Intercity Geometry Seminar 2020    
Perverse filtrations, Hilbert schemes,    
and the P=W conjecture for parabolic Higgs bundles    
after Junliang Shen and Zili Zhang    

What is the Intercity Geometry Seminar?

  The Intercity Geometry Seminar is a series of 4 sessions of 2-4 lectures held each year in the Netherlands on a recent paper of algebraic geometry. This year's edition will focus on Perverse filtrations, Hilbert schemes, and the P=W conjecture for parabolic Higgs bundles (Junliang Shen and Zili Zhang, 2018).
  The first lectures will introduce the general concepts that are needed to read the paper, while the last ones will focus on the text itself and further directions of research. The seminar is intended for PhD students, Post-docs, and senior researchers with a basic knowledge of algebraic geometry.


Abstract

  Simpson established the non-abelian Hodge theorem for a curve of genus at least 2. It gives a canonical diffeomorphism between the moduli space nof rank n stable Higgs bundle M_Dol and the character variety of rank n stable local systems M_B on this curve. Both moduli spaces are complex manifolds, however the canonical diffeomorphism does not preserve the complex structures. In particular the cohomology rings of these moduli spaces are canonicaly isomorphic but not their Hodge structures.
  A striking phenomenon was discovered by de Cataldo, Hausel, and Migliorini. The Simpson correspondence is expected to identify two filtrations: the weight filtration (of M_B) and the perverse filtration associated with the Hitchin fibration (of M_Dol). Such a phenomenon is refered to as the P = W conjecture. Several cases of the conjecture are now proved but the general statement is still open.
  The studied paper provides a proof of the P = W conjecture for the moduli spaces of parabolic Higgs bundles associated with the affine Dykin diagrams A₀, D₄, E₆, E₇, and E₈. Their proof relies on an explicit description of the perverse filtration as M_Dol is isomorphic to a Hilbert Scheme of points in this context.

  The introductory lectures will cover the following topics:

• Moduli spaces of Higgs bunlde, the Hitchin correspondence and the Hitchin Fibration.
• Hilbert scheme of points and their tautological rings.
• Perverse sheaves, perverse filtrations and Mixed Hodge Structures.