Abstract

The homotopy type conjecture (weak form of geometric P=W conjecture) states that: for any (smooth) Betti moduli space $\mathcal{M}_B$ of complex dimension d over a (punctured) Riemann surface, the dual boundary complex $\mathbb{D}\partial\mathcal{M}_B$ is homotopy equivalent to a sphere of dimension d-1. The main goal of this lecture series is to explain a proof of the conjecture for any very generic GL_n(C)-character variety $\mathcal{M}_B$.

By generic, I mean $\mathcal{M}_B$ is a moduli space of GL_n(C)-local systems on a genus g Riemann surface with local monodromies at k punctures in prescribed generic semisimple conjugacy classes; By very generic, I mean that at least one conjugacy class is in addition regular semisimple.

A key ingredient in the proof is to obtain a strong form improving A.Mellit's cell decomposition: $\mathcal{M}_B$ itself is decomposed into locally closed subvarieties of the form $(\mathbb{C}^*)^{d-2b} \times \mathcal{A}$, where $\mathcal{A}$ is stably isomorphic to $\mathbb{C}^b$. We expect that $\mathcal{A}$ is in general a counterexample to the Zariski cancellation problem for dimension b at least 3 in characteristic zero.

When $\mathcal{M}_B$ is only generic, it's a consequence of Mellit's curious Hard Lefschetz theorem that $\mathbb{D}\partial\modulispace_B$ is always a rational homology sphere of the right dimension. In general, it's possible to propose a motivic HLRV conjecture for any generic $M_B$, part of which consists of an integral curious Poincare duality conjecture. The latter implies that $\mathbb{D}\partial\modulispace_B$ is an integral homology sphere of the right dimension.

Finally, time permitting, I will talk about some work in progress.By a folklore conjecture, all smooth character varieties are log CY. Then, the previous main result is also relevant to Kontsevich's conjecture: the dual boundary complex of any log CY variety is a sphere. We will see that some of the ideas above apply and extend to the mirror P=W conjecture for affine log CY varieties. As a reward, this gives a potential mirror interpretation of Kontsevich's conjecture at least for affine log CY varieties.

The plan is as follows:

Lecture 0: overview;

Lecture 1: dual boundary complexes;

Lecture 2: cell decomposition;

Lecture 3: motives;

Lecture 4 (time permitting): mirror P=W phenomena.

Speaker

I'm currently a visiting postdoc at IMS, CUHK. I was a postdoc at YMSC (Jan.2021-Jul.2022), and at ENS Paris (Sep.2018 - Sep.2020). I obtained my math PhD at UC Berkeley in 2018, advised by Vivek Shende and Richard E.Borcherds.

My research interests include microlocal sheaf theory and Legendrian knots, Hodge theory of character varieties and P=W, and mirror symmetry.