Cell decomposition and dual boundary complexes of character varieties

Abstract

The weak geometric P=W conjecture of Katzarkov--Noll--Pandit--Simpson asserts that for any smooth Betti moduli space ${\mathcal{M}}_B$ of complex dimension $d$ associated to a punctured Riemann surface, the dual boundary complex $\mathbb{D}\partial {\mathcal{M}}_B$ is homotopy equivalent to a $(d-1)$-sphere. Here, ${\mathcal{M}}_B$ is a generic $G{L}_n(\mathbb{C})$-character variety of genus $g$, with prescribed semisimple local monodromies at $k$ punctures. In this article, we prove the weak geometric P=W conjecture for all \emph{very generic} ${\mathcal{M}}_B$, namely when at least one conjugacy class is regular semisimple. A key step is a strengthening of Mellit's cell decomposition theorem: we decompose ${\mathcal{M}}_B$ itself (without passing to a vector bundle) into locally closed subvarieties which are stably isomorphic to $({\mathbb{C}}^{\times }{)}^{d-2b}\times {\mathbb{C}}^b$. A second ingredient is a motivic characterization of the integral cohomology of dual boundary complexes, developed in [Su24]. Following Simpson's strategy, the proof reduces to an inductive computation of dual boundary complexes arising from this cell decomposition.

v5: 51 pages; correction and update to the published version. The main theorem on the weak geometric P=W conjecture is unchanged

References in corpus (3)

• Arithmetic harmonic analysis on character and quiver varieties
• The dual complex of Calabi--Yau pairs
• Global geometry on moduli of local systems for surfaces with boundary