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Stable cohomology of universal character varieties

Using nonabelian Hodge theory and Saito's mixed Hodge modules, the authors prove that the Leray-Serre spectral sequence for universal PGL $_n$ -character varieties over moduli spaces of curves degenerates at $E_2$ , thereby establishing the stabilization of their rational cohomology as the genus goes to infinity and computing the stable limit, with analogous results extended to G-character varieties over punctured curves.

Original authors: Ishan Banerjee, Faye Jackson, Anne Larsen, Sam Payne, Xiyan Zhong

Published 2026-02-05

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CC BY 4.0

Original authors: Ishan Banerjee, Faye Jackson, Anne Larsen, Sam Payne, Xiyan Zhong

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an explorer trying to map the shape of a vast, shifting landscape. In mathematics, this landscape is called a character variety. It's a place where you collect all the possible ways to assign "symmetries" (like rotations or flips) to the loops you can draw on a curved surface, like a donut or a pretzel.

The paper you provided is like a detailed guidebook for exploring a specific, massive version of this landscape. Here is the breakdown of what the authors discovered, using simple analogies.

1. The Map and the Terrain

Think of a curve (or a surface) as a piece of rubber with holes in it. The number of holes is called the genus ( $g$ ). A sphere has 0 holes; a donut has 1; a pretzel has 2, and so on.

The Problem: Mathematicians wanted to understand the "shape" (cohomology) of the space that holds all possible symmetry assignments for these surfaces.
The Challenge: As the number of holes ( $g$ ) gets bigger and bigger, the space becomes incredibly complex. It's like trying to predict the weather on a planet with a billion continents.

2. The "Stabilization" Discovery

The authors made a surprising discovery: The complexity stops growing after a certain point.

Imagine you are building a tower of blocks. Usually, as you add more floors, the structure changes in unpredictable ways. But in this specific mathematical world, once the tower gets high enough (once the number of holes $g$ is large enough), adding more floors doesn't change the pattern of the blocks anymore. The structure "stabilizes."

The Result: They proved that if you look at the "shape" of these symmetry spaces for surfaces with many holes, the pattern of their features becomes constant. You can calculate the shape for a surface with 100 holes, and it will look exactly the same as the shape for a surface with 1,000 holes, provided you are looking at features of a certain size.

3. The "Universal" Machine

The paper studies a "Universal Character Variety." Think of this not as a single map, but as a master blueprint that contains the maps for every possible surface at once.

If you pick a specific surface (a specific point in the blueprint), you get the map for that surface.
The authors showed that when you look at this master blueprint, the way the different maps relate to each other is surprisingly simple. The "spectral sequence" (a complex mathematical tool used to build the map layer by layer) stops changing after the second step. It's like a puzzle that solves itself almost immediately, rather than requiring thousands of moves.

4. The Two Proofs: The "High-Tech" and the "Classic"

To prove that the map-solving tool stops changing, the authors used two different methods, like two different ways to fix a broken engine:

The High-Tech Method: They used a sophisticated modern theory called "Mixed Hodge Modules" (think of this as using a high-resolution 3D scanner to see the internal structure of the engine). This was fast and efficient.
The Classic Method: They used older, more traditional tools of geometry and algebra (like using a wrench and a screwdriver). This was longer but relied on fundamental principles everyone understands.

Both methods led to the same conclusion: the engine is stable.

5. The "Punctured" Twist

The authors also looked at surfaces with a puncture (a tiny hole or a missing point).

Imagine a donut with a tiny hole poked in it.
They found that even with this extra complication, the "stabilization" rule still holds. The patterns of symmetry for these punctured surfaces also settle into a predictable, unchanging form as the number of holes increases.

6. The "Census" of Shapes

Finally, the authors didn't just say "it stabilizes"; they actually counted the features.

They provided a formula (a recipe) to calculate exactly how many "holes" or "loops" exist in these symmetry spaces for different sizes.
They included a table (Figure 1 in the paper) that lists these numbers for small cases, acting like a census report for these mathematical shapes.

Summary

In short, this paper proves that the chaotic, ever-changing world of symmetry spaces on curved surfaces has a hidden order. Once the surfaces get big enough, the rules governing their shapes become simple and unchanging. The authors provided the tools to see this order and the exact numbers to describe it, using both modern high-tech math and classic geometric reasoning.

Technical Summary: Stable Cohomology of Universal Character Varieties

Problem Statement
The paper investigates the cohomological stability of universal character varieties over the moduli space of curves, $\mathcal{M}_g$ . Specifically, it studies the universal $PGL_n$ -character variety $\mathcal{M}^{PGL}_{B}(\mathcal{C}_g, n, d) \to \mathcal{M}_g$ , where the fiber over a point $[C]$ is the character variety of $PGL_n(\mathbb{C})$ -local systems on the curve $C$ . The authors also consider universal $G$ -character varieties for $G = GL_n$ and $SL_n$ over the moduli space of punctured curves, $\mathcal{M}_{g,1}$ .

The central question is whether the rational cohomology groups of these universal varieties stabilize as the genus $g \to \infty$ , and if so, to compute the stable limit. This work extends previous stability results for the moduli space of curves (Madsen-Weiss) and Hurwitz spaces (Ellenberg-Venkatesh-Westerland, Landesman-Levy, Putman) to the setting of non-abelian character varieties.

