Cell decomposition and dual boundary complexes of character varieties

The weak geometric P=W conjecture of L. Katzarkov, A. Noll, P. Pandit, and C. Simpson asserts that for any smooth Betti moduli space ${\mathcal{M}}_B$ of complex dimension $d$ over a punctured Riemann surface, the dual boundary complex $\mathbb{D}\mathrm{\partial }{\mathcal{M}}_B$ is homotopy equivalent to a $(d-1)$-dimensional sphere. Here, we consider ${\mathcal{M}}_B$ as a generic $G{L}_n(\mathbb{C})$-character variety defined on a Riemann surface of genus $g$, with local monodromies specified by generic semisimple conjugacy classes at $k$ punctures. In this article, we establish the weak geometric P=W conjecture for all \emph{very generic} ${\mathcal{M}}_B$ in the sense that at least one conjugacy class is regular semisimple. A crucial step is to establish a stronger form of A. Mellit’s cell decomposition theorem, i.e. we decompose ${\mathcal{M}}_B$ (without passing to a vector bundle) into locally closed subvarieties of the form $({\mathbb{C}}^{\times }{)}^{d-2b}\times \mathcal{A}$, where $\mathcal{A}$ is stably isomorphic to ${\mathbb{C}}^b$. A second ingredient involves a motivic characterization of the integral cohomology of dual boundary complexes developed in a subsequent article [Su24]. Following C. Simpson’s strategy, the proof is now an inductive computation of the dual boundary complexes from such a cell decomposition.