Andres Fernandez Herrero, February 25, 2022

Title: Intrinsic construction of moduli spaces via affine Grassmannians

Abstract: Moduli spaces arise as a geometric way of classifying objects of interest in algebraic geometry. For example, there exists a quasiprojective moduli space that parametrizes stable vector bundles on a smooth projective curve C. In order to further understand the geometry of this space, Mumford constructed a compactification by adding a boundary parametrizing semistable vector bundles. If the smooth curve C is replaced by a higher dimensional projective variety X, then one can compactify the moduli problem by allowing vector bundles to degenerate to an object known as a "torsion-free sheaf". Gieseker and Maruyama constructed moduli spaces of semistable torsion-free sheaves on such a variety X. More generally, Simpson proved the existence of moduli spaces of semistable pure sheaves supported on smaller subvarieties of X. All of these constructions use geometric invariant theory (GIT).

For a projective variety X, the moduli problem of coherent sheaves on X is naturally parametrized by an algebraic stack M, which is a geometric object that naturally encodes the notion of families of sheaves. In this talk I will explain a GIT-free construction of the moduli space of Gieseker semistable pure sheaves which is intrinsic to the moduli stack M. Our main technical tools are the theory of Theta-stability introduced by Halpern-Leistner, and some recent techniques developed by Alper, Halpern-Leistner and Heinloth. In order to apply these results, one needs to prove some monotonicity conditions for a polynomial numerical invariant on the stack. We show monotonicity by defining a higher dimensional analogue of the affine grassmannian for pure sheaves. If time allows, I will also explain some applications of these ideas to other moduli problems. This talk is based on joint work with Daniel Halpern-Leistner and Trevor Jones.