Benedict Morrissey

The P=W Seminar is a seminar focusing on mathematics relating to the P=W conjecture. In particular on the Hitchin system side it will go over Ngo's support theorem.

Topic Segments for Talk(s):
  • Ngo's Support Theorem (B.M.)
  • Non-abelian Hodge Theory (Marielle Ong)
  • Deligne Twistor Space (Jia-Choon Lee)
  • Topological Field Theory approach to cohomology of Character Varieties (B.M.)
  • Proof of the SL(2) P=W Conjecture (Sukjoo Lee)
  • Cluster structure following Gross--Hacking--Keel--Kontsevich (Sukjoo Lee)

Some references:

Cohomology of the Hitchin fibration & counting Hitchin points in Hitchin fibers:

Primary:

  • Ngo: Le lemme fondamental pour les algebres de Lie
  • Ngo: Fibration de Hitchin et endoscopie
  • Chaudouard -- Asterisque: Sur les comptage des fibres de Hitchin

Expository:

  • Chaudouard: Geometry of the fundamental lemma
  • Nadler: The Geometric Nature of the Fundamental Lemma
  • Ngo: Abelian Fibrations

Hitchin Fibers:

  • Hitchin: Stable bundles and integrable systems
  • Donagi: Decomposition Spectral covers
  • Donagi--Gaitsgory: Gerbe of Higgs bundles

Further:

  • Chaudouard--Laumon: Series of Articles
  • Cataldo: A support theorem for the Hitchin fibration: the case of SL(n)
  • Yun's survey article ICM: Hitchin type moduli stacks in automorphic representation theory
  • Baraglia: Monodrromy of the SL(n) and GL(n) Hitchin fibrations

Perverse Sheaves;

  • Cataldo notes PCMI (background)
  • Migliorini--Shende: Higher discriminants and the topology of algebraic maps

Spectral curves and knots, perverse filtration on Jacobians:

  • Maulik--Yun
  • Oblomkov--Shende
  • Maulik
  • Migliorini--Shende--Viviani

Counting Points: Schiffman: Indecomposable vector bundles and stable Higgs bundles over smooth projective curves

Link to orbital integrals:

  • Zhiwei Yun -- PCMI: Lectures on Springer Theories and Orbital Integrals

Cohomology Character varieties

  • Hausel--Rodriguez-Villegas: Mixed Hodge polynomials of character varieties
  • Explicit generators and relations SL(2) case:
  • Hausel--Thaddeus:Generators for the cohomology ring of the moduli space of rank 2 Higgs bundles
  • Hausel--Thaddeus:Relations in the cohomology ring of the moduli space of rank 2 Higgs bundles (see comments in Cataldo--Hausel--Migliorini about Hodge filtration on this)

TFT approach:

  • Ben-Zvi--Gunningham--Nadler: The Character Field Theory and Homology of Character Varieties

COHA approach:

  • Davison: Cohomological Hall Algebras and Character Varieties

Relation between each side

Non abelian Hodge theory:

  • Simpson
  • Potier -- Fibrés de Higgs et syst`emes locaux. (Seminaire Bourbaki)
  • Corlette, Donaldson
  • Hitchin -- Self Duality equations on a Riemann Surface

The P=W Correspondence

  • Migliorini's survey
  • Cataldo--Hausel--Migliorini: Topology of Hitchin systems and Hodge theory of character varieties: the case A_1
  • Cataldo--Maulik: The perverse filtration for the Hitchin fibration is locally constant

Further Topics:

  • Mirror symmetry.
  • Geometric P=W
  • Simpson's example of this; The dual boundary complex of the SL2 character variety of a punctured sphere
  • Cluster Structures -- see Gross--Hacking--Keel--Kontsevich
  • Dependence Hitchin fibration on complex surface: Hitchin "Flat connection", Witten et al, Faltings, Ben-Zvi--Frenkel (via Sugawara construction/operator).
  • Link to DT invariants of Calabi Yau fibration -- Diaconescu--Donagi--Pantev