2023/05/21 by Mohammad Farajzadeh Tehrani, Farajzadeh-Tehrani, Mohammad, Charles Frohman +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometry and complex manifolds
paper · doi:10.48550/arxiv.2305.12306
openalex publication_date 2023/05/21 · openalex created_date 2023/05/24 · openalex updated_date 2026/07/28
This paper addresses some conjectures and questions regarding the absolute and relative compactifications of the \SL(2,\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 \SL(2,\C) to compactify the (relative) character varieties. Specifically, we construct a uniform relative compactification over the base space of \Cn and determine its monodromy, addressing a question posed by Simpson.