Wall-Crossing and Algebraic Geometry

We will establish stability conditions and wall-crossing in derived categories as a standard methodology for a wide range of fundamental problems in algebraic geometry. Previous work based on wall-crossing, in particular my joint work with Macri, has led to breakthroughs on the birational geometry of moduli spaces and related varieties. Recent advances have made clear that the power of stability conditions extends far beyond this setting, allowing us to study vanishing theorems or bounds on global sections, Brill-Noether problems, or moduli spaces of varieties. The Brill-Noether problem is one of the oldest and most fundamental questions of algebraic geometry, aiming to classify possible degrees and embedding dimensions of embeddings of a given variety into projective spaces. Recent work by myself, a post-doc (Chunyi Li) and a PhD student (Feyzbakhsh) of mine has established wall-crossing as a powerful new method for such questions. We will push this method further, all the way towards a proof of Green's conjecture, and the Green-Lazarsfeld conjecture, for all smooth curves. We will use similar methods to prove new Bogomolov-Gieseker type inequalities for Chern classes of stable sheaves and complexes on higher-dimensional varieties. In addition to constructing stability conditions on projective threefolds---the biggest open problem within the theory of stability conditions, we will apply them to study moduli spaces of sheaves on higher-dimensional varieties, and to characterise special abelian varieties. We will use the construction of stability conditions for families of varieties in my current joint work to systematically study the geometry of Fano threefolds and fourfolds, in particular their moduli spaces, by establishing relations between different moduli spaces, and describing their Torelli maps. Finally, we will study rationality questions, with a particular view towards a wall-crossing proof of the irrationality of the very general cubic fourfold.