Bridgeland Moduli of Derived Objects on Algebraic Surfaces

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Coherent sheaves are the bread and butter of algebraic geometry. They are the natural extension of vector bundles to a category that is closed under kernels and cokernels. Their naturality and usefulness was first explored in Serre's landmark paper (FAC). Traditionally, the coherent sheaves on a smooth projective variety are broken down in terms of dimension of support and ``stability'' (Geometric Invariant Theory). However, recent work in string theory points to an entire manifold of stability conditions on categories of complexes of vector bundles (D-branes in the physics literature). These resemble perverse sheaves, and like perverse sheaves seem to have extremely nice properties. In joint work with Daniele Arcara, the PI put stability conditions on a rigorous mathematical footing for all complex surfaces, and in the current proposal he will explore the applications of this new theory to ``classical'' problems in algebraic geometry.Algebraic geometry is the study of the shapes of solution sets of systems of polynomial equations in many variables. One crucial tool in this study is the construction of invariants, i.e. auxiliary structures that allow one to distinguish among the different shapes. Rather surprisingly, string theorists have made very significant contributions to algebraic geometry in recent years. In work relevant to this project, they have proposed the existence of a ``stability manifold'' for ``D-branes,'' which seems to be a very powerful new tool for both distinguishing different shapes and for answering classical questions in algebraic geometry (e.g. How many variables does one need in order to embed a particular shape?) The PI will develop this new tool, building on his previous work explaining the two-dimensional case.