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

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.

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).

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.

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.

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.

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.

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.

L-twisted Hitchin systems and vanishing cycles

We'll continue reading Maulik and Shen's paper. I'll state a version of the P=W conjecture for twisted moduli spaces. In future talks, we'll see that the P=W conjecture is easier to show in this case. Meanwhile, I'll reduce the non-twisted P=W conjecture to the twisted one with a beautiful argument involving vanishing cycles.

Global Springer theory

I will explain some geometric properties of the moduli stacks of parabolic Higgs bundles appearing in Yun's global Springer theory. We will then focus on the case of PGL_n and see how a support theorem for the corresponding parabolic Hitchin fibration implies strong perversity of the tautological cohomology classes.

Ngô's fundamental lemma

We will finish the proof of the P=W conjecture. It remains to show that all supports in the decomposition theorem for parabolic Hitchin systems are trivial. Maulik and Shen's proof relies on several ingredients, which are all outsourced to other papers. I'll state and explain all of them; yet, subjectively the most interesting ingredient and definitely the most fundamental is Ngô's fundamental lemma. It gives dimension bounds on the supports of weak abelian fibrations. I'll define weak abelian fibrations and we'll prove Ngô's result modulo some technical statements.