Compactification of $\operatorname{SL}(3,\mathbb{C})$-Character Varieties of Surfaces via Skein Algebras

Abstract

We investigate the filtration structure of the skein algebra induced by the (quantum) trace map. Utilizing the structure, we prove that relative $\operatorname{SL}(3, \mathbb{C})$ character varieties of punctured surfaces admit log Calabi-Yau compactifications and prove the (weak) geometric P=W conjecture.