Methodology
The authors employ a combination of non-abelian Hodge theory, the theory of mixed Hodge modules, and classical Hodge theory. The core technical strategy involves the following steps:

Non-Abelian Hodge Correspondence: The authors utilize the relative non-abelian Hodge correspondence to relate the universal character variety on the Betti side ( $\mathcal{M}^{PGL}_{B}$ ) to a universal moduli space of semistable Higgs bundles on the Dolbeault side ( $\mathcal{M}^{PGL}_{Dol}$ ). While the Betti side is constructed analytically, the Dolbeault side admits an algebraic structure suitable for applying tools from algebraic geometry.
Degeneration of the Leray-Serre Spectral Sequence: A pivotal technical result is proving that the Leray-Serre spectral sequence for the projection $\pi: \mathcal{M}^{PGL}_{B}(\mathcal{C}_g, n, d) \to \mathcal{M}_g$ $π : M_{B}^{P G L} (C_{g}, n, d) \to M_{g}$ degenerates at the $E_2$ $E_{2}$ page.
- Proof 1 (Saito's Theory): Using Saito's theory of mixed Hodge modules, the authors show that the pushforward of the constant sheaf splits as a direct sum of its cohomology sheaves. This relies on the purity of the cohomology of the fibers (Higgs moduli spaces) and the smoothness of the morphism.
- Proof 2 (Classical Hodge Theory): An alternative proof is provided using classical Hodge theory and derived category manipulations. This involves constructing a smooth relative compactification of the Higgs moduli space and analyzing the resulting Gysin sequence to establish the splitting of the pushforward.
Cohomology of Fibers and Base: The authors combine the degeneration result with known generators and relations for the cohomology of the fibers (based on work by Hausel-Thaddeus, Markman, and Hausel-Rodriguez-Villegas) and the stable cohomology of the base moduli spaces $\mathcal{M}_g$ and $\mathcal{M}_{g,1}$ with symplectic coefficients (based on Looijenga, Kawazumi, and Randal-Williams).

Key Contributions and Results

Degeneration Theorem (Theorem 1.3 / Theorem 3.1): For coprime $d$ and $n$ , and $g \ge 2$ , the Leray-Serre spectral sequence for the universal $PGL_n$ -character variety degenerates at $E_2$ . This implies an isomorphism $H^k(\mathcal{M}^{PGL}_{B}(\mathcal{C}_g, n, d); \mathbb{Q}) \cong \bigoplus_{p+q=k} H^p(\mathcal{M}_g; R^q\pi_*\mathbb{Q})$ .
Stable Cohomology Formula (Theorem 1.1): The authors compute the stable limit of the rational cohomology. For $d \in (\mathbb{Z}/n\mathbb{Z})^\times$ and $k \le 2(g-1)(n-1)+2$ , the cohomology is isomorphic to the degree $k$ part of:
$H^*(\mathcal{M}_g; V^\bullet(V_g \otimes \widetilde{W}_n)) \otimes \text{Sym}^\bullet(V^\circ \otimes \widetilde{W}_n)$
where $V_g = H^1(S_g; \mathbb{Q})$ , $V^\circ = \mathbb{Q} \oplus \mathbb{Q}[-2]$ , and $\widetilde{W}_n$ is the reduced cohomology of $\mathbb{P}^{n-1}$ .
Stability and Independence (Corollary 1.2): In the stable range ( $k \le (2g-2)/3$ ), the cohomology groups are independent of $g$ and vanish for odd $k$ . Furthermore, for $k \le 2n-1$ , the groups are independent of $n$ .
Extension to $GL_n$ and $SL_n$ (Theorem 1.4): Similar stability results are established for universal $SL_n$ and $GL_n$ -character varieties over $\mathcal{M}_{g,1}$ with fixed central monodromy around the puncture. The cohomology is expressed in terms of local systems on $\mathcal{M}_{g,1}$ .
Independence of Obstruction Class (Remark 1.6): The rational cohomology of the universal character variety is shown to be independent of the choice of the obstruction class $d \in (\mathbb{Z}/n\mathbb{Z})^\times$ . This is deduced via the non-abelian Hodge correspondence rather than first principles or Galois conjugation arguments, which do not apply to the total space.

Significance
The paper establishes a foundational stability phenomenon for the cohomology of universal character varieties, analogous to the stability of the moduli space of curves. By proving the degeneration of the Leray spectral sequence, the authors reduce the complex problem of computing the cohomology of the total space to the more tractable problem of computing the cohomology of the base with coefficients in specific local systems derived from the fiber.

The work provides explicit formulas for the stable cohomology, linking it to the representation theory of the symplectic group and the cohomology of $\mathcal{M}_g$ . The authors note that while the independence of the cohomology from the obstruction class $d$ is a consequence of their proof, it currently lacks a direct geometric explanation within the Higgs bundle setting, though it aligns with broader conjectures regarding Gopakumar-Vafa invariants. The results also clarify the structural differences between character varieties over $\mathcal{M}_g$ and $\mathcal{M}_{g,1}$ , particularly regarding the definition of relative varieties for $GL_n$ and $SL_n$ .