Cell decomposition, dual boundary complexes and motives of character varieties

Tao Su
Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Algebraic Topology (math.AT), Representation Theory (math.RT)
2023-07-30 16:00:00
The homotopy type conjecture (weak form of the geometric P=W conjecture) states: for any (smooth) Betti moduli space $\mathcal{M}_B$ of complex dimension $d$ over a (punctured) Riemann surface, the dual boundary complex $\mathbb{D}\partial\mathcal{M}_B$ is homotopy equivalent to a sphere of dimension $d-1$. Say, $\mathcal{M}_B$ is a generic $GL_n(\mathbb{C})$-character variety on a genus $g$ Riemann surface with local monodromies at $k$ punctures in prescribed generic semisimple conjugacy classes. First, we prove the homotopy type conjecture for $\mathcal{M}_B$ if it's very generic: at least one conjugacy class is in addition regular semisimple. Second, the main result is obtained by proving a strong form of A. Mellit's cell decomposition: $\mathcal{M}_B$ itself is decomposed into locally closed subvarieties of the form $(\mathbb{C}^{\times})^{d-2b}\times\mathcal{A}$, where $\mathcal{A}$ is stably isomorphic to $\mathbb{C}^b$. We expect that $\mathcal{A}$ is in general a counterexample to the Zariski cancellation problem for dimension $b\geq 3$ in characteristic zero. Third, we propose a conjectural formula $A$ for Voevodsky's motive with compact support of any generic $\mathcal{M}_B$. This directly generalizes the HLRV conjecture. It also suggests an integral curious Poincar\'{e} duality conjecture $B$ for the singular weight cohomology with compact support of the same variety. Some partial results are: $1$. prove a weak form of $A$ for very generic $\mathcal{M}_B$, i.e. the formula for its class in the Grothendieck ring of effective pure Chow motives; $2$. for geneirc $\mathcal{M}_B$, $B$ implies that $\mathbb{D}\partial\mathcal{M}_B$ is an integral homology sphere of the right dimension. Finally, we verify all the conjectures in simple examples.
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