varieties; the main example was K3 surfaces. More precisely, the final goal was to prove a conjecture by Bridgeland which aimed at understanding the birational geometry of a moduli space of stable sheaves on a K3 surface in terms of wall-crossing in the derived category. In two joint papers with Arend Bayer, we proved this conjecture. As a consequence of our result, several conjectures in the field has been solved for the two main families of Irreducible Holomorphic Symplectic varieties, including two well-known conjectures on the structure of their Nef cones (the first was a conjecture by Hassett and Tschinkel and the second by Kawamata and Morrison), and a third conjecture on the existence of Lagrangian fibrations (this conjecture was originally proposed by Bogomolov and Tyurin, and appeared in print in the works of Huybrechts, Sawon, Hassett and Tschinkel). A second direction of work in the project concerned existence and basic properties for Bridgeland stability on three-dimensional projective varieties. The main result achieved by the PI is two joint papers with Arend Bayer, Aaron Bertram, and Yukinobu Toda in which we give a conjectural framework for Bridgeland stability conditions to exist and to be well-behaved. The main conjecture we propose is a generalization of the classical Bogomolov inequality for stable sheaves. It turned out that this conjecture implies well-known open problems in Algebraic Geometry, including the Fujita Conjecture and generalizations of the Castelnuovo inequality, relating the genus and the degree of a curve on a threefold. The PI proved this conjecture in the starting case of the three dimensional projective space. Currently the conjecture is known also for the quadric threefold (this was the work of the PI's PhD student, Benjamin Schmidt), and very recently on abelian threefolds. A third project aimed at understanding triangulated categories arising as non-trivial semi-orthogonal components in the derived category of a cubic hypersurface. Aiming at this goal, the PI, jointly with Marti' Lahoz and Paolo Stellari, studied ACM stable vector bundles on cubic threefolds and fourfolds. Broader impacts of the project included training of PhD students and young postdocs, delivering series of lectures at schools and conferences, organizing seminars and conferences. In particular, two of the PI's PhD students worked on questions related to the project. One student, Benjamin Schmidt already obtained new results, which have been accepted for publication.
