Moduli spaces can be viewed 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).

The moduli problem of sheaves on X is naturally parametrized by a geometric object M called an "algebraic stack". In this talk I will explain an alternative GIT-free construction of the moduli space of semistable pure sheaves which is intrinsic to the moduli stack M. This approach also yields a Harder-Narasimhan stratification of the unstable locus of the stack. 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 some other moduli problems. This talk is based on joint work with Daniel Halpern-Leistner and Trevor Jones. Speaker(s): Andres Fernandez Herrero (Cornell University)