Svetlana Makarova | On moduli of objects in abelian categories
Channel: EIMI, PDMI RAS and Chebyshev Laboratory
Date: 2021-12-24

Seminar on A1-topology, motives and K-theory, December 23, 2021
Svetlana Makarova (University of Pennsylvania)
On moduli of objects in abelian categories

Historically, the first methodical construction of moduli spaces was given by GIT (geometric invariant theory). However, these constructions don’t keep track of automorphisms of objects – for that, we will need to introduce stacks. The language of stacks, in addition, should be viewed as an “intrinsic version of GIT”. In recent years, a theory has emerged that extends the GIT definitions and constructions to the world of stacks — BGIT (beyond GIT). Among notions that have been generalized are good moduli spaces, notions of stability and filtrations, and this is mainly due to Alper, Halpern-Leistner, Hall and Rydh.

The talk will be divided into two parts. In the first part, I will define the main objects of BGIT and illustrate them with a "toy example" of representations of quivers. I will present two new approaches to proving the well-known result that moduli of representations of acyclic quivers are projective. One of the approaches is in fact "GIT-free" and allows one to derive effective bounds on the positivity of a determinantal line bundle. This extends the story from the world of bundles on curves due to Esteves and Popa, and it has not been known for quivers. This part is based on current work in progress with Belmans, Damiolini, Franzen, Hoskins and Tajakka.

In the second part of the talk, I will talk about a more general attempt to construct moduli of objects in abelian categories. We will notice that we can impose certain categorical properties of the category of quiver representations on an abelian category and still expect a projectivity result.