Algebra and Geometry Lecture Series (2022)
I will give a lecture series on cohomology of Hitchin moduli spaces and the P=W conjecture at Yale's Algebra and Geometry Lecture Series.
Location: LOM 214, starting at 4:00 pm.
Title: Cohomology of Hitchin moduli spaces and the P=W conjecture.
Lecture notes:
Lecture 1: (Sep 15) Introduce Dolbeault and Betti moduli spaces, the P=W conjecture, and review the perverse filtration with examples.
Lecture 2: (Sep 22) Describe the cohomology of moduli of Higgs bundles in terms of tautological classes, and reduce P=W to an algebro-geometric statement concerning Chern classes and the perverse filtration.
Lecture 3: (Sep 29) Sheaf-theoretic enhancement for "Perverse = Chern".
Lecture 4: (Oct 6) Support theorem, and applications to Hitchin systems.
References for support theorem: over the elliptic locus [11, Chap 7] (see also [10, Chap 1]); over the total Hitchin base for the stable part [8]; for the parabolic Hitchin map which is really the one needed for P=W [2, Sect. 4].
Lecture 5: (Oct 13) Wrap up everything: support theorem, vanishing cycles, and global Springer theory.
Description:
Hitchin moduli spaces parameterize solutions to certain self-dual equations, underly orbital integrals, and serve as a key player in the study of automorphic forms and the geometric Langlands correspondence. This lecture series will concern the cohomological structure for Hitchin moduli spaces with an emphasize on the P=W conjecture.
In 2010, de Cataldo, Hausel, and Migliorini propose a deep conjecture, which predicts that topology of the Hitchin system is closely connected to Hodge theory of its corresponding character variety under the non-abelian Hodge correspondence; more precisely, they conjectured that, the perverse filtration of the Hitchin system is matched with the weight filtration of the character variety, which is now referred to as the P(erverse) = W(eight) conjecture.
The purpose of this lecture series is to discuss a very recent proof of the P=W conjecture for GL_n for any rank and genus in joint work with Davesh Maulik. We will focus on introducing tools and ingredients in algebraic geometry and representation theory that are helpful for us to understand better the cohomology of Hitchin moduli spaces, which lead to a proof of the conjecture.
Tentative schedule:
Sep 15, 22, 29;
Oct 6, 13.
Prerequisite:
The lecture series will cover a variety of topics; not all detailed proofs may be given.
The audience needs to have basic familiarity with varieties, vector bundles, Chern classes, and moduli spaces. Being familiar with perverse sheaves and the decomposition theorem (e.g. the topics course Math 709 I taught in Spring 2022) will be helpful but not strictly required.
There was also a learning seminar on Hitchin moduli spaces some time ago.
References:
Main references:
M. de Cataldo, T. Hausel, and L. Migliorini, Topology of Hitchin systems and Hodge theory of character varieties: the case A1. Ann of Math. (2) 175 (2012), no. 3. 1329–1407.
D. Maulik and J. Shen, The P=W conjecture for GL_n. preprint on arXiv 2209.02568.
Lecture notes and surveys:
Lecture notes from Geordie Williamson's webpage.
Mark de Cataldo's survey.
Luca Migliorini's survey.
Tamas Hausel's survey. (It contains many other stuffs as well.)
Some relevant research papers:
M. de Cataldo, D. Maulik, and J. Shen, Hitchin fibrations, abelian surfaces, and the P=W conjecture, J. Amer. Math. Soc. 35 (3), 2022, 911-953.
P-H Chaudouard and G. Laumon, Un théorème du support pour la fibration de Hitchin. Ann. Inst. Fourier (Grenoble) 66 (2016), no.2, 711-724.
D. Maulik and J. Shen, Endoscopic decomposition and the Hausel-Thaddeus conjecture, Forum Math. Pi 9 (2021), No. e8, 49pp.
D. Maulik and J. Shen, Cohomological \chi-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles, ArXiv 2012.06627. Geom. Topol. to appear.
B.C. Ngô, Le lemme fondamental pour les algèbres de Lie, Publ. Math. IHES. 111 (2010) 1–169.
Some of my recorded lectures on relevant topics.