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BIMSA AG Seminar
Dual boundary complexes of character varieties and the geometric P=W conjecture
Dual boundary complexes of character varieties and the geometric P=W conjecture
Organizers
Speaker
Time
Thursday, February 22, 2024 3:00 PM - 4:00 PM
Venue
A6-101
Online
Zoom 638 227 8222
(BIMSA)
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. A motivic argument proving that the dual boundary complex of a stably affine space is contractible.