M-Seminar: Cohomological DT theory and nonabelian Hodge theory for stacks III
Speaker: Ben Davison (University of Edinburgh)
Abstract: This is the third of a series of 3 lectures.
The nonabelian Hodge correspondence provides a diffeomorphism between certain coarse moduli spaces of semistable Higgs bundles on a smooth projective curve C (the Dolbeault side) and coarse moduli spaces of representations of the fundamental group of C (the Betti side). In the case of coprime rank and degree, these spaces are smooth, and the famous P=W conjecture states that the isomorphism in cohomology provided by the above diffeomorphism takes the weight filtration on the Betti side to the perverse filtration on the Dolbeault side. The purpose of these talks is to use recent advances in cohomological Donaldson-Thomas theory to extend this story to moduli stacks.
For coprime rank and degree, two key features in the study of classical nonabelian Hodge theory are the perverse filtration with respect to the Hitchin base, and the purity of the cohomology of the Dolbeault moduli space. I will present an extension of the BBDG decomposition theorem to moduli stacks of objects in 2CY categories, which enables us to reproduce both of the above features for stacks in nonabelian Hodge theory.
These results, along with cohomological Hall algebras, allow us to connect the intersection cohomology of coarse moduli spaces with the Borel-Moore homology of the above stacks, providing the connection between three versions of the P=W conjecture: the original conjecture for smooth moduli spaces, the version for intersection cohomology of singular moduli spaces, and a new version for stacks.
Meeting ID: 958 3425 4862
Thursday, October 28 at 1:30pm