Dual boundary complexes of character varieties and the geometric P=W conjecture
Dr. Tao Su
2023-11-13 15:40-16:40
MCM410
Speaker: Dr. Tao SuTime: 15:40-16:40 November 13, 2023 (Monday)
Place: MCM410
Title: Dual boundary complexes of character varieties and the geometric P=W conjecture
Abstract: Aiming at a geometric interpretation of the famous P=W conjecture (now theorem) in nonabelian Hodge theory (NAH), the geometric P=W conjecture of Katzarkov-Noll-Pandit-Simpson predicts that NAH identifies the Hitchin fibration at infinity with another fibration intrinsic to the Betti moduli space M_B, up to homotopy. Its weak form states that: the dual boundary complex of M_B (of complex dimension d) is homotopy equivalent to a sphere of dimension d-1 (the Hitchin base at infinity).
In this talk, I will explain a proof of the weak geometric P=W conjecture for all very generic GL_n(C)-character varieties M_B over any (punctured) Riemann surface. The proof involves two main ingredients: 1. improve A.Mellit's cell decomposition into a strong form: M_B itself is decomposed into locally closed subvarieties of the form $(\mathbb{C}^*)^{d-2b} \times A$, where $A$ is stably isomorphic to $\mathbb{C}^b$; 2. Some motivic argument. Some further aspects and directions are: 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. This is based on https://arxiv.org/abs/2307.16657 and some work in progress.
