Context of the question: Let $G=\pi_1(S_g)$ be the fundamental group of the oriented closed surface of genus $g$. The $P=W$ conjecture states that a certain filtration (of algebraic geometric origin) on the cohomology of the Betti moduli space $M_B(r)$ of all $r$-dimensional complex representations of $G$ is equal to the weight filtration induced by its natural structure of affine variety.
The question. I think this weight filtration has to do with a 'compactification' of $M_B$ which would make it compact Kähler. But I have no idea how to compactify, and even if I had, how the cohomology of the compactification would relate to the original cohomology. I have tried looking up the literature but have found nothing much.
Could anybody give a reference for this sort of questions? Thank you very much.