Non-abelian Hodge correspondence and P=W conjecture

Columbia, Fall 2022

The goal of this seminar is to start from the beginnings of non-abelian Hodge theory and ultimately understand Maulik and Shen's proof of P=W conjecture. On the way there we will study important papers on the structure and cohomology of Betti, de Rham and Dolbeault moduli spaces.

Talks will be on Wednesdays 12pm - 1:30pm in room 417 (Math building) except for the first talk on Oct 5 which will be in room 326 (Uris hall).

Date Speaker Topic References
Wed Oct 5 Anna Abasheva

What is P=W conjecture about?

In the first half of the talk I will introduce the main characters of non-Abelian Hodge correspondence: Betti, de Rham and Dolbeault moduli spaces. We will discuss isomorphisms between them and define perverse (P) and weight (W) filtrations on their cohomology. I will state non-abelian Hodge correspondence and P=W theorem. This part will be an overview and there will not be any proofs. In the second part, we will see how non-Abelian Hodge correspondence and P=W phenomenona work in the simplest possible case of rank 1 bundles. Amal's notes

[Mig17], [GX08, Sections 2-4]
Wed Oct 12 Anna Abasheva

Curious Poincare duality on Betti space

Curious Poincare duality is an identity on mixed Hodge numbers of a Betti space. It is a reflection of a certain symmetry of the cohomology of Betti space, the curious Leschetz duality. Even though the definition of a Betti space is very explicit, the proof of this identity is quite unexpected. It combines a reduction to positive characteristics arguement with representation theoretic results. We will try to understand both ingredients.

[HR08, 2.1-2.3, 3.5, Appendix]
Wed Oct 19 Anna Abasheva, Kevin Chang

Katz's theorem

This is a continuation of my previous talk. I'll give an idea of the proof of Katz's theorem which shows how to compute the E-polynomial of a complex variety with polynomial count by reduction to positive characteristics.

Riemann-Hilbert correspondence

The Riemann-Hilbert correspondence (for smooth projective varieties) is an equivalence between local systems and integrable connections. In this talk, we'll see how Simpson upgrades this to a complex analytic isomorphism of the Betti and de Rham moduli spaces. If there's time, I'll also say a bit about the more general version of the Riemann-Hilbert correspondence (for regular holonomic D-modules on not necessarily projective varieties).

[Sim95, p.23-31]
Wed Oct 26 Andres Fernandez Herrero

Hitchin fibration on Dolbeault space

In this talk we will define the Hitchin fibration and prove that it is an algebraic completely integrable system. We will recall the necessary definitions and constructions, including the description of the symplectic form on Dolbeault moduli space. If time permits, we will explain the BNR correspondence for general spectral curves and the proof of properness of the Hitchin fibration.

[Hit87,2-5.1]
Wed Nov 2 Anna Abasheva

Non-abelian Hodge correspondence

Non-abelian Hodge correspondence establishes a diffeomorphism between the moduli space of Higgs bundles and the space of flat connections. The crucial part in the construction of this diffeomorphism is a theorem by Simpson which states that a Higgs bundle admits a Yang-Mills metric if and only if it is polystable. I'll show how to reduce the correspondence to Simpson's theorem and then I'll prove the easiest direction of the theorem. The opposite direction relies on a description of the limit behavior of solutions to the heat equation. While I'm not going to solve any PDEs, I'll explain to you at least why the heat equation comes up here and (if we have time) what's the strategy for its solution. The talk will be more analytic than usual but I'll introduce all the necessary definitions.

[LP91]
Wed Nov 9 Morena Porzio

Perverse sheaves and Lefschetz hyperplane theorem

The aim of the talk is to give a geometric description of the perverse filtration on the cohomology of a constructible sheaf over an affine quasi-projective variety Y, in terms of a filtration induced by intersecting with hyperplanes in a good position. In order to do so, the talk will consist of two parts. Firstly, since Y is not necessarily non-singular, we will have a crash course on constructible and perverse sheaves in order to state the Strong Weak Lefschetz Theorem (sic!) for intersection cohomology. After that, we will see how to induce a filtration starting with a linear n-flag on a projective space, and compare it with the perverse one in the filtered derived category of abelian groups.

