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.
was designed to bring new ideas from string theory into the mainstream of algebraic geometry, and, more specifically, birational geometry. This was largely successful with the publication of two already highly cited papers, joint with Daniele Arcara and Max Lieblich in the Journal of the European Mathematical Society and joint with Arcara, Izzet Coskun and Jack Huizenga in Advances in Mathematics, both published in 2013. The study of vector bundles on Riemann surfaces goes back at least to the 1960s with important work by Mumford and the Indian school of algebraic geometers. The idea there was that in order to have a nice (i.e. projective) parameter (moduli) space for vector bundles, it was necessary to cut down the universe of vector bundles via the canonical notion of stability. When stability was introduced for higher dimensional varieties, it was no longer canonical, but Mumford and, later, Gieseker, used stabilities that depended upon the choice of an ample divisor class. With the introduction, in string theory, by Michael Douglas, of "pi" stability, it was realized that by moving into the derived category, there were far more natural notions of stability than previously thought. Tom Bridgeland mathematicized this idea, and Arcara and I were among the first to pick this up and apply it systematically to the study of sheaves on surfaces. Our work has been influential. Coskun and his collaborators have proved some powerful results on sheaves on the projective plane, Bayer and Macri have done some marvelous work on the positivity of these moduli spaces, and results on del Pezzo surfaces, Enriques surfaces and K3 surfaces have been coming out at a rapid rate. I would argue that this grant, awarded early and with great foresight by the NSF, helped me to get the ball rolling in this very exciting field.
