Andres Fernandez Herrero (Cornell)

Tuesday, May 3, 2022 - 1:30pm

PDL C-38

**Preseminar 1:30-2:15**

**Title**:

*An introduction to the moduli of sheaves*

**Abstract:**

*We will introduce the moduli problem of torsion-free sheaves on a projective variety, and explain its construction using Geometric Invariant Theory.*

**Seminar 2:30-3:20**

**Title**:

*Intrinsic construction of moduli spaces via affine Grassmannians*

**Abstract**:

*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.*