The notion of stability conditions on derived categories emerged from high energy physics literature in the study of Dirichlet branes in string theory. Besides its original motivation, the theory of stability conditions has a much further reach, with connections to counting invariants, representation theory, homological mirror symmetry, and classical algebraic geometry. This project lies at the nexus of these areas; the theme is to apply Bridgeland stability condition techniques to solve questions arising from birational geometry and mathematical physics. A stability condition identifies a class of objects in the derived category, called stable objects, in an intrinsic way. Understanding if certain objects are stable or not gives important information on the geometry of algebraic varieties. A stability condition depends on the choice of certain parameters. Understanding how stable objects change by varying these parameters is the key to many applications of the theory and is the technical core of this project.
There are three main goals in the project. The first is to investigate the existence problem for Bridgeland stability conditions. A general framework for constructing Bridgeland stability conditions on threefolds was introduced by the investigator and collaborators, via certain conjectural inequalities. The new viewpoint is to look at fibrations in particular products, and to deduce such inequalities from the positivity of a certain divisor class on the base of the fibration. The second goal is to study stability conditions on non-commutative K3 surfaces and to use Bridgeland's theory to reinterpret and generalize geometric constructions, for example for cubic fourfolds, by using moduli spaces of Bridgeland stable objects. The third goal is to investigate whether such moduli spaces are "well-behaved" projective varieties in general and to study their local structure in the case of K3 surfaces. Applications include investigation of a conjecture on derived equivalences of birational irreducible holomorphic symplectic varieties.
