Stratifications for group actions and moduli problems
Many moduli spaces are constructed as quotients of group actions (either in algebraic or symplectic geometry). However, moduli spaces only classify "semistable" objects and to fully understand the moduli problem, we must also study unstable objects. We can stratify moduli problems using Harder--Narasimhan filtrations so that the open stratum parametrises semistable objects. These stratifications can be described in a more concrete way in terms of group actions. More precisely, we will study stratifications in geometric invariant theory due to Kempf and Hesselink and Morse theoretic stratifications in symplectic geometry associated to the norm square of a moment map due to Kirwan and Ness. For a reductive group acting on a projective variety, these stratifications coincide by work of Kirwan and Ness. In fact, this relies on the Kempf-Ness Theorem which says that the GIT quotient is homeomorphic to the symplectic reduction. We will describe these results and analogous statements for a reductive group acting on an affine variety with respect to a character. Then we will compare these stratifications associated to group actions with Harder--Narasimhan stratifications coming from moduli problems. We focus on two moduli problems: moduli of quiver representations and moduli of sheaves over a projective variety.
A 5 part course at the Seoul National University, 7-11th December 2015.
Notes (to be posted after each lecture)
Talk 1 (07/12/15): Geometric Invariant Theory and instability stratifications.
Talk 2 (08/12/15): Symplectic reduction and the norm square of the moment map.
Talk 3 (09/12/15): The Kempf-Ness Theorem and agreement of the stratifications [Kirwan and Ness].
Talk 4 (10/12/15): Stratifications for moduli of quiver representations.
Talk 5 (11/12/15): Stratifications for moduli of sheaves over a projective variety.
Talks 4 and 5 are based on my research:
Stratifications associated to reductive group actions on affine spaces, Q. J. Math. (2014) 65 (3) 1011--1047
(with F. Kirwan) Quotients of unstable subvarieties and moduli spaces of sheaves of fixed Harder-Narasimhan type, Proc. London Math. Soc (2012) 105 (3) 852--890
Stratifications for moduli of sheaves and moduli of quiver representations, arXiv: 1407.4057.