The $P=W$ conjecture for $\mathrm{GL}_n$

The Betti moduli space $M_B$ parametrizes irreducible local systems on a punctured curve with prescribed central monodromy, while $M_{\mathrm{Dol}}$ is the corresponding Dolbeault moduli space. Non-abelian Hodge theory gives a canonical diffeomorphism between them and identifies their cohomology. Let $P_kH^m(M_{\mathrm{Dol}},\mathbb{Q})$ denote the perverse filtration associated with the Hitchin system, and let $W_iH^m(M_B,\mathbb{Q})$ denote the weight filtration on the cohomology of the Betti moduli space. The $P=W$ conjecture for $\mathrm{GL}_n$. For any $k,m\in\mathbb{Z}_{\geq 0}$, one has

$$P_kH^m(M_{\mathrm{Dol}}, \mathbb{Q}) = W_{2k}H^m(M_B, \mathbb{Q}) = W_{2k+1}H^m(M_B, \mathbb{Q}).$$

The conjecture refines the cohomological identification supplied by non-abelian Hodge theory by relating the topology of the Hitchin system to the mixed Hodge structure of the character variety. The compatibility with Galois conjugation on the Betti side has been proven, implying that the statement does not depend on the degree $d$ as long as it is coprime to $n$.