The Geometric P=W conjecture and Thurston's compactification

2025/07/09 by Ashwin Ayilliath-Kutteri, Ayilliath-Kutteri, Ashwin, Mohammad Farajzadeh Tehrani +3
Mathematics · #14M35 #57K31 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2507.07211

openalex publication_date 2025/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we use new results together with established facts about Thurston's compactification of Teichmüller space to address the geometric P=W conjecture for SL(2,ℂ), which concerns projective compactifications of character varieties of closed surfaces. In particular, we construct a projective compactification of the SL(2,ℂ)-character variety of any closed surface of genus g>1, in which the boundary divisors are toric varieties and the dual intersection complex is a sphere. A main technical step, of independent interest, is the derivation of an explicit formula for a well-known embedding of the set of isotopy classes of multicurves on a closed surface of genus g into ℕ^9g-9.

Citations

• Sliced skein algebras and geometric Kauffman bracket
• Degenerations Of Skein Algebras And Quantum Traces
• On compactifications of the SL(2,C) character varieties of punctured surfaces
• Dimension and Trace of the Kauffman Bracket Skein Algebra
• On the geometric P=W conjecture
• Skein algebras of surfaces
• Harmonic Maps to Buildings and Singular Perturbation Theory
• Projective completions of affine varieties via degree-like functions
• Topology of Hitchin systems and Hodge theory of character varieties: the case A₁
• Multicurves and regular functions on the representation variety of a surface in SU(2)

Related

• On the geometric P=W conjecture
• On compactifications of the SL(2,C) character varieties of punctured surfaces
• Cell decomposition and dual boundary complexes of character varieties
• On compactifications of character varieties of n-punctured projective line
• P=W conjectures for character varieties with symplectic resolution