Deligne's weight spectral sequence and tautological cohomology of the moduli space of curves

1. Problem Statement

The paper addresses the computation of the weight-graded pieces of the rational cohomology (specifically compactly supported cohomology) of the moduli spaces of pointed smooth curves, denoted as $\mathcal{M}_{g,n}$.

• Context: For $g, n \geq 0$ with $2g-2+n > 0$, the cohomology of $\mathcal{M}_{g,n}$ carries a weight filtration $W_*$ induced by Deligne's mixed Hodge structures (or $\ell$-adic Galois representations).
• Goal: To compute the decomposition of the graded pieces $\text{gr}^q H^r(\mathcal{M}_{g,n})$ into irreducible representations of the symmetric group $S_n$ (which permutes the marked points) and pure rational Hodge structures (or Galois representations).
• Specific Challenge: While low-genus cases ($g=1, 2, 3, 4$) and specific low-marked-point cases were previously known, higher genus cases (specifically $g=5$) and higher marked point counts (e.g., $g=3, n=3$) remained computationally intractable due to the complexity of the boundary stratification in the Deligne-Mumford compactification $\overline{\mathcal{M}}_{g,n}$.
• Assumption: The authors work under the assumption that Pixton's relations constitute all relations in the tautological ring, a conjecture widely believed to be true in the ranges studied.

2. Methodology

The authors developed a computer program (implemented in SageMath, extending the admcycles module) to implement Deligne's weight spectral sequences. The methodology relies on the following theoretical and algorithmic pillars:

A. Deligne's Spectral Sequences

The computation utilizes two dual spectral sequences that relate the cohomology of the open moduli space $\mathcal{M}_{g,n}$ to the cohomology of the boundary strata in $\overline{\mathcal{M}}_{g,n}$:

1. Pushforward Spectral Sequence: Converges to the weight-graded pieces of $H^*(\mathcal{M}_{g,n})$. The $E_1$ page is built from the cohomology of products of moduli spaces associated with stable graphs, twisted by the determinant of the edge set.
2. Pullback Spectral Sequence: The Poincaré dual version, converging to the compactly supported cohomology $H^*_c(\mathcal{M}_{g,n})$. This is identified with the Getzler-Kapranov complex.

Both sequences degenerate at the $E_2$ page. The differentials ($d_1$) are explicitly constructed using gluing maps and automorphism groups of stable graphs.

B. Tautological Cohomology

The authors restrict their computations to the tautological ring $RH^*(\mathcal{M}_{g,n})$.

• Justification: In the range $2g+n \leq 12$ (and specifically for the cases $g=5$ and $g=3, n=3$), the cohomology groups appearing in the $E_1$ page of the spectral sequences are generated by tautological classes.
• Pixton's Relations: The program uses Pixton's relations to define the quotient of the free algebra of decorated strata classes, effectively computing the dimension and structure of the tautological ring.

C. Algorithmic Implementation

The core algorithm involves:

1. Graph Enumeration: Iterating over isomorphism classes of stable graphs $\Gamma$ with a fixed number of edges.
2. Symmetry Handling: Computing invariants under the action of the automorphism group $\text{Aut}(\Gamma)$ and the symmetric group $S_n$ (specifically subgroups $S_\lambda$ corresponding to partitions $\lambda \vdash n$).
3. Differential Computation: Explicitly calculating the image of the $d_1$ differential by mapping classes from a graph $\Gamma$ to graphs $\Gamma'$ obtained by contracting edges (pushforward) or splitting vertices/adding loops (pullback).
4. Optimization:
   • Orbit Representatives: Instead of computing over the full orbit of graphs, the program computes on a single representative and symmetrizes the results.
   • Symmetry Reduction: Utilizing subgroups like $S_2^k$ (swapping half-edges of self-loops) to reduce the number of decorated stratum classes.
   • Perfect Pairing: For high degrees, using Poincaré duality to compute cohomology via multiplication matrices rather than solving large systems of Pixton relations.
5. Parallelization: A custom task scheduler manages memory-intensive computations, handling failures due to memory overload and dynamically adjusting parallel tasks.

3. Key Contributions

1. Software Implementation: Creation of a robust computer program that implements Deligne's weight spectral sequences for moduli spaces of curves, integrating and optimizing the admcycles SageMath module.
2. Handling High Complexity: Successfully navigating the combinatorial explosion of stable graphs and the algebraic complexity of Pixton's relations for cases previously out of reach.
3. Extension to Compact Type and Rational Tails: The program also computes cohomology for $\mathcal{M}^{ct}_{g,n}$ (curves of compact type) and $\mathcal{M}^{rt}_{g,n}$ (curves with rational tails) by restricting the spectral sequence sums to specific graph types (no bridges, no rational tails, respectively).

4. Main Results

The authors determined the weight-graded pieces and $S_n$-equivariant structure for two specific cases where the cohomology is of Tate type (meaning odd weights vanish, and even weights are sums of tensor powers of the Tate class $L$).

Case 1: Genus 5, No Marked Points ($\mathcal{M}_5$)

• Result: The cohomology is purely Tate type.
• Decomposition:
  • $H^0 = \mathbb{Q}$
  • $H^2 = L$
  • $H^4 = L^2$
  • $H^5 = L^3$
  • $H^6 = L^3$
  • $H^7 = L^4$
  • $H^{14} = L^{12}$
• Significance: This confirms a conjecture regarding the rational cohomology of the open locus of general curves ($O_5$) and resolves the structure of $H^*(\mathcal{M}_5)$, which was previously unknown.

Case 2: Genus 3, Three Marked Points ($\mathcal{M}_{3,3}$)

• Result: The cohomology is of Tate type with non-trivial $S_3$-action.
• Decomposition: Expressed in terms of irreducible representations $s_\lambda$ of $S_3$ and the Tate class $L$:
  • $H^0 = s_3$
  • $H^2 = (2s_3 + s_{2,1})L$
  • $H^4 = (s_3 + s_{2,1})L^2$
  • $H^5 = 2(s_3 + s_{2,1})L^3$
  • $H^6 = s_3 L^6$
  • $H^8 = 2(s_3 + s_{2,1})L^7$
• Significance: Provides the first complete description of the equivariant cohomology for this specific high-complexity case.

5. Significance and Future Work

• Verification of Conjectures: The results verify previous conjectures (e.g., by Tommasi and Zheng) regarding the cohomology of specific loci within $\mathcal{M}_{g,n}$.
• Tate Type Confirmation: The results support the broader conjecture that for $3g+2n < 25$, the cohomology of $\mathcal{M}_{g,n}$ is of Tate type (odd weights vanish).
• Methodological Advancement: The paper demonstrates that Deligne's spectral sequences, when combined with modern computational algebraic geometry (tautological rings), are a viable tool for computing high-genus cohomology, a task previously limited to low genus.
• Future Applications: The authors note that the program could be extended to compute weight 11 contributions (which are non-tautological but known for $g=1$) and potentially other ranges if the behavior of non-tautological classes under gluing maps is understood.

In summary, this paper represents a significant breakthrough in the computational topology of moduli spaces, moving from theoretical bounds to explicit, verified computations for genus 5 and complex marked-point configurations using advanced spectral sequence techniques and high-performance computing.