Log Calabi-Yau compactifications of $SL(2,\mathbb{C})$ character varieties

Technical Summary: Log Calabi–Yau Compactifications of SL(2, C) Character Varieties

Problem Statement
The paper addresses a folklore conjecture predicting that character varieties of surfaces admit log Calabi–Yau compactifications with relatively mild singularities. Specifically, the authors investigate the $SL(2, \mathbb{C})$ character varieties of compact oriented surfaces ($Ch_{g}^{SL(2, \mathbb{C})}$) and the relative $SL(2, \mathbb{C})$ character varieties of punctured surfaces ($Ch_{g,n}^{SL(2, \mathbb{C}), t}$).

The core problem is to establish the existence of a compactification $(Y, D)$ for these affine varieties such that:

1. $Y$ is a normal projective variety and $D$ is a reduced effective Weil divisor with $Y \setminus D$ isomorphic to the character variety.
2. The pair $(Y, D)$ is log Calabi–Yau, meaning the log canonical divisor class is trivial ($K_Y + D \sim 0$).
3. The singularities of the pair are "mild," specifically satisfying the log canonical (lc) or divisorial log terminal (dlt) conditions.

While the conjecture is known for toric cases (e.g., $GL(1, \mathbb{C})$) and specific low-dimensional non-abelian cases (e.g., $(g,n)=(0,4)$ and $(1,1)$), a general proof for arbitrary genus $g$ and puncture count $n$ in the $SL(2, \mathbb{C})$ setting was lacking.

Methodology
The authors' approach relies on a two-step strategy combining general algebraic geometry with specific combinatorial constructions:

1. Filtration-Based Compactifications: The paper utilizes projective compactifications of affine varieties constructed from filtrations of their algebras of regular functions. Specifically, they analyze the compactifications recently constructed by Kutteri–Tehrani–Frohman (for the compact case) and Tehrani–Frohman (for the punctured case). These constructions define the compactification as the Proj of a Rees algebra associated with a specific filtration.

2. General Log Calabi–Yau Criterion (Theorem C): The authors establish a general sufficient condition (Theorem C/Theorem 3.7) for a compactification arising from a filtration to be log canonical and log Calabi–Yau. The criteria require:

   • The underlying affine variety to be normal.
   • The filtration's degree-like function to be a "subdegree."
   • The associated graded algebra to be isomorphic to the algebra of a cone complex $\Omega$.
   • The cone complex $\Omega$ to have a level set homeomorphic to a sphere.

3. Degeneration Argument: To prove the general criterion, the authors construct a canonical one-parameter degeneration of the compactification $(Y, D)$. The central fiber of this degeneration is shown to be a "compactified vertex," which is a stable toric variety associated with a polyhedral complex. By verifying that this central fiber is semi-log canonical and log Calabi–Yau, and using deformation theory (specifically properties of Cohen–Macaulay schemes and semi-log canonical pairs), they deduce that the general fiber (the original compactification) shares these properties.

Key Contributions and Results

• Theorem A (Main Result): The authors prove Conjecture 1.1 for $G = SL(2, \mathbb{C})$.

  • Compact Case: For every integer $g \ge 0$, the $SL(2, \mathbb{C})$-character variety $Ch_{g}^{SL(2, \mathbb{C})}$ admits a dlt (divisorial log terminal) log Calabi–Yau compactification.
  • Punctured Case: For every $g \ge 0, n > 0$ and $t \in \mathbb{C}^n$, the relative character variety $Ch_{g,n}^{SL(2, \mathbb{C}), t}$ admits a log canonical log Calabi–Yau compactification. Furthermore, if the boundary traces $t$ are "Kostov generic" (a condition ensuring smoothness of the variety), the compactification is dlt.

• Theorem B (Geometric Properties): The authors verify that the specific compactifications constructed by Kutteri–Tehrani–Frohman and Tehrani–Frohman satisfy the geometric requirements. They show that for any pair of pants decomposition (compact case) or ideal triangulation (punctured case), the resulting compactification is log canonical and log Calabi–Yau.

• Theorem C (General Criterion): A new independent result providing sufficient conditions for filtrations to yield log Calabi–Yau compactifications. This theorem connects the combinatorial structure of the associated graded algebra (via cone complexes) to the birational geometry of the resulting compactification.

Significance and Claims
The paper claims to provide the first examples of log Calabi–Yau compactifications for $SL(2, \mathbb{C})$ character varieties where $g \ge 2$ in the compact case, and beyond the elementary $(0,4)$ and $(1,1)$ cases in the punctured setting.

The significance of the work is framed within several contexts:

• Minimal Model Program (MMP): It advances the understanding of character varieties as objects within the MMP, specifically regarding the existence of compactifications with controlled singularities.
• Dual Complexes and the P=W Conjecture: The existence of these compactifications is a prerequisite for investigating the geometric P=W conjecture, which predicts that the dual complex of such a compactification is homeomorphic to a sphere and identifies with the sphere at infinity in the base of the Hitchin fibration.
• Goldman Volume Form: The authors note that the existence of a log Calabi–Yau compactification implies the existence of a unique algebraic volume form with first-order poles along the boundary. They suggest (as a direction for future work) that this form likely coincides with the Goldman volume form, potentially implying that character varieties admit log symplectic compactifications.

The authors explicitly state that their results rely on the specific constructions of Kutteri–Tehrani–Frohman and Tehrani–Frohman, applying their general criterion to these specific cases to resolve the conjecture for $SL(2, \mathbb{C})$. They do not claim to have constructed new compactifications but rather to have proven the singularity properties of existing ones.