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 simply reads: the dual boundary complex of M_B is homotopy equivalent to a sphere of dimension one less (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, each being a product of an algebraic torus and a stably affine space; 2. give a motivic characterization of the integral cohomology of dual boundary complexes, and demonstrate that the dual boundary complex of any stably affine space (of positive dimension) is contractible.