$P=W$ $~$ for $~$ $GLn$

These notes were taken during talk given by Davesh Maulik.

Notations used throughout the talk
$C-$ non-singular projective irreducible curve
genus of curve $g(C)\geq 2$
$n-$ rank
$d-$ degree and $(n,d)=1$

Moduli space of Higgs Bundle on the curve

$K_c$ is canonical bundle and $\theta$ is Higgs field (twisted endomorphism of vector bundle)

Twisted character variety of the curve

Non-abelian Hodge theorem
There exists diffeomorphism between $\mathcal{M}^{Higgs}$ and $\mathcal{M}^{char}$

Example: When n=1
Jacobian$(C)\times \mathbb{C}^{g}\cong(S^1)^{2g}\times\mathbb{R}^{2g}\cong(\mathbb{C}^*)^{2g}$

We have the identification between singular cohomology $H^{*}(\mathcal{M}^{Higgs},\mathbb{Q})$ and $H^{*}(\mathcal{M}^{char}, \mathbb{Q})$

$H^{*}(\mathcal{M}^{char}, \mathbb{Q})$ has mixed Hodge structure and in particular weight filtration $W^kH^*$.

P=W conjecture (dCHM): Perverse filtration on Higgs moduli space $P_{\leq k}H^{*}(\mathcal{M}^{Higgs})$ matches up with weight filtration $W_{\leq 2k}H^*(\mathcal{M}^{char}).$

Result proved for

1. $n=2, g\geq 2$ (2010)
2. $n\geq 2, g=2$ (2019)
3. for all n and for all g (2022)

Don’t know if this conjecture is true for other cases like

1. for other cases like $SLn$
2. when $(n,d)\neq 1$