The central theme of the project is to apply derived category techniques to solve questions arising from Birational Geometry and String Theory. In particular, the P.I. has three main goals: (1) To study moduli spaces of stable sheaves on surfaces, by using wall-crossing with respect to Bridgeland stability conditions. (2) To prove a conjectural bound on Chern classes of certain stable complexes on threefolds, which generalizes a classical result by Bogomolov and Gieseker. (3) To study sheaves on projective spaces, cubic hypersurfaces, and the Grothendieck-Knudsen moduli space of stable n-pointed rational curves. Far-reaching applications would include Le Potier's Strange Duality Conjecture, the existence of Bridgeland stability conditions on Calabi-Yau threefolds, the Fujita Conjecture on adjoint linear series for threefolds, and new results in the theory of counting invariants.
The broader context of this project is the area of Algebraic Geometry. The central objects of interest in Algebraic Geometry are algebraic varieties, namely the loci of solutions of polynomial equations in many variables. Roughly, the idea is to study algebraic varieties "indirectly," by using certain "homological" invariants associated to geometric objects on them -- for example, differential forms. The technique of the derived category was developed in Verdier's thesis, under the guidance of Grothendieck, in 1967. The original motivation was the need to find a proper foundation for Grothendieck's duality theory, which provides non-trivial relations among the above mentioned invariants. More recently, the theory of derived categories has found important and deep applications beyond the original motivation and even outside Algebraic Geometry; in particular, to Representation Theory, Symplectic Geometry, and High Energy Physics.The present proposal builds indeed on the interaction with these disciplines -- notably on the work of Kontsevich, Bondal, Orlov, Bridgeland, Kuznetsov among others -- to deduce new results in Algebraic Geometry and other areas.