[CM09, 2], [CM10, Thm. 4.1.1]
Wed Nov 16 Morena Porzio and Kevin Chang

Perverse sheaves and Lefschetz hyperplane theorem

End of the previous talk.

P=W for tautological classes

In this talk, we'll reduce the P=W conjecture for all cohomology to a statement about Chern classes of the universal family (tautological classes). The two key inputs are (1) Markman's result saying that the tautological classes generate the cohomology of the Dolbeault moduli space (2) Shende's computation of the weights of the tautological classes. After discussing the reduction by Maulik-Shen, I'll say a bit about how Shende gets the weights.

[MS22, 1-2.1], [Sh17], [CMS22, Lemma 4.6]
Wed Nov 23 No seminar (Thanksgiving)
Wed Nov 30 Kevin Chang

Vanishing cycles

I'll state some of the basic definitions and properties of nearby/vanishing cycles. These appear in the proof of P=W by allowing us to relate the cohomology of twisted Hitchin systems.

Wed Dec 7 Anna Abasheva L-twisted Hitchin systems and vanishing cycles [MS22]
Sat Dec 10 (Unusual time!) Andres Fernandez Herrero Global Springer theory [MS22]
Wed Dec 14 Anna Abasheva Wrap-up [MS22, 3]

References

[CMS22] Mark A. de Cataldo, Davesh Maulik und Junliang Shen. "Hitchin Fibrations, abelian surfaces, and the P=W conjecture". In J. Am. Math. Soc., Vol. 35, N. 3 (2022), 911-953. arXiv: 1909.11885.
[CM09] Mark Andrea A. de Cataldo and Luca Migliorini. "The decomposition theorem, perverse sheaves and the topology of algebraic maps". In: Bull. Am. Math. Soc., New Ser. 46.4 (2009), 535-633. arXiv:0712.0349.
[CM10] Mark Andrea A. de Cataldo and Luca Migliorini. "The perverse filtration and the Lefschetz hyperplane theorem". In Annals of Mathematics, 171 (2010), 2089-2113, arXiv:0805.4634.
[GX08] William M. Goldman and Eugene Z. Xia. Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces. Bd. 904. Providence, RI: American Mathematical Society (AMS), 2008, 69. arXiv:math/0402429.
[HR08] Tamás Hausel and Fernando Rodriguez-Villegas. "Mixed Hodge polynomials of character varieties. With an appendix by Nicholas M. Katz". In: Invent. Math. 174.3 (2008), 555-624. arXiv:math/0612668.
[Hit87] Nigel J. Hitchin. "Stable bundles and integrable systems." In: Duke Math. J. 54 (1987), 91-114.
[MS21] Davesh Maulik and Junliang Shen. "Endoscopic decompositions and the Hausel-Thaddeus conjecture". In Forum Math. Pi, Vol 9:e8 (2021), 1-49. arXiv:2008.08520.
[MS22] Davesh Maulik and Junliang Shen. "The P=W conjecture for GLn". arXiv:2209.02568.
[Mig17] Luca Migliorini. "Recent results and conjectures on the non abelian Hodge theory of curves". In: Boll. Unione Mat. Ital. 10.3 (2017), 467-485.
[LP91] Joseph Le Potier. "Fibr&eactue;s de Higgs et systèmes locaux". In Astérisque, tome 201-202-203 (1991), Séminaire Bourbaki, exp. no 737, p. 221-268
[Sh17] Vivek Shende. "The Weights of the Tautological Classes of Character Varieties". In Int. Math. Res. Not., Vol. 2017, N. 22 (2017), 6832-6840. arXiv:1411.4975.
[Sim95] Carlos T. Simpson. "Moduli of representations of the fundamental group of a smooth projective variety. II". In: Publ. Math., Inst. Hautes Étud. Sci. 80 (1995), 5-79.