On compactifications of the SL(2,C) character varieties of punctured surfaces

This paper addresses some conjectures and questions regarding the absolute and relative compactifications of the $\text{SL}(2,\text{C})$-character variety of an $n$-punctured Riemann surface without boundary. We study a class of projective compactifications determined by ideal triangulations of the surface and prove explicit results concerning the boundary divisors of these compactifications. Notably, we establish that the boundary divisors are toric varieties and confirm a well-known conjecture asserting that the (dual) boundary complex of any (positive dimensional) relative character variety is a sphere. In a different vein, we enhance and streamline Komyo’s compactification method, which utilizes a projective compactification of $\text{SL}(2,\text{C})$ to compactify the (relative) character varieties. Specifically, we construct a uniform relative compactification over the base space of ${\text{C}}^n$ and determine its monodromy, addressing a question posed by Simpson.