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The Geometric P=W conjecture and Thurston's compactification

Ashwin Ayilliath-Kutteri, Mohammad Farajzadeh-Tehrani, Charles Frohman

TL;DR

The paper advances the geometric P=W program for SL(2,C) character varieties of closed surfaces by constructing a projective compactification with toric boundary divisors whose dual complex is a sphere, and by establishing a precise toric degeneration framework via a new 9g-9 intersection-coordinate embedding of multicurves. Central to the approach are the skein algebra Sk_g, modified Dehn–Thurston coordinates, and a refined Thurston compactification that yields a polyhedral sphere complex P; the authors prove that the graded coordinate ring of the character variety is isomorphic to a Stanley–Reisner ring of the image of multicurves, with the boundary decomposing into toric strata matching P. A key technical achievement is explicit formulas for intersection numbers with dual curves in terms of DT coordinates, enabling the ConeTheorem and the leading-term analysis that connect the algebraic degeneration to Thurston’s boundary. The genus-2 example concretely demonstrates the toric boundary structure and the sphere dual complex, offering a concrete model for the broader construction and its implications for the geometric and cohomological P=W conjectures.

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 $\mathrm{SL}(2,\mathbb{C})$, which concerns projective compactifications of character varieties of closed surfaces. In particular, we construct a projective compactification of the $\mathrm{SL}(2,\mathbb{C})$-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 $\mathbb{N}^{9g-9}$.

The Geometric P=W conjecture and Thurston's compactification

TL;DR

The paper advances the geometric P=W program for SL(2,C) character varieties of closed surfaces by constructing a projective compactification with toric boundary divisors whose dual complex is a sphere, and by establishing a precise toric degeneration framework via a new 9g-9 intersection-coordinate embedding of multicurves. Central to the approach are the skein algebra Sk_g, modified Dehn–Thurston coordinates, and a refined Thurston compactification that yields a polyhedral sphere complex P; the authors prove that the graded coordinate ring of the character variety is isomorphic to a Stanley–Reisner ring of the image of multicurves, with the boundary decomposing into toric strata matching P. A key technical achievement is explicit formulas for intersection numbers with dual curves in terms of DT coordinates, enabling the ConeTheorem and the leading-term analysis that connect the algebraic degeneration to Thurston’s boundary. The genus-2 example concretely demonstrates the toric boundary structure and the sphere dual complex, offering a concrete model for the broader construction and its implications for the geometric and cohomological P=W conjectures.

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 , which concerns projective compactifications of character varieties of closed surfaces. In particular, we construct a projective compactification of the -character variety of any closed surface of genus , 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 into .

Paper Structure

This paper contains 7 sections, 9 theorems, 138 equations, 18 figures.

Table of Contents

  1. Introduction and the main results
  2. Preliminaries in skein algebra and DT coordinates
  3. Thurston's compactification
  4. Compactification of $\mathcal{X}_g$
  5. Proof of Theorem \ref{['f-functions']}
  6. Proof of Theorem \ref{['ThmC']}
  7. Example of genus $2$

Key Result

Theorem 1.3

For $g\geq 0$ and $n>0$ satisfying $2g+n\geq 3$, associated to any "ideal triangulation" of $\Sigma_{g,n}$, there is a normal compactification $\overline{\mathcal{X}}_{g,n}$ of $\mathcal{X}_{g,n}$ such that the boundary divisor $D_{g,n}=\partial \overline{\mathcal{X}}_{g,n}$ is an irreducible ample

Figures (18)

  • Figure 1: A folded pair of pants that includes two copies of the curve $a$ on its boundary.
  • Figure 2: Two pants decompositions of a genus two surface.
  • Figure 5: Decomposition of $\Sigma_g$ into annuli $A_i$ and shrunken pants.
  • Figure 6: The picture in an annulus when $n_i=3$ and $t_i=2$.
  • Figure 7: Left: the dual curve $a'$ when $a$ belongs to two different pairs of pants. Right: the dual curve $a'$ when $a$ belongs to only one pair of pants.
  • ...and 13 more figures

Theorems & Definitions (24)

  • Conjecture 1.1: MMS
  • Remark 1.2
  • Theorem 1.3: FF23
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 14 more